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Scattering of elastic waves in multilayered media with irregular interfaces
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Scattering of elastic waves in multilayered media with irregular interfaces
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Content
SCATTERING OF ELASTIC WAVES IN MULTILAYERED MEDIA WITH IRREGULAR
INTERFACES
by
Gang Ding
A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(MECHANICAL ENGINEERING)
December 2024
Copyright 2024 Gang Ding
Dedication
This dissertation is dedicated to my parents, who have always set high expectations for my educational
journey. While I may have let my father down in spirit, I am determined to honor my 85-year-old mother
and fulfill her hopes for me.
ii
Acknowledgments
I would like to extend my heartfelt gratitude to my former advisor, Professor M. Dravinski, for his
unwavering guidance and support throughout this research endeavor. I also wish to express my profound
appreciation to Professors S. Sadhal and V. Lee for taking over as my advisors and providing invaluable
guidance and support in completing this research after Professor Dravinski’s retirement. Special thanks
go to Professor S. Sadhal for graciously serving as the Chairman of my dissertation guidance committee.
Without the dedicated support of Professors Sadhal and Lee, I would not have been able to resume my
doctoral research after such a long interruption. I am also deeply thankful to Professor H. Flashner for
devoting his time to serve on my dissertation guidance committee at the qualifying examination as well
as the defense. I would also like to express my sincere gratitude to Professors A. Oberai and P. Plucinsky
for their generous contributions to my guidance committee.
I also appreciate many professors in the Departments of Aerospace and Mechanical Engineering, Civil
and Environmental Engineering, and Mathematics with whom I had the privilege to study and work.
In addition, gratitude extends to my friends Ramdass Keshavamurthy, Tiao Zhou, Xuqiang Wu, Houfei
Fang, Tao Zheng, and Yuqing Niu for their assistance and beneficial discussion.
I wish to thank everybody in my family, especially my wife, for their support, encouragement, and
patient expectations and to whom I dedicate this dissertation.
Finally, I greatly appreciate the Teaching Assistantship from the Department of Mechanical Engineering at the University of Southern California, and the partial support under NSF grant CMS-9412759.
iii
Table of Contents
Dedication ii
Acknowledgments iii
List of Tables vii
List of Figures ix
Abstract xx
Chapter 1: Introduction 1
Chapter 2: General Statement of Problem 7
Chapter 3: Antiplane Strain Model Using Half-Space Green’s Functions 10
3.1 Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.2 Steady-State Solution of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.2.1 Free Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.2.2 Scattered Wave Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.2.3 Source Intensities for Scattered Wave Field . . . . . . . . . . . . . . . . . . . . . . . 18
3.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.3.1 Error Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.3.2 One-Layer Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.3.2.1 Parametric Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.3.2.2 Steady-State Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.3.2.3 Transient Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.3.3 Two-Layer Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.3.3.1 Parametric Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.3.3.2 Steady-State Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.3.3.3 Transient Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
Chapter 4: Antiplane Strain Model Using Full-Space Green’s Functions 38
4.1 Statement of Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.2 Steady-State Solution of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.2.1 Scattered Wave Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.2.2 Source Intensities for the Scattered Wave Field . . . . . . . . . . . . . . . . . . . . 41
4.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
iv
4.3.1 Error Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.3.2 One-Layer Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.3.2.1 Parametric Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.3.2.2 Steady-State Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.3.3 Two-Layer Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.3.3.1 Parametric Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.3.3.2 Steady-State Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
Chapter 5: Plane Strain Model 88
5.1 Statement of Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
5.2 Steady-State Solution of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.2.1 Free Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.2.2 Scattered Wave Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.2.3 Source Intensities for the Scattered Wave Field . . . . . . . . . . . . . . . . . . . . 95
5.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.3.1 Error Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.3.2 One-Layer Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.3.2.1 Parametric Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.3.2.2 Steady-State Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
Chapter 6: Three-Dimensional Model 109
6.1 Statement of Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.2 Steady-State Solution of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.2.1 Free Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
6.2.2 Scattered Wave Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.2.3 Source Intensities for the Scattered Wave Field . . . . . . . . . . . . . . . . . . . . 115
6.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6.3.1 Error Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
6.3.2 One-Layer Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
6.3.2.1 Parametric Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 123
6.3.2.2 Steady-State Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
6.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
Chapter 7: Summary and Conclusions 155
Bibliography 158
Appendices 170
Chapter A: Antiplane Strain Model 171
A.1 Half-Space Green’s Function and Matrix System . . . . . . . . . . . . . . . . . . . . . . . . 171
A.2 Full-Space Green’s Function and Matrix System . . . . . . . . . . . . . . . . . . . . . . . . 175
A.3 Free Field for Incident Love Surface Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
A.3.1 One-Layer Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
A.3.2 Two-Layer Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
Chapter B: Plane Strain Model 186
v
B.1 Full-Space Green’s Functions and Matrix System . . . . . . . . . . . . . . . . . . . . . . . . 186
B.2 Free Field for Plane Strain Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
B.2.1 Plane Harmonic P or SV Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
B.2.2 Rayleigh Wave for One-layer Model . . . . . . . . . . . . . . . . . . . . . . . . . . 198
B.2.3 Rayleigh Wave for Pure Half-Space Model . . . . . . . . . . . . . . . . . . . . . . . 203
B.3 Influence of the SV Incident Angle on the Reflected P wave . . . . . . . . . . . . . . . . . . 205
Chapter C: Three-Dimensional Model 207
C.1 Full-Space Green’s Functions for the 3D Model . . . . . . . . . . . . . . . . . . . . . . . . . 207
C.2 Matrix System of the 3D Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
vi
List of Tables
3.1 The optimum distance (dr) between the auxiliary surface and the interface based on the
minimization of the error Er (3.26). Parameters: N = 60, M = L = 24; Xn = 2;
Xm = Xl = 2.1; vertical incidence θ0 = 0o
. . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.2 Minimum number of collocation points N for a one-layer model which results in the
relative error Er less than 0.001 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.1 The optimum spacing ds between two adjacent collocation points based on the minimum
error Er for the one-layer antiplane strain model using the full-space Green’s functions
based on the critical error Er = 10−5
. The angles of incidence are 0
o
, 30o
, 60o
and 85o
.
h = 1, a = 0.5, d = 0.2, dr = d0 = 3ds, 2ws = 2wc = 2 . . . . . . . . . . . . . . . . . . . 52
4.2 The optimum length 2wc of the interface C1 based on the minimum error Er for the
one-layer antiplane strain model using the full-space Green’s functions. The angles of
incidence are 0
o
, 30o
, 60o
and 85o
. h = 1, a = 0.5, d = 0.2. 2ws = 2wc, ds = 0.0122,
dr = d0 = 3ds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.3 The optimum spacing ds between two adjacent collocation points based on the minimum
error Er for the two-layer antiplane strain model using the full-space Green’s functions.
The critical Er is assumed to be 0.0001. The angles of incidence are 0
o
, 30o
, 60o
and 85o
.
a = 0.5, h1 = 1, h2 = 0.5, d1 = d2 = 0.2, dr = d0 = 3ds, 2ws = 2wc = 2 . . . . . . . . . 72
4.4 The optimum length 2wc of the interfaces and the top surface based on the minimum error
Er for the two-layer antiplane strain model using the full-space Green’s functions. The
angles of incidence are 0
o
, 30o
, 60o
and 85o
. a = 0.5, h1 = 1, h2 = 0.5, d1 = d2 = 0.2.
ds = 0.0125, dr = d0 = 3ds, N1 = N2 = P = 11m, M = L11 = L21 = L12 = 5m,
L22 = 3m , m = 1, 2, ..., 25; 2wc = 2ws = (N1 − 1)ds . . . . . . . . . . . . . . . . . . . . 73
4.5 Reduction of Love wave amplitude (Alove) at x3 → ∞, 2-layer free field . . . . . . . . . . . 81
5.1 Parameters for the steady-state response for the one-layer plane strain model using the
full-space Green’s functions. Ω = 1 − 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.2 Peak displacement amplitude comparison between the one-layer model with scatterer and
the corresponding flat-layer model for incident Rayleigh waves . . . . . . . . . . . . . . . 104
vii
5.3 Comparison of Rayleigh wave surface displacements between transparency one-layer and
pure half-space models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6.1 Parameters for the steady-state response for the one-layer three-dimensional model using
the full-space Green’s functions. Ω = 4 − 5; Angles of incidence are 0
o − 85o
. . . . . . . 125
viii
List of Figures
2.1 Geometry of a three-dimensional multilayer model with irregular interfaces subjected to
an incident plane harmonic body wave or Love, Rayleigh surface wave . . . . . . . . . . . 9
3.1 2D model geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.2 Distribution of sources and collocation points using half-space Green’s functions for top
layer DJ . The number of collocation points along the interface CJ is denoted by NJ , the
length of the interface CJ along which the collocation points are distributed is defined
by 2w
J
c
, and the length of the irregular part of the interface is represented by 2w
J
i
. The
length of the auxiliary surface CJ1 along which the sources of order LJ1 are allocated is
given by 2w
J1
s
. The spacing between two adjacent collocation points is denoted by d
J
s
and that between the interface CJ and the auxiliary surface CJ1 by d
J1
r
. Star: position of
a line source; circle: position of a collocation point . . . . . . . . . . . . . . . . . . . . . . . 14
3.3 Distribution of sources and collocation points using half-space Green’s functions for the
layer Dj where j = 1, 2, ..., J. The numbers of collocation points along the interface Cj
and Cj+1 are denoted by Nj and Nj+1, respectively, the lengths of the interface Cj and
Cj+1 along which the collocation points are distributed are defined by 2w
j
c and 2w
j+1
c ,
respectively, and the lengths of the irregular part of the interface are represented by 2w
j
i
and 2w
j+1
i
, respectively. The lengths of the auxiliary surface Cj1 and Cj2 along which the
sources of order LJ1 and Lj2 are allocated are given by 2w
J1
s
and 2w
J2
s
, respectively. The
spacing between two adjacent collocation points is defined by d
j
s on the interface Cj and
that between the interface Cj and the auxiliary surface Cjk is denoted by d
jk
r , k = 1, 2.
Star: position of a line source; circle: position of a collocation point . . . . . . . . . . . . . 15
3.4 Distribution of sources and collocation points using half-space Green’s functions for
bottom half-space D0. The number of collocation points along the interface C1 is denoted
by N1, the length of the interface C1 along which the collocation points are distributed is
defined by 2w
1
c
, and the length of the irregular part of the interface is represented by 2w
1
i
.
The length of the auxiliary surface C00 along which the sources of order M are allocated
is given by 2w
00
s
. The spacing between two adjacent collocation points is denoted by d
1
s
and that between the interface C1 and the auxiliary surface C00 by d
00
r
. Star: position of a
line source; circle: position of a collocation point . . . . . . . . . . . . . . . . . . . . . . . . 16
ix
3.5 Geometry of the one-layer model with a cosine-shaped interface and the corresponding
distribution of sources and collocation points using half-space Green’s functions. Here,
h = 1, 2wi = 1, d = 0.1 and the material properties are β0 = ρ0 = 1 in domain D0
and β1 = ρ1 = 0.5 in domain D1. Star: position of a line source; circle: position of a
collocation point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.6 Error Er as a function of the distance dr between the interface C and the auxiliary
surfaces C00 and C11 for the one-layer model subjected to a vertically incident plane
harmonic SH wave. N = 60, M = L = 24; 2wc = 2, 2ws = 2.1; Ω = 3; dr = 0.102 is
the optimum dr proposed by the relationship (dr
.= 3ds) . . . . . . . . . . . . . . . . . . . 23
3.7 Error Er as a function of the collocation points N for the one-layer model and a vertically
incident SH wave. 2wc = 2, 2ws = 2.1; dr = 3ds; θ0 = 0o
; Ω = 4; M = L;
2N/(M + L) = 2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.8 Error Er as a function of length 2wc over which the collocation points are applied for
the one-layer model and a vertically incident SH wave. θ0 = 0o
; ds = 0.0339; dr = 3ds;
Ω = 12.5; M = L; 2N/(M + L) = 2.5; 2wc = (N − 1)ds . . . . . . . . . . . . . . . . . . 27
3.9 Comparison of normalized surface displacement and phase delay between the present
approach (solid) and the Aki-Larner technique (circle) for various angles of incidence.
The material properties are defined as β0 = 4.0km/ sec, ρ0 = 3.3g/cm3
in the bottom
half-space (D0) and β1 = 3.0km/ sec, ρ1 = 2.8g/cm3
in the top layer (D1), and the depth
of layer h = 25km. The length and the max deviation of the cosine-shaped scatterer are
50km and 5km, respectively. The wavelength of the incident SH wave in the half-space
is 10km which corresponds to circular frequency ω = 2.5133 sec−1
. . . . . . . . . . . . . 28
3.10 Normalized surface response for the one-layer model and vertically incident waves as
a function of dimensionless frequency Ω and spatial variable x1. θ0 = 0o
; 2wc = 2,
2ws = 2.1; dr = 3ds, N = 40 and M = L = 16. . . . . . . . . . . . . . . . . . . . . . . . 30
3.11 Transient surface response of the one-layer model due to a vertically incident Ricker
wavelet. tp = 1.25 sec, ts = 1.5 sec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.12 Geometry of the two-layer model with two cosine-shaped interfaces and the corresponding
distribution of sources and collocation points using half-space Green’s functions. Here,
h1 = 1, h2 = 0.5; 2w
(1)
i = 2w
(2)
i = 1; d1 = d2 = 0.1 and the material properties are
β2 = ρ2 = 0.25; β1 = ρ1 = 0.5 and β0 = ρ0 = 1. Star: position of a line source; circle:
position of a collocation point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.13 Normalized surface response of the two-layer model for a vertically incident plane
harmonic SH wave as a function of dimensionless frequency Ω and spatial variable
x1. θ0 = 0o
; Ω = 0 − 4; N1 = 60, M = L11 = 24; N2 = 30, L12 = L21 = 12.
Throughout the parametric error analysis of the two-layer model, 2wc = 2, 2ws = 2.1;
d
(11)
r = d
(00)
r = 3d
(1)
s , d(12)
r = d
(21)
r = 3d
(2)
s . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
x
3.14 Transient surface response of the two-layer model due to a vertically incident SH Ricker
wavelet. tp = 5 sec, ts = 15 sec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.1 Distribution of sources and collocation points using full-space Green’s functions for the
top layer DJ . An additional auxiliary surface CJ2 is introduced above the top surface
SF to impose traction-free conditions on the top surface SF . Definitions of the other
parameters in this figure are the same as in Chapter 3. Star: position of a line source;
circle: position of a collocation point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.2 Geometry of the one-layer model with a cosine-shaped interface using the full-space
Green’s functions. The distribution of sources (stars) on the auxiliary surfaces C00, C11,
and C12 and collocation points (circles) on the interface C1 and the top surface SF are
shown as well. 2wi = 2a = h = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.3 Full-space Green’s functions. Transparency test for one-layer antiplane strain model
subjected to incident plane harmonic SH waves with different angles of incidence and
circular frequencies. The material of the top layer is the same as that of the half-space, i.e.,
µ1 = µ0 = 1, β1 = β0 = 1, ρ1 = ρ0 = 1, ν1 = ν0 = 1/3. N = P = 77, M = L1 = 35 ,
L2 = 21 , L = L1 + L2 , h = 1, d = 0.2, a = 0.5, 2ws = 2wc = 2, d0 = dr = 0.3 . . . . . 46
4.4 Full-space Green’s functions. Flat-layer test for one-layer antiplane strain model subjected
to incident harmonic SH waves with different angles of incidence and circular frequencies.
N = P = 77, M = L1 = 35, L2 = 21, L = L1 + L2, h = 1, a = 0.5, d = 0,
2ws = 2wc = 2, dr = d0 = 0.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.5 Full-space Green’s functions. Testing for optimum rm (equation (4.24 )). The error Er
is shown as a function of number of series m (equation (4.25)) for the one-layer model
subjected to an incident antiplane harmonic wave with a frequency Ω = 4. The angles
of incidence are 0
o
, 30o
, 60o
and 85o
. h = 1, a = 0.5, d = 0.2, dr = d0 = 0.0732,
2ws = 2wc = 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.6 Full-space Green’s functions. Estimate of auxiliary surface location dr(d0). Error Er is
shown as a function of the spacing dr between the auxiliary surfaces C0, C11 and the
interface C1 for the one-layer model. The dash line represents the optimum location
(dr = 3ds) proposed by J. E. Luco. The angles of incidence are 0
o
, 30o
, 60o
and 85o
. The
circular frequency Ω = 4. h = 1, a = 0.5, d = 0.2. N = P = 165, L1 = M = 75,
L2 = 45. 2wc = 2ws = 4. ds = 2wc/(N − 1). d0 = dr. . . . . . . . . . . . . . . . . . . . . 50
4.7 Testing of the optimum spacing ds between two adjacent collocation points using the
full-space Green’s functions approach. Error Er is shown as a function of the number of
collocation points for the one-layer model subjected to an incident antiplane harmonic
wave with a frequency Ω = 2. The angles of incidence are 0
o
, 30o
, 60o
and 85o
. h = 1,
a = 0.5, d = 0.2, 2ws = 2wc = 2. ds = 2ws/(N − 1). dr = d0 = 3ds . . . . . . . . . . . . 51
4.8 Full-space Green’s functions. Testing for optimum 2wc. Error Er is shown as a function
of the length 2wc of the interface C1 for the one-layer model subjected to an incident
antiplane harmonic wave with a frequency Ω = 2. The angles of incidence are 0
o
, 30o
,
60o
and 85o
. h = 1, a = 0.5, d = 0.2. 2ws = 2ws. ds = 0.0122, dr = d0 = 3ds. . . . . . . 53
xi
4.9 Comparison of normalized surface displacement between indirect boundary integral
approach using the full-space Green’s functions (solid) and Aki-Larner results (circles) [1].
β0 = 4.0 km/ sec, ρ0 = 3.3 gm/cm3
in the half-space and β1 = 3.0 km/ sec, ρ1 = 2.8
gm/cm3
in the overlayed single layer. The wavelength of the incident SH wave in the
half-space is 10 km which corresponds to a circular frequency ω = 0.8π s−1
. . . . . . . . 55
4.10 SH wave surface displacements for the one-layer antiplane strain model. d = 0.2;
Ω = 1, 2, 4. Solid line: using full-space Green’s functions; Dash lines: using half-space
Green’s functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.11 SH wave surface displacements for the one-layer antiplane strain model. d = 0.2;
Ω = 3, 5, 7. Solid line: using full-space Green’s functions; Dash lines: using half-space
Green’s functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.12 SH wave surface response for the one-layer model with various deviations d of the
irregular interface. Ω = 1, θ0 = 0o
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.13 SH wave surface response for the one-layer model with various deviations d of the
irregular interface. Ω = 2, θ0 = 0o
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.14 SH wave surface response for the one-layer model with various deviations d of the
irregular interface. Ω = 4, θ0 = 0o
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.15 Love wave surface response for the one-layer model with various deviations d of the
irregular interface. Ω = 0.96 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.16 Love wave surface response for the one-layer model with various deviations d of the
irregular interface. Ω = 2.96 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.17 Love wave surface response for the one-layer model with various deviations d of the
irregular interface. Ω = 3.44 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.18 Love wave surface response for the one-layer model with various deviations d of the
irregular interface. Ω = 5.60 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.19 SH wave surface displacement at (x1, x3) = (0, 0) versus frequency for the one-layer
model. θ0 = 0o
, d = 0.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.20 Love wave surface displacement at (x1, x3) = (0, 0) versus frequency for the one-layer
model. d = 0.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.21 Geometry of the two-layer antiplane strain model with two cosine-shaped interfaces
and the corresponding distribution of sources and collocation points using the full-space
Green’s functions. Here, h1 = 1, h2 = 0.5; 2w
(1)
i = 2w
(2)
i = 1. Star: position of a line
source; circle: position of a collocation point. . . . . . . . . . . . . . . . . . . . . . . . . . . 64
xii
4.22 Transparency test for the two-layer antiplane strain model and incident plane harmonic
SH waves with different angles of incidence and frequencies using the full-space
Green’s functions. Material of the three domains are assumed to be the same, i.e.,
µ2 = µ1 = µ0 = 1, β2 = β1 = β0 = 1, ρ2 = ρ1 = ρ0 = 1, ν2 = ν1 = ν0 = 1/3.
N1 = N2 = P = 121, M = L11 = L21 = L12 = 55 , L22 = 22 , a = 0.5, h1 = 1,
h2 = 0.5, d1 = d2 = 0.2, 2ws = 2wc = 6 , dr = d0 = 0.1 . . . . . . . . . . . . . . . . . . . 66
4.23 Flat-layer test for the two-layer antiplane strain model and incident harmonic SH waves
with different angles of incidence and frequencies using the full-space Green’s functions.
N1 = N2 = P = 121, M = L11 = L21 = L12 = 55 , L22 = 22 , a = 0.5, h1 = 1,
h2 = 0.5, d1 = d2 = 0, 2ws = 2wc = 6 , dr = d0 = 0.1 . . . . . . . . . . . . . . . . . . . . 67
4.24 Testing for the optimum matrix squeness rm. Error Er is shown as a function of m for
the two-layer antiplane strain model subjected to an incident antiplane harmonic wave
with a frequency Ω = 4 using the full-space Green’s functions. The angles of incidence
are 0
o
, 30o
, 60o
and 85o
. N1 = N2 = P = 231, M = L11 = L21 = L12 = 5m, L22 = 3m,
m = 1, 2, ..., 48; a = 0.5, h1 = 1, h2 = 0.5, d1 = d2 = 0.2, 2ws = 2wc = 4, dr = d0 = 3ds 68
4.25 Estimate of auxiliary surface location dr. Error Er is shown as a function of the spacing
dr between the auxiliary surfaces C0, C11, C12, C21 and the interface C1, C1, C2, C2,
respectively for the two-layer antiplane strain model using the full-space Green’s function
approach. The angles of incidence are 0
o
, 30o
, 60o
and 85o
. The dimensionless frequency
Ω = 4. N1 = N2 = P = 121, M = L11 = L21 = L12 = 55, L22 = 33; a = 0.5, h1 = 1,
h2 = 0.5, d1 = d2 = 0.2, 2ws = 2wc = 4 , d0 = dr. The dash line represents the optimum
location of auxiliary surfaces d0 = dr = 3ds proposed by J. E. Luco . . . . . . . . . . . . . 70
4.26 Testing of the optimum spacing of collocation points ds. Error Er is shown as a function
of m for the two-layer antiplane strain model subjected to an incident antiplane harmonic
wave using the full-space Green’s functions. The angles of incidence are 0
o
, 30o
, 60o
and 85o
. Ω = 2. N1 = N2 = P = 11m, M = L11 = L21 = L12 = 5m, L22 = 3m ,
m = 1, 2, ..., 25; a = 0.5, h1 = 1, h2 = 0.5, d1 = d2 = 0.2, 2ws = 2wc = 2 , dr = d0 = 3ds 71
4.27 Testing of the optimum length 2wc using full-space Green’s functions - Error Er as a
function of 2wc for the two-layer antiplane strain model subjected to an incident antiplane
harmonic wave with a dimensionless frequency Ω = 2. The angles of incidence are 0
o
,
30o
, 60o
and 85o
. a = 0.5, h1 = 1, h2 = 0.5, d1 = d2 = 0.2. ds = 0.0125; dr = d0 = 3ds,
N1 = N2 = P = 11m, M = L11 = L21 = L12 = 5m, L22 = 3m , m = 1, 2, ..., 25,
2wc = 2ws = (N1 − 1)ds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.28 SH wave surface displacements for the two-layer antiplane strain model. d1 = d2 = 0.2;
Ω = 1, 2, 4. Solid line: using full-space Green’s functions; Dash line: using half-space
Green’s functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.29 SH wave surface displacements for the two-layer antiplane strain model. d1 = d2 = 0.2;
Ω = 3, 5, 7. Solid line: using full-space Green’s functions; Dash line: using half-space
Green’s functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
xiii
4.30 Surface displacement for the two-layer model for a vertical SH incidence (θ0 = 0o
) with
different combinations of d1 and d2 values. Ω = 1. . . . . . . . . . . . . . . . . . . . . . . . 77
4.31 Surface displacement for the two-layer model for a vertical SH incidence (θ0 = 0o
) with
different combinations of d1 and d2 values. Ω = 2. . . . . . . . . . . . . . . . . . . . . . . . 77
4.32 Surface displacement for the two-layer model for a vertical SH incidence (θ0 = 0o
) with
different combinations of d1 and d2 values. Ω = 4. . . . . . . . . . . . . . . . . . . . . . . . 78
4.33 Surface displacement at (x1, x3) = (0, 0) for the two-layer model for a vertical SH
incidence (θ0 = 0o
) as a function of dimensionless frequency Ω. d1 = d2 = 0.1 . . . . . . . 78
4.34 Love wave surface displacement for the two-layer model with different combinations of
d1 and d2 values. Ω = 1.008. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.35 Love wave surface displacement for the two-layer model with different combinations of
d1 and d2 values. Ω = 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.36 Love wave surface displacement for the two-layer model with different combinations of
d1 and d2 values. Ω = 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.37 Further Love wave surface displacement test for the two-layer model with various d1
values for different fixed d2 values. Ω = 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.38 Transparency test. Love wave surface displacement for the two-layer model with different
combinations of d1 and d2 values. Ω = 0.96. . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.39 Transparency test. Love wave surface displacement for the two-layer model with different
combinations of d1 and d2 values. Ω = 2.96. . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.40 Transparency test. Love wave surface displacement for the two-layer model with different
combinations of d1 and d2 values. Ω = 3.44. . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.41 Transparency test. Love wave surface displacement for the two-layer model with different
combinations of d1 and d2 values. Ω = 5.60. . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.42 Love wave surface displacement at (x1, x3) = (0, 0) versus dimensionless frequency Ω
for the two-layer model. d1 = 0.1, d2 = −0.1 . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.1 Steady-state surface displacement for a one-layer plane strain model subjected to incident
P waves with different angles of incidence. Ω = 5. a = 0.5, h = 1. Solid: d = 0.2, Dash:
d = 0.1, Points: d = 0.0, Dash-Dot: d = −0.1, Dot: d = −0.2. . . . . . . . . . . . . . . . . 101
5.2 Steady-state surface displacement for a one-layer plane strain model subjected to incident
P waves with different angles of incidence. Ω = 6. a = 0.5, h = 1. Solid: d = 0.2, Dash:
d = 0.1, Points: d = 0.0, Dash-Dot: d = −0.1, Dot: d = −0.2. . . . . . . . . . . . . . . . . 101
xiv
5.3 Steady-state surface displacement for a one-layer plane strain model subjected to incident
SV waves with different angles of incidence. Ω = 4. a = 0.5, h = 1. Solid: d = 0.2, Dash:
d = 0.1, Points: d = 0.0, Dash-Dot: d = −0.1, Dot: d = −0.2. . . . . . . . . . . . . . . . . 102
5.4 Steady-state surface displacement for a one-layer plane strain model subjected to incident
SV waves with different angles of incidence. Ω = 5. a = 0.5, h = 1. Solid: d = 0.2, Dash:
d = 0.1, Points: d = 0.0, Dash-Dot: d = −0.1, Dot: d = −0.2. . . . . . . . . . . . . . . . . 102
5.5 Steady-state surface displacement for a one-layer plane strain model subjected to incident
Rayleigh waves for various frequencies. a = 0.5, h = 1. Solid: d = 0.2, Dash: d = 0.1,
Points: d = 0.0, Dash-Dot: d = −0.1, Dot: d = −0.2. . . . . . . . . . . . . . . . . . . . . . 103
5.6 Steady-state surface displacement at x1 = 0 for the one-layer plane strain model subjected
to a vertically incident P wave as a function of dimensionless frequencies. a = 0.5, h = 1,
d = 0.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.7 Steady-state surface displacement at x1 = 0 for the one-layer plane strain model subjected
to a vertically incident SV wave as a function of dimensionless frequencies. a = 0.5,
h = 1, d = 0.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.8 Steady-state surface displacement at x1 = 0 for the one-layer plane strain model subjected
to an incident Rayleigh wave as a function of dimensionless frequencies. The first sharp
resonance occurs at Ω1 = 5.12 while the second one at Ω2 = 5.92. a = 0.5, h = 1, d = 0.1 106
6.1 Line segments AA′
and BB′
on which the three-dimensional surface response is evaluated 117
6.2 A one-layer three-dimensional model with collocation points (circles) distributed on the
interface C1 and the top surface SF . Azimuthal angle φ0 = 0o
. . . . . . . . . . . . . . . . 121
6.3 A one-layer three-dimensional model with point sources (stars) distributed on the
auxiliary surfaces C00, C11, and C12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
6.4 Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model subjected to incident P wave, θ0 = 0o
. Ω = 3. h = 1, a = 0.5,
d = 0.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
6.5 Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model subjected to incident P wave, θ0 = 30o
. Ω = 3. h = 1, a = 0.5,
d = 0.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
6.6 Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model subjected to incident P wave, θ0 = 60o
. Ω = 3. h = 1, a = 0.5,
d = 0.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
6.7 Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model subjected to incident P wave, θ0 = 85o
. Ω = 3. h = 1, a = 0.5,
d = 0.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
xv
6.8 Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model subjected to incident SV wave, γ0 = 0o
. Ω = 3. h = 1, a = 0.5,
d = 0.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
6.9 Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model subjected to incident SV wave, γ0 = 30o
. Ω = 3. h = 1, a = 0.5,
d = 0.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
6.10 Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model subjected to incident SV wave, γ0 = 60o
. Ω = 3. h = 1, a = 0.5,
d = 0.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.11 Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model subjected to incident SV wave, γ0 = 85o
. Ω = 3. h = 1, a = 0.5,
d = 0.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.12 Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model subjected to incident SH wave, θ0 = 0o
. Ω = 3. h = 1, a = 0.5,
d = 0.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
6.13 Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model subjected to incident SH wave, θ0 = 30o
. Ω = 3. h = 1, a = 0.5,
d = 0.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
6.14 Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model subjected to incident SH wave, θ0 = 60o
. Ω = 3. h = 1, a = 0.5,
d = 0.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
6.15 Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model subjected to incident SH wave, θ0 = 85o
. Ω = 3. h = 1, a = 0.5,
d = 0.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
6.16 Steady-state surface displacement along the x1−axis and x2−axis for a one-layer
three-dimensional model subjected to incident Love wave. Ω = 3.04. h = 1, a = 0.5,
d = 0.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
6.17 Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model subjected to incident Rayleigh wave. Ω = 3. h = 1, a = 0.5,
d = 0.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
6.18 Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model subjected to incident P wave, θ0 = 0o
. Ω = 4. h = 1, a = 0.5,
d = 0.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
6.19 Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model subjected to incident P wave, θ0 = 30o
. Ω = 4. h = 1, a = 0.5,
d = 0.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
xvi
6.20 Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model subjected to incident P wave, θ0 = 60o
. Ω = 4. h = 1, a = 0.5,
d = 0.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
6.21 Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model subjected to incident P wave, θ0 = 85o
. Ω = 4. h = 1, a = 0.5,
d = 0.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
6.22 Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model subjected to incident SV wave, γ0 = 0o
. Ω = 4. h = 1, a = 0.5,
d = 0.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
6.23 Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model subjected to incident SV wave, γ0 = 30o
. Ω = 4. h = 1, a = 0.5,
d = 0.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
6.24 Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model subjected to incident SV wave, γ0 = 60o
. Ω = 4. h = 1, a = 0.5,
d = 0.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
6.25 Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model subjected to incident SV wave, γ0 = 85o
. Ω = 4. h = 1, a = 0.5,
d = 0.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
6.26 Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model subjected to incident SH wave, θ0 = 0o
. Ω = 4. h = 1, a = 0.5,
d = 0.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
6.27 Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model subjected to incident SH wave, θ0 = 30o
. Ω = 4. h = 1, a = 0.5,
d = 0.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
6.28 Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model subjected to incident SH wave, θ0 = 60o
. Ω = 4. h = 1, a = 0.5,
d = 0.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
6.29 Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model subjected to incident SH wave, θ0 = 85o
. Ω = 4. h = 1, a = 0.5,
d = 0.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
6.30 Steady-state surface displacement along the x1−axis and x2−axis for a one-layer
three-dimensional model subjected to incident Love wave. Ω = 4. h = 1, a = 0.5, d = 0.2 141
6.31 Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model subjected to incident Rayleigh wave. Ω = 4. h = 1, a = 0.5,
d = 0.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
xvii
6.32 Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model subjected to incident P wave, θ0 = 0o
. Ω = 5. h = 1, a = 0.5,
d = 0.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
6.33 Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model subjected to incident P wave, θ0 = 30o
. Ω = 5. h = 1, a = 0.5,
d = 0.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
6.34 Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model subjected to incident P wave, θ0 = 60o
. Ω = 5. h = 1, a = 0.5,
d = 0.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
6.35 Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model subjected to incident P wave, θ0 = 85o
. Ω = 5. h = 1, a = 0.5,
d = 0.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
6.36 Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model subjected to incident SV wave, γ0 = 0o
. Ω = 5. h = 1, a = 0.5,
d = 0.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
6.37 Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model subjected to incident SV wave, γ0 = 30o
. Ω = 5. h = 1, a = 0.5,
d = 0.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
6.38 Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model subjected to incident SV wave, γ0 = 60o
. Ω = 5. h = 1, a = 0.5,
d = 0.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
6.39 Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model subjected to incident SV wave, γ0 = 85o
. Ω = 5. h = 1, a = 0.5,
d = 0.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
6.40 Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model subjected to incident SH wave, θ0 = 0o
. Ω = 5. h = 1, a = 0.5,
d = 0.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
6.41 Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model subjected to incident SH wave, θ0 = 30o
. Ω = 5. h = 1, a = 0.5,
d = 0.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
6.42 Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model subjected to incident SH wave, θ0 = 60o
. Ω = 5. h = 1, a = 0.5,
d = 0.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
6.43 Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model subjected to incident SH wave, θ0 = 85o
. Ω = 5. h = 1, a = 0.5,
d = 0.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
xviii
6.44 Steady-state surface displacement along the x1−axis and x2−axis for a one-layer
three-dimensional model subjected to incident Love wave. Ω = 5.04. h = 1, a = 0.5,
d = 0.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
6.45 Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model subjected to incident Rayleigh wave. Ω = 5. h = 1, a = 0.5,
d = 0.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
6.46 Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model with various scatterer deviations d subjected to a vertically incident P
wave. Ω = 4. h = 1, a = 0.5. Dash: d = −0.4, dash-dot: d = −0.2, solid: d = 0, plus:
d = 0.2, dot: d = 0.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
6.47 Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model with various scatterer deviations d subjected to a vertically incident
SV wave. Ω = 4. h = 1, a = 0.5. Dash: d = −0.4, dash-dot: d = −0.2, solid: d = 0, plus:
d = 0.2, dot: d = 0.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
6.48 Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model with various scatterer deviations d subjected to a vertically incident
SH wave. Ω = 4. h = 1, a = 0.5. Dash: d = −0.4, dash-dot: d = −0.2, solid: d = 0, plus:
d = 0.2, dot: d = 0.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
6.49 Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model with various scatterer deviations d subjected to an incident Love
surface wave. Ω = 4. h = 1, a = 0.5. Dot: d = −0.4, dash-dot: d = −0.2, point: d = 0,
dash: d = 0.2, solid: d = 0.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
6.50 Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model with various scatterer deviations d subjected to an incident Rayleigh
surface wave. Ω = 4. h = 1, a = 0.5. Dash: d = −0.4, dash-dot: d = −0.2, solid: d = 0,
plus: d = 0.2, dot: d = 0.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
A.1 Love wave speed c versus circular frequency ω for the one-flat-layer model depicted
by Figure 3.5 with d = 0 and material properties defined by (4.20)-(4.21). The left c-ω
relationship is used for surface displacement evaluation of the Love wave . . . . . . . . . . 181
A.2 Wavelength λ versus wave speed c for the Love wave, one-layer model . . . . . . . . . . . 181
A.3 Love wave speed c versus circular frequency ω for the two-flat-layer model depicted by
Figure 4.21 with d1 = d2 = 0 and material properties defined by (4.34). The left c-ω
relationship is used for surface displacement evaluation of the Love wave . . . . . . . . . . 185
A.4 Wavelength λ versus wave speed c for the Love wave, two-layer model . . . . . . . . . . . 185
xix
Abstract
Steady-state scattering of elastic waves in multilayered media with irregular interfaces is investigated by using an indirect boundary integral equation method for antiplane strain, plane strain, and threedimensional models. The material is assumed to be elastic, homogeneous, and isotropic. The wave field in
each domain is expressed as a superposition of the free field of the corresponding flat-layer model and the
scattered wave field caused by irregularities. The scattered wave field is generated by a series of sources
distributed near the irregular interfaces. The half-space Green’s functions are first applied to model the
scattered wave field of the antiplane strain model while later the full-space Green’s functions are applied
to represent the sources of all three different models. The source intensities are determined by satisfying
the boundary and continuity conditions in the least square sense.
For each of the three models mentioned above, systematic parametric error analysis is conducted to
determine the optimum parameters for the steady-state response. Numerical results are then presented
based on the parametric error analysis. Transient response is studied in detail in the antiplane strain
model using the half-space Green’s function approach. The resonant feature of the multilayered structure
is investigated in the antiplane strain and plane strain models, respectively.
It is found that the steady-state response strongly depends on the type, frequency, and angle of the
incident wave, the shape of the irregularity, and the site location. The pattern of the surface response
amplitude of the three-dimensional model is more complicated than those of the two-dimensional models
since the transverse variation of the model structure plays an important role in the steady-state response.
xx
Chapter 1
Introduction
The study of elastic wave scattering in multilayered media with irregular interfaces has many practical
applications. One of them is the nondestructive evaluation of materials. The relationship between the layer
structure and the surface response can provide information about the nature of the multilayered media.
Another application is in geophysics where site effect studies require the analysis of the scattering of
seismic waves in irregular layered media.
Wave propagation in multilayered media with flat interfaces have been studied by many investigators
in the past (e.g. Thomson [87], 1950; Haskell [41], 1953; Alterman and Karal [3], 1968; Watson [92], 1970;
Kennet and Kerry [45], 1979; Luco and Apsel [61], 1983; Apsel and Luco [4], 1983; Schmidt and Jensen [85],
1985; Lekner [58], 1990; Li et al. [59], 1991; Rudgers [80], 1992; Kim [48], 1991). So far, however, there have
been very few investigations of the two-dimensional scattering of elastic waves in multilayered media
of open boundary with irregular interfaces. Aki and Larner ( [1], 1970) studied the steady-state surface
response of a one-layer lateral varying structure with moderately shallow slopes subjected to an incident
harmonic SH-wave. They used a technique based on the Rayleigh hypothesis. Bouchon and Aki ( [14],
1977) and Koketsu et al. ( [51], 1991) extended the Aki-Larner technique to a multilayered case.
Kennet ( [43], 1984a and [44], 1984b) introduced the concept of reflection and transmission operators
in the context of interface problems. The operator approach, which may need a large number of discrete
1
wavenumber terms for higher frequency problems, becomes particularly useful in complex problems since
it allows a physically based description of the process of wave propagation. This work was later generalized
by Chen ( [19], 1991) to synthesize the seismograms of the multilayered media by incorporating the Tmatrix method (Waterman [91], 1969).
Scattering of elastic waves in multilayered media consisting of finite dipping layers was investigated
for both two- and three-dimensional models by Dravinski ( [27], 1983), Eshraghi and Dravinski ( [36], 1989a
and [37], 1989b), Mossessian and Dravinski ( [70], 1990a) using the boundary integral equation approach.
Numerical methods are employed to study most of the practical problems since analytical solutions
of the scattering problems are very limited to simple geometries (Trifunac [88], 1971 and [89], 1973; Wong
and Trifunac [95], 1974; Lee [55], 1982). The most commonly used numerical techniques are finite difference (FD) and finite element (FE) methods, ray methods, boundary integral equation methods (BIE), wave
function expansion approach, and the Aki-Larner ( [1], 1970) technique.
Finite difference and finite element methods (Alterman and Karal [3], 1968; Smith [86], 1975), which
have the advantage for solving problems with complex geometries and varying material properties (Drake
[23], 1972; Smith [86], 1975), appear to be not very efficient in involving large-scale geophysical problems because they require discretization in the entire solution domain and they do not exactly satisfy the
radiation condition at infinity.
Recently, the boundary integral equation methods have been shown to be very effective in pursuing the solutions of the wave scattering in unbounded large-scale geophysical problems (Wong and Jennings [97], 1975; Dravinski [24], 1980, [25], 1982a and [26], 1982b; Wong [94], 1982; Dravinski [27], 1983;
Dravinski and Mossessian [28], 1987; Mossessian and Dravinski [67], 1987). These methods are very efficient for lower frequencies. They require only the discretization of the boundary of scatterers thus greatly
reducing the size of the problems, and the radiation condition at infinity is satisfied exactly. The half-space
Green’s functions incorporated in these methods satisfy the traction-free conditions on the surface of the
2
half-space. These methods in general require a large amount of computational effort for evaluation of the
Green’s functions (Dravinski and Mossessian [29], 1988).
To overcome the shortcomings of both the finite element and the boundary integral equation methods,
some hybrid methods (Berg [9], 1984; Mossessian and Dravinski [67], 1987 and [72], 1992) have been
developed which utilize the versatility of the finite element methods for detailed modeling in the near field
and effectiveness of the boundary integral equation methods in the far field. However, these methods have
convergence difficulties at higher frequencies and can not avoid the evaluation of the Green’s functions.
Besides, the size of the problem is still large.
For wave scattering problems in the high-frequency range, the ray methods based on the asymptotic
techniques in approximating the wave field and Gaussian beam technique are found to be very effective
(Moczo et al. [66], 1987; Nowack and Aki [75], 1984). These methods are relatively fast and can be applied
to inhomogeneous basins with complex geometries.
A different boundary method suitable for elastic wave scattering problems in an infinite media is
the so-called wave function expansion approach (Eshraghi and Dravinski [36], 1989a and [37], 1989b;
Sanchez-Sesma [83], 1983). This approach makes use of the complete family of wave functions which can
be evaluated efficiently. However, these functions do not, in general, satisfy the traction-free conditions
on the surface of the half-space which have to be imposed as additional conditions in the problem.
In addition to the above numerical approaches, Aki and Larner ( [1], 1970) devised a technique using
the Rayleigh hypothesis to calculate elastic wave field in a half-space overlayed by a single layer with an
irregular interface. This technique is applicable to wavelengths that are either larger than or of the order
of the dimension of the irregularities. In this method, the scattered field is described as a superposition of
plane waves. The application of the continuity conditions at the interface yields coupled integral equations
in the spectral coefficients. The equations are satisfied in the wavenumber domain. The interface shape
is made periodic and the equations are Fourier transformed and truncated. Complex frequencies are used
3
to reduce lateral interference associated with the periodic interface shape. The Aki-Larner technique may
suffer from the so-called Rayleigh ansatz error for problems involving large interface slopes (Aki and
Larner [1], 1970).
Bouchon ( [11], 1973) and Bard and Bouchon ( [5], 1980a and [6], 1980b) used the Aki-Larner technique
to study the seismic response of sediment-filled valley subjected to incident SH, P, and SV waves. It has
been shown that the accuracy of the Aki-Larner technique decreases as the angle of incidence increases
(Bard and Bouchon [5], 1980a).
With the exception of the ray theory approach, all the numerical methods discussed above are generally used to evaluate the steady-state response. The transient response of the problems is calculated by
two approaches: (1) direct methods and, (2) indirect methods. In direct methods, the transient solution is
formulated as a function of space and time. The main difficulty with these methods is the accumulation of
errors with increase of time (Rizzo et al. [78], 1985). Smith ( [86], 1975) used the finite element approach
and Boore et al. ( [10], 1971) applied the finite difference technique to solve the antiplane strain problem
directly in the time domain. For indirect methods, the transient response is obtained from the steady-state
solution through the use of the Fourier or the Laplace transform (Kobayashi [49], 1983). Niwa et al. ( [73],
1986) used the Fourier transform and the BIE approach for transient response evaluation for completely
embedded irregularities subjected to incident P and SV waves. Bard and Bouchon ( [5], 1980a, [6], 1980b
and [7], 1983) used the Fourier transform technique to extend the Aki-Larner method to the transient
analysis of alluvial valleys for antiplane and plane strain problems.
This work is an extension of the investigation done by Dravinski ( [27], 1983) and Dravinski and
Mossessian ( [28], 1987) who considered the scattering of SH, P, SV and Rayleigh waves by multiple finite
dipping layers of arbitrary shape. Here the problem is extended to infinitely long layers of irregular shape.
The present work places no restriction on the interface slopes and it does not utilize the Rayleigh hypothesis. Half-space Green’s function approach is first introduced to evaluate the scattered wave field for the
4
antiplane strain model. Closed-form full-space Green’s functions [49](1983) are then utilized to evaluate
the scattering field caused by the interface irregularity for the antiplane strain model again and for plane
strain and three-dimensional models. Traction-free conditions on the top surface of the half-space are not
satisfied by the full-space Green’s functions and need to be imposed as additional conditions( [102], 2000).
The problem is first defined in Chapter 2. A general formulation for antiplane strain models is given
in Chapter 3 using half-space Green’s functions for SH incident waves. The wave field in each domain
is expressed as a superposition of the free field of the corresponding flat-layer model and the scattered
wave field caused by the irregularities of interfaces. Parametric error analysis is conducted to determine
the optimum choices of parameters for steady-state response. Numerical results, including steady-state
and transient surface responses, are presented for one- and two-layer models. In particular, the transient
surface response for a vertically incident SH Ricker wavelet in the one-layer model clearly reveals the
location and amplitude of the irregularity and the thickness of the top layer away from the scatterer. This
implies that the study of both steady-state and transient surface responses provides broad application
prospects for experimental earthquake engineering research, including site amplification and subsurface
structures ( [33], 2003 and [34], 1996), etc.
The general antiplane strain model is formulated again using the full-space Green’s functions for both
SH and Love incident waves in Chapter 4. Good agreement is achieved for the one- and two-layer steadystate surface response results between the two different Green’s function approaches for SH incidences.
An additional transparency test on a two-layer model, where the material properties of the two lower
domains are made identical to simulate a one-layer model for the same Love incident wave, shows a good
agreement between the two models. This further supports the validity of the one- and two-layer antiplane
strain formulations.
Chapter 5 deals with the plane strain models using the full-space Green’s functions. Parametric error
analysis and steady-state results are presented for plane P, SV and Rayleigh incidences. An additional
5
transparency test on a one-layer model, where the material properties of the two domains are made identical to simulate a pure half-space model for the same Rayleigh incident wave, shows a good agreement
between the two models. This further supports the validity of the one-layer and pure half-space plane
strain models.
Three-dimensional models using the full-space Green’s functions are investigated in Chapter 6. Detailed parametric error analysis is given followed by the steady-state responses for P, SV, SH, Love and
Rayleigh incident waves. Certain mode conversions that occur in the three-dimensional surface response
cannot be captured by two-dimensional models. This highlights the necessity of three-dimensional modeling to accurately describe motion amplification in irregular layered media.
This research is summarized and concluded in Chapter 7.
6
Chapter 2
General Statement of Problem
The geometry of the problem is illustrated by Figure 2.1. The model consists of elastic layers over a halfspace. The interfaces between the layers are assumed to be irregular but sufficiently smooth without any
sharp corners being present. The spatial domains are denoted by Dj , j = 0, 1, 2, ..., J while the irregular
interfaces are designated Ci
, i = 1, 2, ..., J. The top free surface is denoted by SF . Each irregular interface
is assumed to vary around the corresponding flat reference interface with depth hi, i = 1, 2, ..., J. The
flat reference interfaces are denoted by Si, i = 1, 2, ..., J. If the i
th irregular interface is flat, then Ci will
coincide with the reference interface Si
.
The material of each domain is assumed to be elastic, homogeneous, and isotropic. The global Cartesian
coordinates with origin at O are denoted by x1, x2, and x3. The system is subjected to an incident plane
harmonic P, SV, SH, Love or Rayleigh wave. The incident angle with respect to x3-axis and an azimuthal
angle with respect to x1-axis are denoted by θo and φo, respectively.
In the absence of body forces, the equations of motion for steady-state elastic wave propagation are
specified by [64]
(λj + µj )∇∇ · uj + µj∇2uj + ρjω
2uj = 0; r ∈ Dj ; j = 0, 1, ..., J (2.1)
7
where
∇ = i
∂
∂x1
+ j
∂
∂x2
+ k
∂
∂x3
; ∇2 =
∂
2
∂x2
1
+
∂
2
∂x2
2
+
∂
2
∂x2
3
(2.2)
Here, uj = (uj , vj , wj ) is the displacement vector with the Cartesian displacement components uj , vj and
wj , r is the position vector, ρj is the mass density, λj and µj are Lame’s constants, ω is circular frequency
and i, j and k denote the unit vectors along the x1−, x2− and x3−axes, respectively. Throughout this
study, the subscript j = 0, 1, 2, ..., J corresponds to the j
th layer.
The traction-free condition on the top surface SF can be expressed by
σ
(J)
i3 = 0; r ∈ SF ; i = 1, 2, 3 (2.3)
The continuity of the displacements and tractions along the irregular interface Cj is given by
uj = uj−1; r ∈ Cj ; j = 1, 2, ..., J (2.4)
t
n
j = t
n
j−1
; r ∈ Cj ; j = 1, 2, ..., J (2.5)
where t
n
j
is the traction vector at the interface Cj with unit normal n.
In addition, the scattered wave field must satisfy appropriate radiation conditions at infinity [49].
To simulate an outgoing line source, the time factor e
iωt is understood in both antiplane strain and
plane strain models, as the asymptotic expansions for the adopted line sources are in the form of e
−ikσ for
the antiplane strain model, and e
−iζσ and e
−iησ for the plane strain model.
For the analysis of a three-dimensional model, since the spatial factors in the three-dimensional Green’s
functions are in the form of e
iζσ and e
iησ [50], the time factor has to be e
−iωt to simulate an outgoing point
source.
8
Here, k, ζ, and η are the wavenumbers associated with SH, P, and SV waves, respectively, and σ denotes
the distance from the observation point to the line or point source.
Figure 2.1: Geometry of a three-dimensional multilayer model with irregular interfaces subjected to an
incident plane harmonic body wave or Love, Rayleigh surface wave
9
Chapter 3
Antiplane Strain Model Using Half-Space Green’s Functions
3.1 Statement of the Problem
The geometry for this problem is depicted in Figure 3.1 with all the parameters defined in Chapter 2.
Figure 3.1: 2D model geometry
10
The system is subjected to a plane harmonic SH incident wave with an off-vertical angle of incidence
θ0. For the antiplane strain problem, the motion of the media takes place along the x2-axis only, i.e.,
u = (0, v, 0). Therefore, in the absence of body force, the equation of motion is given by [64]
(∇2 + k
2
j
)vj (r, ω) = 0; j = 0, 1, ..., J; ∇2 =
∂
2
∂x2
1
+
∂
2
∂x2
3
(3.1)
where
kj =
ω
βj
; β
2
j =
µj
ρj
; λ
(j)
β =
2πβj
ω
(3.2)
Here v is the only non-zero component of the displacement field acting along the x2-axis, kj , βj and λ
(j)
β
represent wavenumber, shear wave velocity and wavelength for the j
th layer, respectively, ω is the circular
frequency, µj is the shear modulus, and ρj is the density.
The perfect bonding conditions along the interface Cj can be written as
vj = vj−1; r ∈ Cj ; j = 1, 2, ..., J (3.3)
µj
∂vj
∂n
= µj−1
∂vj−1
∂n
; r ∈ Cj ; j = 1, 2, ..., J (3.4)
where n denotes the unit normal to Cj . The traction-free boundary condition (2.3) on the top surface SF
reduces to
∂vJ
∂x3
= 0; r ∈ SF (3.5)
The incident wave is assumed of the form
11
v
inc = e
−iko(x1 sin θo−x3 cos θo)+iωt; i =
√
−1 (3.6)
The factor e
iωt will be omitted in the following.
3.2 Steady-State Solution of the Problem
The total wave field in the multi-layer media can be expressed as a superposition of the free field and
scattered wave field according to
vj = v
f f
j + v
s
j
; j = 0, 1, 2, ..., J (3.7)
in which the superscript ff denotes the free field of the flat-layer system, and s denotes the scattered wave
field caused by the irregular interfaces.
3.2.1 Free Field
The free field in each layer is given by
v
f f
j = Aje
−ikj (x1 sin θj−x3 cos θj ) + Bje
−ikj (x1 sin θj+x3 cos θj )
; (x1, x3) ∈ Dj; j = 0, 1, ..., J (3.8)
where Aj and Bj are the amplitudes of up and down propagating SH waves in the domain Dj . The
parameters Aj , Bj and θj are known [2].
3.2.2 Scattered Wave Field
The unknown scattered wave field can be expressed in terms of single-layer potentials [54] [90]:
12
v
s
J =
Z
CJ1
qJ1(r0)GJ (r, r0
)dr0; r ∈ DJ (3.9)
v
s
j =
Z
Cj1
qj1(r0)Gj (r, r0
)dr0 +
Z
Cj2
qj2(r0)Gj (r, r0
)dr0; r ∈ Dj , j = 1, 2, ..., J − 1 (3.10)
v
s
0 =
Z
C00
q00(r0)G0(r, r0
)dr0; r ∈ D0 (3.11)
where q00, qj1 and qj2 are the unknown density functions. The functions Gj are the Green’s functions for
the line load in the half-space which satisfy the following equations [27]:
(∇2 + k
2
j
)Gj (r, r0) = δ(|r − r0|); j = 0, 1, 2, ..., J (3.12)
∂Gj
∂x3
= 0 at x3 = 0 (3.13)
with δ(.) being the Dirac delta function. The Green’s functions are given by [64]
Gj (r, r0) = i
4
[H
(2)
0
(kjσ1) + H
(2)
0
(kjσ2)]; j = 0, 1, 2, ..., J (3.14)
σ1 =
p
(x1 − x10)
2 + (x3 − x30)
2; σ2 =
p
(x1 − x10)
2 + (x3 + x30)
2; (3.15)
in which H
(2)
0
(.) denotes the Hankel function of the second kind and order zero. Figure 3.2 shows the
auxiliary surface CJ1 (below the interface CJ ) for the top layer DJ , Figure 3.3 shows the auxiliary surfaces
Cj1 (below the interface Cj ) and Cj2 (above the interface Cj+1) for the layer Dj , and the auxiliary surface
C00 (above the corresponding interface C1) is displayed by Figure 3.4 [27]. It should be noted that the
13
unknown scattered wave field representation (Equations (3.9)-(3.11)) involves improper integrals since the
interfaces Cj extend to infinity.
O
O
O O
O O
O O O O O O O
* * * * * *
* * *
dr
ds
2w
D
2wc
1
C 1
1
C 2wi
J
J
J
J
J
J
J
J
J
h
SF O
X
X
1
3
S
Figure 3.2: Distribution of sources and collocation points using half-space Green’s functions for top layer
DJ . The number of collocation points along the interface CJ is denoted by NJ , the length of the interface
CJ along which the collocation points are distributed is defined by 2w
J
c
, and the length of the irregular part
of the interface is represented by 2w
J
i
. The length of the auxiliary surface CJ1 along which the sources of
order LJ1 are allocated is given by 2w
J1
s
. The spacing between two adjacent collocation points is denoted
by d
J
s
and that between the interface CJ and the auxiliary surface CJ1 by d
J1
r
. Star: position of a line
source; circle: position of a collocation point
14
O
O
O O
O O
O O O O O O O
*
* *
* *
* * * * * *
* * *
*
*
O O O O O
O O O
O O O O O
* *
dr
ds
2w
D
j
j
j
2wc
j
1
C 1
j
j
j 1
dr
j 2
2wS
j2
2wc
j+1
2wi
2wi
j+1
Cj2
Cj+1
Cj
S
Figure 3.3: Distribution of sources and collocation points using half-space Green’s functions for the layer
Dj where j = 1, 2, ..., J. The numbers of collocation points along the interface Cj and Cj+1 are denoted
by Nj and Nj+1, respectively, the lengths of the interface Cj and Cj+1 along which the collocation points
are distributed are defined by 2w
j
c and 2w
j+1
c , respectively, and the lengths of the irregular part of the
interface are represented by 2w
j
i
and 2w
j+1
i
, respectively. The lengths of the auxiliary surface Cj1 and
Cj2 along which the sources of order LJ1 and Lj2 are allocated are given by 2w
J1
s
and 2w
J2
s
, respectively.
The spacing between two adjacent collocation points is defined by d
j
s on the interface Cj and that between
the interface Cj and the auxiliary surface Cjk is denoted by d
jk
r , k = 1, 2. Star: position of a line source;
circle: position of a collocation point
15
O
O
O O O O
O O O O O O O
*
* *
* *
*
* * *
d
2w00
00 00
r
ds
1
2wi
1
2w1
C1
c
D0
s
C
Figure 3.4: Distribution of sources and collocation points using half-space Green’s functions for bottom
half-space D0. The number of collocation points along the interface C1 is denoted by N1, the length of
the interface C1 along which the collocation points are distributed is defined by 2w
1
c
, and the length of the
irregular part of the interface is represented by 2w
1
i
. The length of the auxiliary surface C00 along which
the sources of order M are allocated is given by 2w
00
s
. The spacing between two adjacent collocation points
is denoted by d
1
s
and that between the interface C1 and the auxiliary surface C00 by d
00
r
. Star: position of
a line source; circle: position of a collocation point
If the scattered wave field is assumed in terms of discrete line sources, then for the j
th layer
q00 = amδ(|r − rm|); r ∈ D0, rm ∈ C00; m = 1, 2, ..., M (3.16)
qj1 = blj1
δ(|r − rlj1
|); r ∈ Dj , rlj1 ∈ Cj1;
lj1 = 1, 2, ..., Lj1; j = 1, 2, ..., J − 1 (3.17)
16
qj2 = blj2
δ(|r − rlj2
|); r ∈ Dj , rlj2 ∈ Cj2;
lj2 = Lj1 + 1, Lj1 + 2, ..., Lj1 + Lj2; j = 1, 2, ..., J − 1 (3.18)
qJ1 = blJ1
δ(|r − rlJ1
|); r ∈ DJ , rlJ1 ∈ CJ1; lJ1 = 1, 2, ..., LJ1 (3.19)
where am, blj1
and blj2
denote the unknown source intensities, and M, Lj1 and Lj2 represent the order of
approximation, that is the number of discretized sources, along the auxiliary surfaces C00, Cj1 and Cj2,
respectively. Therefore the infinite integrals in Equations (3.9)-(3.11) are replaced by the finite integrals
and, consequently, the number of sources which represents the scattered waves become finite as well. The
orders of approximations M, Lj1 and Lj2 are still to be determined.
Substitution of equations (3.16)-(3.19) into (3.9)-(3.11) yields the scattered wave field in the following
form:
v
s
J =
X
LJ1
lJ1=1
blJ1GJ (r, rlJ1
); r ∈ DJ ; rlJ1 ∈ CJ1 (3.20)
v
s
j =
X
Lj1
lj1=1
blj1Gj (r, rlj1
) +
Lj
X1+Lj2
lj2=Lj1+1
blj2Gj (r, rlj2
);
r ∈ Dj , j = 1, 2, ..., J − 1; rlj1 ∈ Cj1; rlj2 ∈ Cj2 (3.21)
v
s
0 =
X
M
m=1
amG0(r, rm); r ∈ D0; rm ∈ C00 (3.22)
17
The approach used in this work avoids the problem associated with singularities of Green’s functions
at the cost of introducing the so-called auxiliary surfaces.
3.2.3 Source Intensities for Scattered Wave Field
The scattered wave field (3.20) - (3.22) satisfies the equation of motion (3.1) and the traction-free boundary
condition (3.5). The unknown source intensities of the scattered wave field are determined by imposing
the displacement and traction continuity conditions (3.3) and (3.4). This implies that
v
s
j − v
s
j+1 = v
f f
j+1 − v
f f
j
(3.23)
µj
∂vs
j
∂n
− µj+1
∂vs
j+1
∂n
= µj+1
∂vf f
j+1
∂n
− µj
∂vf f
j
∂n
r ∈ Cj ; j = 1, 2, ..., J (3.24)
Choosing Nj collocation points along interfaces Cj , the continuity conditions (3.23)-(3.24) can be
written in the following form:
Aa = f (3.25)
which can be solved for the unknown coefficient vector a in the least-squares sense using the QR decomposition [74].
Therefore, the boundary conditions (3.5) on the surface of the half-space SF are satisfied exactly, while
the continuity conditions (3.3)-(3.4) are satisfied in the least-squares sense (3.25). Since the number of collocation points changes with frequency (at least ten collocation points per wavelength of the incident wave)
18
so does the number of sources. Consequently, the continuity conditions (3.25) are frequency dependent as
well.
3.3 Numerical Results
The numerical results are presented for one- and two-layer antiplane strain models. These models incorporated most of the physical characteristics of the problem with the minimum computational effort. For
each model, the numerical results consist of parametric error analysis, the steady-state response, and the
transient response.
Parametric error analysis deals with the role of different parameters upon the surface response. The
following parameters are investigated: the auxiliary surfaces Cj1 and Cj2, the number of sources M ,
Lj1 and Lj2, and the number of collocation points Nj along each interface Cj . These parameters are
determined by minimizing an error based on the surface response iterations. The steady-state surface
response is then determined using the optimum values of the required parameters obtained in the error
analysis. The transient response is obtained through the Fourier transform using the FFT technique.
3.3.1 Error Criteria
For the error analysis, all spatial variables are normalized with respect to the corresponding flat-layer
depth of the lowest interface which is assumed to be unity. For the steady-state problem, the surface
displacement field is normalized with respect to the corresponding free-field solution.
Since there are no exact solutions available, the surface response error is defined as the relative error
of the two consecutive iterations of the surface response according to
Er =
PNs
i=1 (|v
n
J
(ri) − v
n−1
J
(ri)|)
2
PNs
i=1 |v
n
J
(ri)|
2
; ri ∈ SF (3.26)
19
where the subscript J indicates the top layer DJ , Ns is the number of observation points on the surface SF ,
and the superscripts n and n − 1 represent two consecutive iterations of the surface response. When the
numerical calculations converge, the error Er should approach zero. In practice, the minimum of the error
Er is being sought. Therefore, the choice of auxiliary surfaces Cj1 and Cj2, and the number of sources
and collocation points M, Lj1, Lj2 and Nj will be determined by minimizing the error Er.
Throughout the tests, the number of observation points used to compute the error Er is chosen to be
Ns = 40. The observation points are equally spaced over the range of x1 ∈ (−2, 2).
The dimensionless frequency is defined by
Ω = 2hJω
πβJ
(3.27)
where hJ and βJ are the corresponding flat-layer depth and shear wave velocity of the top layer, and ω
denotes the circular frequency of the incident wave.
3.3.2 One-Layer Model
The geometry of this model is depicted in Figure 3.5. The interface C between the layer and the half-space
is cosine-shaped defined by
x3 =
1 for |x1| > 0.5
1 + 0.05(1 + cos 2πx1) for |x1| ≤ 0.5
(3.28)
The number of sources along the auxiliary surfaces C00 and C11 are denoted by M and L, respectively.
The number of collocation points along the interface is taken to be N, while the length of the interface on
which the collocation points are distributed is specified by 2wc. The length of the auxiliary surfaces C00
and C11 over which the corresponding sources are distributed is denoted by 2ws, and 2wi
is the length
20
of the irregular part of the interface. The maximum deviation of the irregular interface from a flat-layer
model is denoted by d, and h is the depth of the flat layer. The spacing between two adjacent collocation
points is marked by ds and that between the auxiliary surfaces C00/C11 and the interface C by dr. Unless
stated differently, the material properties are those defined in Figure 3.5. Since there is only one interface
between the layer and half-space, the index j associated with multiple interfaces is omitted.
O
O
O O
O O
O O O O O O O
* * * * * *
* * *
dr
ds
2w
D
2wc
C 1
2wi
C
h
SF O
X
X
1
3
S
* * * * * *
* * *
2wS
1
1
1
C00
D0
dr
d
q0
Incident plane harmonic wave
Figure 3.5: Geometry of the one-layer model with a cosine-shaped interface and the corresponding distribution of sources and collocation points using half-space Green’s functions. Here, h = 1, 2wi = 1,
d = 0.1 and the material properties are β0 = ρ0 = 1 in domain D0 and β1 = ρ1 = 0.5 in domain D1.
Star: position of a line source; circle: position of a collocation point
21
3.3.2.1 Parametric Error Analysis
First, the transparency test for a zero-scattering condition is performed. In this test, the material properties
of the layer are assumed to be the same as those of the half-space. For an initial choice of the parameters
(θ0 = 0o
; N = 60, M = L = 24; 2wc = 2, 2ws = 2.1, ds = 0.0339, dr = 0.1017, Ω = 2 and Ω = 4)
the contribution of the scattered field is found to be negligible, and the normalized surface displacement is
calculated to be equal to one. The transparency test also provides the ratio of the number of rows versus
the number of columns in matrix A (equation (3.25)). The best accuracy is achieved for 2N/(M +L) = 2.5
which relates the number of collocation points (N) and the number of sources (M + L). Since the former
is a function of the frequency (at least ten collocation points per wavelength of the incident wave) the ratio
correlates the number of collocation points and the number of sources with frequency.
The second test investigates the location of the auxiliary surfaces C00 and C11 relative to the interface C. J. E. Luco (private communication) proposed the following relationship between the sources and
collocation points:
dr
.= 3ds (3.29)
where dr is the distance between the interface C and the auxiliary surface C00 or C11, and ds represents
the spacing between two adjacent collocation points (see Figure 3.5).
To test the relationship (3.29), the distance ds between the two adjacent collocation points is assumed
to be fixed. At the same time, the location of auxiliary surfaces C00 and C11, specified by the parameter dr,
is varied. All other parameters, such as the number of sources M and L, the number of collocation points
N, the length of auxiliary surfaces 2ws, and the length of interface 2wc, remain the same. Therefore, from
the surface displacements, the corresponding error Er has been calculated for a range of auxiliary surfaces
(0.05 ⩽ dr ⩽ 0.5). Figure 3.6 illustrates the results of this test for one frequency. The minimum error has
22
been obtained for dr
.= 0.1 which agrees very well with the optimum distance dr = 0.102 predicted by
equation (3.29). For different frequencies, the results of the same test are summarized in Table 3.1.
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
10−5
10−4
10−3
10−2
10−1
dr
Er
Ω=3.0
Figure 3.6: Error Er as a function of the distance dr between the interface C and the auxiliary surfaces
C00 and C11 for the one-layer model subjected to a vertically incident plane harmonic SH wave. N = 60,
M = L = 24; 2wc = 2, 2ws = 2.1; Ω = 3; dr = 0.102 is the optimum dr proposed by the relationship
(dr
.= 3ds)
23
Table 3.1: The optimum distance (dr) between the auxiliary surface and the interface based on the minimization of the error Er (3.26). Parameters: N = 60, M = L = 24; Xn = 2; Xm = Xl = 2.1; vertical
incidence θ0 = 0o
Ω dr
0.5 0.075
1.0 0.075
2.5 0.100
3.0 0.100
4.0 0.150
12.5 0.100
20.0 0.125
Apparently, for a wide range of frequencies the optimum value for dr agrees well with that obtained by
equation (3.29). Consequently, this relationship has been accepted through further study of the problem.
The third test deals with the estimate of the minimum number of collocation points N. For that purpose, the error Er is calculated for different numbers of collocation points N with the other parameters
being fixed. The results of the calculations are depicted in Figure 3.7. It is interesting to note that the error
Er maintains minimum values for a wide range of N. Similar tests are done for various incident angles
and frequencies. For the error Er ⩽ 0.001, these results are summarized in Table 3.2. As expected, the
number of collocation points Nmin increases with the increase in frequency Ω. Namely, with an increase
in frequency, the wavelength of the incident and scattered waves becomes smaller. Therefore, the distance
between two adjacent collocation points ds has to be sufficiently small to account for the shorter wavelengths in the problem. It is found in this study that there should be at least ten collocation points per
wavelength of the incident wave.
24
The fourth test deals with the length of the interface 2wc on which the collocation points are distributed
(Figure 3.5). For this test, the maximum distance between the adjacent collocation points ds is assumed to
be
ds max = 2/Nmin (3.30)
where Nmin is the number of collocation points given in Table 3.2. One should recall that 2 represents the
value of the interface length 2wc used in the third test. Thus, the separation of the collocation points is
fixed, ds = ds max, while the parameters 2wc, N, M, and L are varied. For vertical incidence, the results
are shown in Figure 3.8. It has been shown that 2wc = 2, which is twice the length of the irregular part of
the interface, is sufficient for convergence of the test. Similar results are obtained for other cases of Table
3.2,
Therefore, the convergence tests can be summarized as follows:
• The transparency tests provide the initial values of the parameters in the model.
• Luco’s relationship (3.29) yields the acceptable error in the scattered wave field.
• Initially, there should be at least ten collocation points per wavelength of the incident wave along the
interface, and the number of collocation points N should be increased with increase of frequency.
• The length of interface 2wc needed for convergence of the results is about twice the length of the
irregular part of the interface.
25
10 20 30 40 50 60 70 80 90 100 110
10−8
10−7
10−6
10−5
10−4
10−3
N
Er
Ω=4.0
Figure 3.7: Error Er as a function of the collocation points N for the one-layer model and a vertically
incident SH wave. 2wc = 2, 2ws = 2.1; dr = 3ds; θ0 = 0o
; Ω = 4; M = L; 2N/(M + L) = 2.5
Table 3.2: Minimum number of collocation points N for a one-layer model which results in the relative
error Er less than 0.001
Nmin Ω = 0.5 Ω = 1 Ω = 2 Ω = 3 Ω = 4 Ω = 12.5 Ω = 20
θ0 = 0o 15 25 35 35 20 45 55
θ0 = 30o 15 20 20 30 30 50 70
θ0 = 60o 20 20 25 30 35 55 70
θ0 = 85o 20 25 30 35 35 55 85
26
0 0.5 1 1.5 2 2.5 3 3.5 4
10−4
10−3
10−2
10−1
100
2w c
Er
Ω=12.5
Figure 3.8: Error Er as a function of length 2wc over which the collocation points are applied for the onelayer model and a vertically incident SH wave. θ0 = 0o
; ds = 0.0339; dr = 3ds; Ω = 12.5; M = L;
2N/(M + L) = 2.5; 2wc = (N − 1)ds
As the final test, the response of a model considered by Aki and Larner [1] is reconsidered next. Aki
and Larner devised a technique to calculate the elastic wave field in a half-space overlaid by a single layer
with an irregular interface. The incident plane SH wave propagating from below was assumed.
The geometry of the problem studied by Aki and Larner [1] consists of a single layer where the interface
between the layer and the half-space is defined (in km) by
x3 =
25 for |x1| > 25
25 + 2.5(1 + cos πx1
25 ) for |x1| ≤ 25
(3.31)
The velocities and densities of the layer and the half-space roughly correspond to those of the crust
and upper mantle, respectively. The plane SH wave with a wavelength of 10km in the half-space is incident
from below at various incident angles.
27
The comparison of the numerical results obtained by the method proposed in this work with those of
Aki and Larner [1] is shown in Figure 3.9. Relatively good agreement for the surface displacement between
the two approaches is achieved. The phase delay of the surface displacement field is in excellent agreement
for the two results. Therefore, the results of Figure 3.9 confirm the accuracy of the present method.
−2 0 2 4
0
0.5
1
1.5
2
42
o
−2 0 2 4
−0.5
0
0.5
42
o
−2 0 2 4
0
0.5
1
1.5
2
Normalized Surface Displacement
48
o
−2 0 2 4
−0.5
0
0.5
Phase Delay
48
o
−2 0 2 4
0
0.5
1
1.5
2
x
1
( 1 unit = 25 km)
55
o
−2 0 2 4
−0.5
0
0.5
55
o
x
1
( 1 unit = 25 km)
Figure 3.9: Comparison of normalized surface displacement and phase delay between the present approach
(solid) and the Aki-Larner technique (circle) for various angles of incidence. The material properties are
defined as β0 = 4.0km/ sec, ρ0 = 3.3g/cm3
in the bottom half-space (D0) and β1 = 3.0km/ sec, ρ1 =
2.8g/cm3
in the top layer (D1), and the depth of layer h = 25km. The length and the max deviation of
the cosine-shaped scatterer are 50km and 5km, respectively. The wavelength of the incident SH wave in
the half-space is 10km which corresponds to circular frequency ω = 2.5133 sec−1
.
This completes the parametric error analysis for the one-layer model. Steady-state results for different
frequencies and angles of incidence are presented next.
28
3.3.2.2 Steady-State Response
Based on the convergence criteria developed above, the steady-state surface response for the one-layer
model with a vertical SH incidence is shown in Figure 3.10. The following can be observed from the
steady-state results.
• A small deviation along the flat-layer interface may cause large surface motion amplification (up to
several times of the corresponding free-field motion).
• The surface displacement strongly depends upon the incident wave and the position of the site.
• For larger values of |x1|, the normalized surface displacement approaches unity which corresponds
to the flat-layer response. This is expected since, away from the irregularity, the scattered wave field
should decrease in amplitude and the total wave field approaches the values of the free field.
• For observation sites directly above the irregular interface, the surface response exhibits resonance
features [104]. The resonance frequencies are closely related to the ones for a flat-layer model. For
a vertical incidence, the first five frequencies at the site (x1, x3) = (0, 0) are found to be Ωi = 1.00,
2.50, 4.75, 6.50 and 8.50. For the corresponding flat-layer model, the resonance frequencies are
Ωi = 1, 3, 5, 7 and 9. The difference between the two is attributed to the presence of the irregularity.
• The surface motion is found to be symmetric for a vertically incident wave, thus lending further
support for the validity of the calculated results.
This concludes the steady-state analysis for a one-layer model. The transient results are considered
next.
29
−4
−2
0
2
4
0
1
2
3
4
5
0
1
2
3
4
5
x
1
θ
0
= 0 o
Ω
Normalized surface displacement
Figure 3.10: Normalized surface response for the one-layer model and vertically incident waves as a function of dimensionless frequency Ω and spatial variable x1. θ0 = 0o
; 2wc = 2, 2ws = 2.1; dr = 3ds,
N = 40 and M = L = 16.
3.3.2.3 Transient Response
The transient response for the one-layer model (Figure 3.5) is obtained from the steady-state response
through the Fourier synthesis. Here, the response is studied for an incident SH Ricker wavelet [77] defined
by
f(t) = (√
π/2)(τ − 0.5)e
−τ
; τ = (π(t − ts)/tp)
2
(3.32)
where ts corresponds to the peak amplitude in the time domain and tp corresponds to the circular frequency ωp = 2π/tp which is associated with the peak amplitude in the frequency domain. Throughout
the transient analysis, the response is calculated at 41 observation points equally spaced in the range of
x1 ∈ [−4, 4] on the surface SF (see Figure 3.1). Figure 3.11 displays the transient surface response of a
30
one-layer model with an irregular interface due to a vertically incident SH Ricker wavelet. The response
can be regarded as a superposition of motion from the flat-layer model and that caused by the scattering
from the layer irregularities. The scattered surface wave can be observed travelling with the shear wave
velocity of the layer β1. This is indicated by the lines (A1 → B1) and (A2 → B2) in Figure 3.11. The
delay in the peaks of the transient response for sites near the origin is caused by the presence of the irregularity in the layer. Namely, the time between two adjacent peaks along the transient response at x1 = 0
corresponds to the double time required by a vertically propagating wave to traverse the distance from the
surface to the lowest position of the irregular interface. This time is larger here than that in the flat-layer
case, thus causing the gradual delay of the peaks at x1 = 0.
Figure 3.11: Transient surface response of the one-layer model due to a vertically incident Ricker wavelet.
tp = 1.25 sec, ts = 1.5 sec .
As expected, the peak transient surface response amplitude gradually attenuates with time due to
backscattering of the waves into the half-space.
This concludes the numerical results for the one-layer model. The two-layer model is considered next.
31
3.3.3 Two-Layer Model
The two-layer model is depicted by Figure 3.12. This model consists of the two cosine-shaped interfaces
C1 and C2 defined by
x3 =
1 for |x1| > 0.5
1 + 0.05(1 + cos 2πx1) for |x1| ≤ 0.5
; r ∈ C1 (3.33)
and
x3 =
0.5 for |x1| > 0.5
0.5 + 0.05(1 + cos 2πx1) for |x1| ≤ 0.5
; r ∈ C2 (3.34)
The number of sources along the auxiliary surfaces C00, C11, C12 and C21 are denoted by M, L11, L12
and L21, respectively. The number of collocation points along the two interfaces Cj are taken to be Nj .
To simplify the model, the length of the two interfaces on which the collocation points are distributed is
specified by 2wc, while that of the auxiliary surfaces C00, C11, C12 and C21 over which the corresponding
sources are distributed are represented by 2ws. The length of the irregular part of the interface Cj is taken
to be 2w
(j)
i
. The maximum deviation of the irregular interface from the flat layer is denoted by dj , and hj is
the depth of the corresponding flat layer. The spacing between two adjacent collocation points is marked
by d
(j)
s and the locations of the auxiliary surfaces C00, C11, C12, C21 are represented by d
(00)
r , d
(11)
r , d
(12)
r
d
(21)
r , respectively. Here, the subscript j = 1, 2 corresponds to the j
th interface.
32
Figure 3.12: Geometry of the two-layer model with two cosine-shaped interfaces and the corresponding
distribution of sources and collocation points using half-space Green’s functions. Here, h1 = 1, h2 = 0.5;
2w
(1)
i = 2w
(2)
i = 1; d1 = d2 = 0.1 and the material properties are β2 = ρ2 = 0.25; β1 = ρ1 = 0.5
and β0 = ρ0 = 1. Star: position of a line source; circle: position of a collocation point
3.3.3.1 Parametric Error Analysis
First, the transparency test for a zero-scattering condition is performed. In this test, the material properties
of the two layers D0 and D1 are assumed to be the same as those of the half-space. The number of the
sources along the auxiliary surfaces C00, C11, C12 and C21 and the collocation points along the interfaces
C1 and C2 were adjusted until the contribution of the scattered field is found to be negligible, and the
normalized surface displacement was found to be equal to one.
33
Based on the parametric error analysis of the one-layer model, Luco’s relationship (3.29) and the length
of the interfaces 2wc = 2(2wi) are assumed in this model as well. Therefore, the next test is conducted to
determine the minimum number of collocation points N1 and N2, along the interfaces C1 and C2, respectively. For that purpose, four different approaches for error calculations have been designed: (i) changing
first the number of collocation points N1 and then, by repeating the error calculations for different numbers
of collocations points N2; (ii) simultaneously increasing N1 and N2 with N1 = N2; (iii) simultaneously
increasing N1 and N2 with N1 = 2N2; and (iv) simultaneously increase of N1 and N2 with N2 = 2N1.
It was concluded from these tests that, for a range of frequencies 0 < Ω ≤ 4, the surface response
converges when N1 is between 20 to 80 while N2 can vary from 30 and 35.
Based on the above parametric tests, the steady-state results for different frequencies and angles of
incidence are presented next.
3.3.3.2 Steady-State Response
For a vertical incidence, the results are depicted in Figure 3.13. The first three frequencies at the site
(x1, x3) = (0, 0) are found to be Ωi = 1.2, 2.5 and 3.4. For the corresponding flat-layer model, the
resonance frequencies are Ωi = 0.88, 2 and 3.12. The difference in the resonance frequencies between
the two is attributed to the presence of the layer irregularities.
As in the one-layer model, the scattered wave field decreases in amplitude and the total wave field
approaches the free-field values away from the region directly atop the layer irregularities. The surface
response is found to be strongly affected by the incident wave and the site location.
This concludes the steady-state analysis of the two-layer problem. Corresponding transient results are
considered next.
34
Figure 3.13: Normalized surface response of the two-layer model for a vertically incident plane harmonic
SH wave as a function of dimensionless frequency Ω and spatial variable x1. θ0 = 0o
; Ω = 0−4; N1 = 60,
M = L11 = 24; N2 = 30, L12 = L21 = 12. Throughout the parametric error analysis of the two-layer
model, 2wc = 2, 2ws = 2.1; d
(11)
r = d
(00)
r = 3d
(1)
s , d(12)
r = d
(21)
r = 3d
(2)
s .
3.3.3.3 Transient Response
The transient results are obtained from the steady-state response through the Fourier synthesis. As before,
the response is evaluated for an incident SH Ricker wavelet at 41 observation points equally spaced in the
range x1 ∈ [−4, 4].
Figure 3.14 displays the transient surface responses of a two-layer model with irregular interfaces for
a vertical Ricker wavelet incidence. The results for obliquely incident waves are omitted for the sake of
brevity. Comparison with the transient response of the one-layer model leads to the following conclusion:
• The peak amplitude decreases with time and angle of incidence for both models while the motion
amplitude of the two-layer model appears larger than that of the one-layer model.
35
• The response patterns of the two-layer model are much more complex than those of the one-layer
model.
Therefore, it appears that the presence of an additional layer may change significantly the surface
response when compared to the single-layer model.
Figure 3.14: Transient surface response of the two-layer model due to a vertically incident SH Ricker
wavelet. tp = 5 sec, ts = 15 sec
3.4 Conclusions
An indirect boundary integral equation approach was used to evaluate the surface response of a multilayered half-space with irregular layers to an incident plane harmonic SH wave. The convergence of the
method was determined based on a parametric error analysis of the problem and through comparison with
the results obtained by Aki and Larner [1]. Subsequently, both steady-state and transient surface responses
were determined for one- and two-layer models. The presented results show that the presence of layer
36
irregularities may cause significant changes in the surface response when compared with the corresponding flat-layer model response. The scattered waves affect the surface ground motion most significantly for
the sites directly atop the irregularities. Away from the irregularities, the amplitude of the scattered waves
decreases and the response approaches the free-field one. The surface motion was found to be very sensitive to the nature of the incident wave (angle of incidence and frequency), the location of the observation
site, and the geometry and material properties of the layered medium.
37
Chapter 4
Antiplane Strain Model Using Full-Space Green’s Functions
4.1 Statement of Problem
The geometry of the problem is shown by Figure 3.1. The model is the same as in Chapter 3. However, the
problem will be solved this time using the full-space Green’s functions. Such an approach will facilitate
the solution of the corresponding plane strain and general three-dimensional models. Namely, for these
models, the full-space Green’s functions can be evaluated explicitly [50] [100] [102] while the evaluation
of the corresponding half-space Green’s functions is much more difficult to achieve.
In addition to the plane harmonic SH incident wave in Chapter 3, the response to the incident Love
surface wave is also investigated in this chapter.
The incident Love wave is assumed of the form [40] [56]
v
inc = e
iωt−iκx1−κx3
q
1−(
c
β0
)
2
; κ =
ω
c
; i =
√
−1 (4.1)
Here, β0 is the S-wave speed in domain D0 while κ and c denote the wavenumber and speed for the
Love wave in all domains of the media, respectively. β0 > c is assumed.
The factor e
iωt will be omitted in the following.
38
4.2 Steady-State Solution of the Problem
As in Chapter 3, the total wave field for the plane harmonic SH incident wave in the media can be expressed
as a superposition of the free field and the scattered field (3.7). The free field for the plane harmonic SH
incident wave has been given by equation (3.8) in each flat layer.
By assuming that the SH- and Love-wave speed values satisfy β0 > β1,..., > c > βJ [40], then the free
field in each flat layer for the incident Love wave is defined by
v
f f
J = AJ e
−iκx1+iκx3
q
(
c
βJ
)
2−1
+ BJ e
−iκx1−iκx3
q
(
c
βJ
)
2−1
; (x1, x3) ∈ DJ, (4.2)
v
f f
j = Aje
−iκx1−κx3
q
1−(
c
βj
)
2
+ Bje
−iκx1+κx3
q
1−(
c
βj
)
2
; (x1, x3) ∈ Dj; j = 0, 1, ..., J − 1 (4.3)
where the parameters Aj , Bj and c are known [2].
For the bottom half-space domain D0, the free field of the Love wave is that of the incident wave (4.1),
i.e., A0 = 1 and B0 = 0.
4.2.1 Scattered Wave Field
As before, it is assumed that the scattered wave field can be expressed in terms of single-layer potentials
[54] [90]:
v
s
0 =
Z
C00
q00(r0)G0(r, r0
)dr0; r ∈ D0 (4.4)
v
s
j =
Z
Cj1
qj1(r0)Gj (r, r0
)dr0 +
Z
Cj2
qj2(r0)Gj (r, r0
)dr0; r ∈ Dj ; j = 1, 2, ..., J (4.5)
where C00, Cj1 and Cj2 are the auxiliary surfaces defined in Figure 3.3 and Figure 3.4 [27] [102], q00, qj1
and qj2 are the corresponding unknown density functions. Since the full-space Green’s functions do not
39
satisfy the traction-free boundary condition (3.5) on the top surface SF , an additional auxiliary surface CJ2
should be introduced above the top surface SF (see Figure 4.1) to impose the traction-free conditions [102].
2w
D
2wc
SF O
X
X
1
S
C
O
O
O O
O O
O O O O O O O
* * * * *
* * *
dr ds
C 2wi
h
3
C
d
*
O O O O O O O O O O O O O
* * * * *
d0
J
J1
J2
J
J
J2
J
J
2wS
J
J1
J1
Figure 4.1: Distribution of sources and collocation points using full-space Green’s functions for the top
layer DJ . An additional auxiliary surface CJ2 is introduced above the top surface SF to impose tractionfree conditions on the top surface SF . Definitions of the other parameters in this figure are the same as in
Chapter 3. Star: position of a line source; circle: position of a collocation point
The Gj are now the full-space Green’s functions for the line load with the material properties of domains Dj , j = 0, 1, 2, ..., J, and they satisfy the motion equation (3.12). For the antiplane strain model,
the full-space Green’s functions are given by [50]
Gj (r, r0
) = i
4
H
(2)
0
(kjσ); j = 0, 1, 2, ..., J (4.6)
r = (x1, 0, x3), r0 = (x10, 0, x30), σ =| r − r0 |
where H
(2)
0
(.) denotes the Hankel function of the second kind and order zero.
40
If the scattered wave field is assumed in terms of discrete line sources, then
q0 = amδ(r − rm); r ∈ D0, rm ∈ C00; m = 1, 2, ..., M (4.7)
qj1 = blj1
δ(r − rlj1
); r ∈ Dj , rlj1 ∈ Cj1; lj1 = 1, 2, ..., Lj1 (4.8)
qj2 = blj2
δ(r − rlj2
); r ∈ Dj , rlj2 ∈ Cj2; lj2 = Lj1 + 1, Lj1 + 2, ..., Lj1 + Lj2 (4.9)
where am, blj1
and blj2
denote the unknown source intensities, and M, Lj1 and Lj2 are the orders of
approximation along the auxiliary surfaces C00, Cj1 and Cj2, respectively.
Substitution of equations (4.7), (4.8) and (4.9) into (4.4) and (4.5) yields the scattered wave field in the
following form
v
s
0 =
X
M
m=1
amG0(r, rm); r ∈ D0; rm ∈ C00 (4.10)
v
s
j =
X
Lj1
lj1=1
bljlGj (r, rlj1
) +
Lj
X1+Lj2
lj2=Lj1+1
blj2Gj (r, rlj2
); r ∈ Dj ; rlj1 ∈ Cj1; rlj2 ∈ Cj2 (4.11)
4.2.2 Source Intensities for the Scattered Wave Field
The free field for the incident SH wave (3.8) or the incident Love wave (4.2)-(4.3) and the scattered wave
field (4.10)-(4.11) satisfy the equation of motion (3.1). The unknown source intensities of the scattered
wave field are determined by imposing the continuity conditions (3.3) - (3.4) along interfaces Cj and the
traction-free condition (3.5) on the top surface SF .
v
s
j−1 − v
s
j = v
f f
j − v
f f
j−1
; r ∈ Cj ; j = 1, 2, ..., J (4.12)
41
µj−1
∂vs
j−1
∂n
− µj
∂vs
j
∂n
= µj
∂vf f
j
∂n
− µj−1
∂vf f
j−1
∂n
; r ∈ Cj ; j = 1, 2, ..., J (4.13)
µJ
∂vs
J
∂x3
= 0; r ∈ SF (4.14)
Choosing Nj collocation points along interface Cj and P collocation points along the top surface SF ,
the continuity conditions (4.12)-(4.13) and the traction-free condition (4.14) can be written in the following
form
Aa = f (4.15)
Detailed expressions for the matrix A, the forcing vector f, and the unknown vector a are given in
Appendix A. The unknown coefficient vector a can be determined by solving the matrix equation (4.15)
using the least-square method with the QR decomposition [74].
4.3 Numerical Results
The numerical results for the full-space Green’s functions are presented for one- and two-layer models.
Parametric error analysis is performed first to investigate the role of different parameters upon the surface response. In addition to the parameters investigated in Chapter 3, such as the auxiliary surfaces C00,
Cj1 and Cj2, the number of sources M, Lj1 and Lj2, and the number of collocation points Nj along interface Cj , the number of collocation points P along the top surface SF should also be studied as well.
These parameters are determined by minimizing an error based on the surface response iterations. Subsequently, the steady-state surface response is determined using the optimum parameters obtained in the
error analysis.
42
4.3.1 Error Criteria
The error Er is defined according to equation (3.26). The choice of the computational parameters will be
determined by minimizing the error Er.
Throughout the tests of this chapter, the number of observation points used to compute the error Er
on the surface SF is chosen to be Ns = 151. These observation points are equally spaced over the range
|x1| ≤ 4.
The dimensionless frequency Ω is defined according to equation (3.27).
4.3.2 One-Layer Model
The geometry of the model is depicted by Figure 4.2.
Figure 4.2: Geometry of the one-layer model with a cosine-shaped interface using the full-space Green’s
functions. The distribution of sources (stars) on the auxiliary surfaces C00, C11, and C12 and collocation
points (circles) on the interface C1 and the top surface SF are shown as well. 2wi = 2a = h = 1.
43
The interface C1 between the top layer and the half-space is cosine-shaped defined by
x3 =
h for |x1| > a
h +
d
2
(1 + cos πx1
a
) for |x1| ≤ a
(4.16)
where h is the corresponding flat-layer depth of the top layer, a(= wi) is the half-width of the scatterer
and d is the maximum deviation of the cosine-shaped scatterer from a flat-layer model. As in Chapter 3,
the index j associated with multiple interfaces is omitted.
The geometry selected here is symmetric about the x2x3−plane. This should produce a symmetric
character of the surface response which can be used to verify the formulation of the antiplane strain model
in this study. This feature will be investigated in detail later.
The number of sources along the auxiliary surfaces C00, C11 and C12 are denoted by M, L1 and L2,
respectively. The length of auxiliary surfaces C00, C11, and C12 over which the corresponding sources are
distributed is denoted by 2ws. The number of collocation points along the interface C1 and the top surface
SF are taken to be N and P, respectively, while the length of the interface C1 and the top surface SF on
which the collocation points are distributed is specified by 2wc. The depth of the flat layer is denoted by h.
The spacing between two adjacent collocation points is marked by ds. The distance between the auxiliary
surfaces C00, C11 and the interface C1 is represented by dr and that between auxiliary surface C12 and
top surface SF by d0. Therefore, the location of auxiliary surfaces can be expressed by
C00:
x3 =
h − dr for a < |x1| ≤ ws
h − dr +
d
2
(1 + cos πx1
a
) for |x1| ≤ a
(4.17)
C11:
44
x3 =
h + dr for a < |x1| ≤ ws
h + dr +
d
2
(1 + cos πx1
a
) for |x1| ≤ a
(4.18)
C12:
x3 = −d0; |x1| ≤ ws
(4.19)
Unless stated differently, the material properties of this one-layer model are those defined in Figure 3.5,
i.e.,
D0 : µ0 = 1, β0 = 1, ρ0 = 1, ν0 = 1/3 (4.20)
D1 : µ1 = 0.125, β1 = 0.5, ρ1 = 0.5, ν1 = 1/3 (4.21)
4.3.2.1 Parametric Error Analysis
Six convergence and two comparison tests are performed to estimate the various parameters of the problem. The following relationships are accepted in this test:
d0 = dr (4.22)
2wc = 2ws (4.23)
Parametric Error Analysis is performed for the one-layer model of incident SH wave only.
1. Transparency Test. In this test, the material properties of the top layer are assumed to be the same
as those of the half-space. For an initial choice of the sources along the auxiliary surfaces C00, C11 and
C12, and the collocation points along the interface C1 and the top surface SF , the surface response for
45
the zero-scattering condition is depicted by Figure 4.3. As expected, the contribution of the scattered field
is found to be negligible, and the surface displacement tends to the corresponding half-space free-field
response.
−4 −2 0 2 4
0
2
4
0
o
ω = π / 2 (s−1)
−4 −2 0 2 4
0
2
4
30 o
−4 −2 0 2 4
0
2
4
|v|
60 o
−4 −2 0 2 4
0
2
4
85 o
x
1
−4 −2 0 2 4
0
2
4
0
o
ω = π (s−1)
−4 −2 0 2 4
0
2
4
30 o
−4 −2 0 2 4
0
2
4
60 o
|v|
−4 −2 0 2 4
0
2
4
85 o
x
1
Figure 4.3: Full-space Green’s functions. Transparency test for one-layer antiplane strain model subjected
to incident plane harmonic SH waves with different angles of incidence and circular frequencies. The
material of the top layer is the same as that of the half-space, i.e., µ1 = µ0 = 1, β1 = β0 = 1, ρ1 = ρ0 = 1,
ν1 = ν0 = 1/3. N = P = 77, M = L1 = 35 , L2 = 21 , L = L1 + L2 , h = 1, d = 0.2, a = 0.5,
2ws = 2wc = 2, d0 = dr = 0.3
2. Flat-Layer Test. In this test, the maximum deviation of the cosine-shaped scatterer is assumed to be
zero. Using the same initial choice of the sources and collocation points used in the transparency test, the
surface response for this zero-deviation case is shown in Figure 4.4. The surface displacement is found to
approach the values corresponding to the flat-layer model.
46
−4 −2 0 2 4
0
2
4
0
o
Ω = 2
−4 −2 0 2 4
0
2
4
30 o
−4 −2 0 2 4
0
2
4
|v|
60 o
−4 −2 0 2 4
0
2
4
85 o
x
1
−4 −2 0 2 4
0
2
4
0
o
Ω = 4
−4 −2 0 2 4
0
2
4
30 o
−4 −2 0 2 4
0
2
4
60 o
|v|
−4 −2 0 2 4
0
2
4
85 o
x
1
Figure 4.4: Full-space Green’s functions. Flat-layer test for one-layer antiplane strain model subjected to
incident harmonic SH waves with different angles of incidence and circular frequencies. N = P = 77,
M = L1 = 35, L2 = 21, L = L1 + L2, h = 1, a = 0.5, d = 0, 2ws = 2wc = 2, dr = d0 = 0.3
3. Matrix Squeness Test. This test investigates the ratio of the number of rows versus the number of
columns in matrix A (equation (A.10)) which is defined as
rm =
2N + P
L1 + M + L2
(4.24)
In this test, the number of collocation points N and P are assumed to be fixed while the number of
sources is varied according to
N = P = 165; L1 = M = 5m; L2 = 3m; m = 1, 2, ..., 38 (4.25)
All other parameters remain the same. Corresponding error Er is calculated for a range of parameters
(1 ≤ m ≤ 38, equation (4.25)). Figure 4.5 illustrates the results of this test for one frequency. It is found that
47
the parameters 13 ≤ m ≤ 29 result in the minimum error which corresponds to matrix ratios rm = 2.9
to 1.3.
The similar results have been obtained for different frequencies (Ω = 2, 4) , different angles of incidence
(0
o
, 30o
, 60o
and 85o
) and different maximum deviations (d = 0.1, 0.2) of the cosine-shaped irregularity.
Therefore, the matrix ratio of the number of rows versus the number of columns is chosen to be
rm ≃ 2.5 (4.26)
0 10 20 30 40
10−10
10−5
100
Er
0
o
0 10 20 30 40
10−10
10−5
100
Er
30 o
0 10 20 30 40
10−10
10−5
100
m
Er
60 o
0 10 20 30 40
10−10
10−5
100
m
Er
85 o
Figure 4.5: Full-space Green’s functions. Testing for optimum rm (equation (4.24 )). The error Er is shown
as a function of number of series m (equation (4.25)) for the one-layer model subjected to an incident
antiplane harmonic wave with a frequency Ω = 4. The angles of incidence are 0
o
, 30o
, 60o
and 85o
. h = 1,
a = 0.5, d = 0.2, dr = d0 = 0.0732, 2ws = 2wc = 4.
48
4. Locations dr and d0 of the Auxiliary Surfaces Test. This test deals with the estimate of the location
(dr) of the auxiliary surfaces C00 and C11 relative to the interface C1 as well as the location d0 of the
auxiliary surface C12 relative to the top surface SF . To perform this test, the locations dr and d0 of auxiliary
surfaces are varied according to following rule
d0 = dr = 0.02i, i = 1, 2, ..., 48 (4.27)
while all other parameters remain the same. The relative error Er based on the surface response is then
computed for a range of auxiliary surfaces (0.02 ⩽ dr ⩽ 0.96). Figure 4.6 demonstrates the results of
this test for one frequency. The optimum results are found to be 3ds ⩽ dr ⩽ 9ds for frequency Ω = 4
which agrees with the relationship (3.29) proposed by J. E. Luco (private communication) at dr = 0.0732
since the spacing between two adjacent collocation points is ds = 0.0244. It is interesting to find that the
optimum range of the dr reduces following the reduction of the numbers of collocation points and sources.
However, the optimum results of dr always starts from dr = 3ds. Therefore, the Luco’s relationship (3.29)
is accepted throughout this analysis.
49
0 0.2 0.4 0.6 0.8 1
10−10
10−5
100
Er
0
o
0 0.2 0.4 0.6 0.8 1
10−10
10−5
100
Er
30 o
0 0.2 0.4 0.6 0.8 1
10−10
10−5
100
dr
Er
60 o
0 0.2 0.4 0.6 0.8 1
10−10
10−5
100
dr
Er
85 o
Figure 4.6: Full-space Green’s functions. Estimate of auxiliary surface location dr(d0). Error Er is shown
as a function of the spacing dr between the auxiliary surfaces C0, C11 and the interface C1 for the one-layer
model. The dash line represents the optimum location (dr = 3ds) proposed by J. E. Luco. The angles of
incidence are 0
o
, 30o
, 60o
and 85o
. The circular frequency Ω = 4. h = 1, a = 0.5, d = 0.2. N = P = 165,
L1 = M = 75, L2 = 45. 2wc = 2ws = 4. ds = 2wc/(N − 1). d0 = dr.
5. Optimum Spacing ds of the Collocation Points Test. This test deals with the estimate of the
optimum spacing ds between two adjacent collocation points which is defined by
ds =
2wc
N − 1
(4.28)
For this purpose, the collocation points and sources vary according to
N = P = 11m; L1 = M = 5m; L2 = 3m; m = 1, 2, ..., 30 (4.29)
so to maintain the size ratio rm ≃ 2.5 while keeping the other parameters unchanged. The relative error
Er is calculated for different numbers of m. Typical results of this test are depicted in Figure 4.7. For
50
different cases, the results of the same test are summarized in Table 4.1. It can be seen that the higher the
frequency, the smaller the spacing (ds) between two adjacent collocation points.
Based on the tests, the parameters m and ds are chosen to be 6 and 0.0308, respectively for all cases
of Ω = 1 − 4 and θ0 = 0o − 85o
according to Table 4.1. Furthermore, it is reasonable to approximately
extend the frequency range to 0 < Ω ≤ 8 by conservatively choosing m = 15 and ds = 0.0122.
0 10 20 30
10−10
10−8
10−6
10−4
10−2
Er
0
o
0 10 20 30
10−10
10−8
10−6
10−4
10−2
Er
30 o
0 10 20 30
10−10
10−8
10−6
10−4
10−2
m
Er
60 o
0 10 20 30
10−10
10−8
10−6
10−4
10−2
m
Er
85 o
Figure 4.7: Testing of the optimum spacing ds between two adjacent collocation points using the full-space
Green’s functions approach. Error Er is shown as a function of the number of collocation points for the
one-layer model subjected to an incident antiplane harmonic wave with a frequency Ω = 2. The angles
of incidence are 0
o
, 30o
, 60o
and 85o
. h = 1, a = 0.5, d = 0.2, 2ws = 2wc = 2. ds = 2ws/(N − 1).
dr = d0 = 3ds
51
Table 4.1: The optimum spacing ds between two adjacent collocation points based on the minimum error
Er for the one-layer antiplane strain model using the full-space Green’s functions based on the critical
error Er = 10−5
. The angles of incidence are 0
o
, 30o
, 60o
and 85o
. h = 1, a = 0.5, d = 0.2, dr =
d0 = 3ds, 2ws = 2wc = 2
Ω m ds
1 ≥ 4 ≤ 0.0465
2 ≥ 5 ≤ 0.0370
4 ≥ 6 ≤ 0.0308
6. Optimum Length 2wc and 2ws of the Interface C1, the Top Surface SF and the Auxiliary Surfaces (C00, C11 and C12) Test. This test is used to determine the optimum length 2wc of the interface
and the top surface where the collocation points are distributed (see Figure 4.2) together with the optimum length 2ws of the auxiliary surfaces. The numbers of collocation points and sources vary according
to equation (4.29) while the spacing ds between two adjacent collocation points and all other parameters
are fixed. Here a small value ds = 0.0122, which is less than ds = 0.0308 determined by the fifth test, is
chosen for this test.
Figure 4.8 illustrates the relative error Er with respect to the parameter m while 2wc can be expressed
by
2wc = (N − 1)ds, N = 11m, m = 1, 2, ..., 30 (4.30)
52
0 10 20 30
10−7
10−6
10−5
10−4
Er
0
o
0 10 20 30
10−6
10−5
10−4
Er
30 o
0 10 20 30
10−6
10−5
10−4
m
Er
60 o
0 10 20 30
10−6
10−5
10−4
m
Er
85 o
Figure 4.8: Full-space Green’s functions. Testing for optimum 2wc. Error Er is shown as a function of the
length 2wc of the interface C1 for the one-layer model subjected to an incident antiplane harmonic wave
with a frequency Ω = 2. The angles of incidence are 0
o
, 30o
, 60o
and 85o
. h = 1, a = 0.5, d = 0.2.
2ws = 2ws. ds = 0.0122, dr = d0 = 3ds.
For different cases, the results of the same test are summarized in Table 4.2.
Table 4.2: The optimum length 2wc of the interface C1 based on the minimum error Er for the one-layer
antiplane strain model using the full-space Green’s functions. The angles of incidence are 0
o
, 30o
, 60o
and
85o
. h = 1, a = 0.5, d = 0.2. 2ws = 2wc, ds = 0.0122, dr = d0 = 3ds
Ω m 2wc
1 14 ≤ m ≤ 25 1.9 ≤ 2wc ≤ 3.3
2 10 ≤ m ≤ 18 1.3 ≤ 2wc ≤ 2.4
4 10 ≤ m ≤ 25 1.3 ≤ 2wc ≤ 3.3
The accepted optimum length of the interface, auxiliary surfaces, and top surface is 2wc = 2ws =
2(2a).
53
By inspection of Table 4.2, it is reasonable to apply 2wc = 2ws = 2(2a) to incident SH or Love waves
with dimensionless frequency up to Ω = 8.
Therefore, the following conclusions can be made from the above convergence tests for the one-layer
model using full-space Green’s functions :
• Both transparency tests and flat-layer tests are successfully satisfied in the antiplane strain model.
• The optimum ratio of the number of rows versus the number of columns in matrix A (equation
(A.10)) is found to be rm
.= 2.5
• The optimum value of dr and d0 is found to be dr = d0 = 3ds.
• The higher the frequency, the smaller the spacing (ds) between two adjacent collocation points. The
spacing ds = 0.0122 is chosen for the range of frequencies 0 < Ω ≤ 8.
• The optimum length 2wc of the collocations along the interface and the top surface and the optimum
length 2ws of the sources along the auxiliary surfaces are found to be about two times the length of
the irregular interface 2a.
Comparison Tests. Finally, a comparison test is also performed to confirm the validity of the proposed
approach. The response of a one-layer model considered by Aki and Larner [1] is reconsidered here,
referring to the model defined by expression (3.31).
The comparison of the numerical results obtained by the method proposed in this work with those of
Aki and Larner [1] is shown by Figure 4.9. Good agreement between the two approaches is achieved for
incident angles less than or equal to 64o
. A relatively large difference is found between the two approaches
for larger angles of incidence, say, 78o
and 89.9
o
, since the accuracy of the Aki-Larner technique may
decrease as the angle of incidence increases [5]. The results of Figure 4.9 confirm the accuracy of the
present results.
54
This completes the parametric error analysis of the antiplane SH model using full-space Green’s functions. Steady-state results for different frequencies and angles of incidence are presented next.
−2 0 2 4
0
0.5
1
1.5
2
42 o
−2 0 2 4
0
0.5
1
1.5
2
48 o
−2 0 2 4
0
0.5
1
1.5
2
55 o
Normalized surface displacement of |v|
−2 0 2 4
0
0.5
1
1.5
2
64 o
−2 0 2 4
0
1
2
3
78 o
x
1
(1 unit = 25 km)
−2 0 2 4
0
1
2
3
89.9 o
x
1
(1 unit = 25 km)
Figure 4.9: Comparison of normalized surface displacement between indirect boundary integral approach
using the full-space Green’s functions (solid) and Aki-Larner results (circles) [1]. β0 = 4.0 km/ sec,
ρ0 = 3.3 gm/cm3
in the half-space and β1 = 3.0 km/ sec, ρ1 = 2.8 gm/cm3
in the overlayed single
layer. The wavelength of the incident SH wave in the half-space is 10 km which corresponds to a circular
frequency ω = 0.8π s−1
.
4.3.2.2 Steady-State Response
Based on the convergence criteria developed earlier, the parameters used for the steady-state surface displacements for a one-layer antiplane strain model are
a = 0.5, h = 1, 2wc = 2ws = 2,
ds = 0.0122, dr = d0 = 3ds;
N = P = 165, L1 = M = 75, L2 = 45.
The steady-state displacements for the one-layer model using the full-space Green’s functions are
depicted by Figure 4.10 and Figure 4.11 for SH incident waves. Corresponding steady-state displacements
55
for the same model using the half-space Green’s functions applied in Chapter 3 are also incorporated for
comparison. It is shown that a good agreement is achieved between the two approaches.
Here, the parameters for the one-layer model using the half-space Green’s functions are identical to
those for the same model using the full-space Green’s functions except P = L2 = 0 to account for highfrequency SH incidences. The parameters provided in Figure 3.10 are valid for the one-layer model using
the half-space Green’s functions only when Ω ≤ 5.
To further investigate the effect of the deviation d of the irregularity on the surface response, the surface
displacements are also calculated for various d values for three frequencies, Ω = 1, 2, 4 (Figure 4.12 to
Figure 4.14). In addition, the surface response for a vertical incidence at one location (x1 = 0) is evaluated
to study the resonance for a range of frequencies as depicted by Figure 4.19.
The surface response is also calculated for the Love wave by reasonably applying the convergence
parameters developed for the one-layer model with SH incidence. Figure 4.15 through Figure 4.18 displays
the Love wave surface displacement for various deviation d of the scatterer for four frequencies, Ω =
0.96, 2.96, 3.44, 5.60. In addition, the resonance feature is also studied for the Love wave for the one-layer
model. Figure 4.20 depicts the Love wave surface displacement at location x1 = 0 for the one-layer model
with a scatterer deviation of d = 0.1 for a range of frequencies 0.08 ≤ Ω ≤ 5.92. The only resonance
frequency is Ω = 1.13, while the corresponding flat-layer resonance frequency is Ω = 1.17.
56
Figure 4.10: SH wave surface displacements for the one-layer antiplane strain model. d = 0.2; Ω = 1, 2, 4.
Solid line: using full-space Green’s functions; Dash lines: using half-space Green’s functions.
Figure 4.11: SH wave surface displacements for the one-layer antiplane strain model. d = 0.2; Ω = 3, 5, 7.
Solid line: using full-space Green’s functions; Dash lines: using half-space Green’s functions.
57
Figure 4.12: SH wave surface response for the one-layer model with various deviations d of the irregular
interface. Ω = 1, θ0 = 0o
Figure 4.13: SH wave surface response for the one-layer model with various deviations d of the irregular
interface. Ω = 2, θ0 = 0o
58
Figure 4.14: SH wave surface response for the one-layer model with various deviations d of the irregular
interface. Ω = 4, θ0 = 0o
Figure 4.15: Love wave surface response for the one-layer model with various deviations d of the irregular
interface. Ω = 0.96
59
Figure 4.16: Love wave surface response for the one-layer model with various deviations d of the irregular
interface. Ω = 2.96
Figure 4.17: Love wave surface response for the one-layer model with various deviations d of the irregular
interface. Ω = 3.44
60
Figure 4.18: Love wave surface response for the one-layer model with various deviations d of the irregular
interface. Ω = 5.60
Figure 4.19: SH wave surface displacement at (x1, x3) = (0, 0) versus frequency for the one-layer model.
θ0 = 0o
, d = 0.1
61
Figure 4.20: Love wave surface displacement at (x1, x3) = (0, 0) versus frequency for the one-layer model.
d = 0.1
These results can be summarized as follows:
• A small deviation along the flat-layer interface may cause large surface motion amplification.
• The surface displacement strongly depends upon the frequency and angle of the incidence for the
incident waves, the site location and the shape of the interface such as the amplitude of irregularity
deviation.
• For large values of |x1|, the surface displacement approaches the response of a flat-layer model. This
is expected since away from the irregularity, the scattered wave field should decrease in amplitude
and the total wave field approaches the free-field one.
• The surface displacement at x1 = ±4 is very small relative to the maximum amplitude around x1 =
0, thus an observation range of x1 = [−4, 4], which is eight times of the length of the irregularity,
is sufficient to detect any appreciable scattered wave field.
62
• For observation sites directly above the irregular interface, the surface response exhibits resonance
features [104]. The resonance frequencies are closely related to the ones for a flat-layer model. For a
vertical SH incidence, the first four frequencies at the site (x1, x3) = (0, 0) with a deviation d = 0.1
of the irregular interface are found to be Ωi = 0.998, 2.956, 4.878, and 6.760. The corresponding
flat-layer model’s resonance frequencies are Ωi = 1, 3, 5, and 7. For an incident Love wave, the sole
resonance frequency at the site (x1, x3) = (0, 0) with a scatterer deviation of d = 0.1 is found to be
Ω = 1.13 while the resonance frequency for the corresponding flat-layer model is Ω = 1.17. The
difference between the models with and without scatterer is attributed to the presence of the irregularity. However, for SH incidence, the amplification of surface motion of the resonance frequencies
becomes significantly different for the two models at higher resonance frequencies.
• For vertically incident SH waves, the surface motion is found to be symmetric for a plane-symmetric
geometry with respect to x2x3−plane thus lending further confidence about the validity of the calculated results.
This concludes the numerical work for the one-layer antiplane strain model. The two-layer model is
considered next.
4.3.3 Two-Layer Model
The two-layer model depicted by Figure 4.21 has its two interfaces defined by
x3 =
h1 for |x1| > a
h1 +
d1
2
(1 + cos πx1
a
) for |x1| ≤ a
; r ∈ C1 (4.31)
and
63
x3 =
h2 for |x1| > a
h2 +
d2
2
(1 + cos πx1
a
) for |x1| ≤ a
; r ∈ C2 (4.32)
where hj , dj and a are the corresponding flat-layer depth, the maximum deviation and the half-length of
the cosine-shaped scatterer of the interface Cj , respectively.
Figure 4.21: Geometry of the two-layer antiplane strain model with two cosine-shaped interfaces and the
corresponding distribution of sources and collocation points using the full-space Green’s functions. Here,
h1 = 1, h2 = 0.5; 2w
(1)
i = 2w
(2)
i = 1. Star: position of a line source; circle: position of a collocation point.
The following assumptions are made throughout the study of the two-layer model:
64
h1 = 1, h2 = 0.5, 2w
(1)
i = 2w
(2)
i = 2a = 1
d
(1)
s = d
(2)
s = ds (4.33)
d
(00)
r = d
(11)
r = d
(12)
r = d
(21)
r = dr
The material properties in the three domains are defined by
D0 : µ0 = 1, β0 = 1, ρ0 = 1, ν0 = 1/3
D1 : µ1 = 0.125, β1 = 0.5, ρ1 = 0.5, ν1 = 1/3 (4.34)
D2 : µ2 = 0.015625, β2 = 0.25, ρ2 = 0.25, ν2 = 1/3
4.3.3.1 Parametric Error Analysis
As in the one-layer model, six convergence tests and one comparison test are performed to determine the
optimum parameters for the problem. The relationships (4.22) - (4.23) are understood in this parametric
error analysis.
Parametric Error Analysis is performed for the two-layer model of incident SH wave only.
1. Transparency Test. In this test, the material properties of the two layers D1 and D2 are assumed to
be the same as those of the half-space D0 while the maximum deviation dj of the cosine-shaped scatterers
is not zero. This model should produce a zero-scattering. Figure 4.22 shows the surface response for the
zero-scattering condition for an initial choice of the sources along the auxiliary surfaces C00, Cj1 and
Cj2, and the collocation points along the interface C1, C2 and the top free surface SF . As expected, the
65
contribution of the scattered field is found to be negligible, and the surface displacement tends to the
half-space free field response.
−5 0 5
1
2
3
ω = π/2 (s −1)
0 o
−5 0 5
1
2
3
30 o
−5 0 5
1
2
3
60 o
|v|
−5 0 5
1
2
3
85 o
x
1
−5 0 5
1
2
3
ω = π (s −1)
0 o
−5 0 5
1
2
3
30 o
−5 0 5
1
2
3
60 o
|v|
−5 0 5
1
2
3
85 o
x
1
Figure 4.22: Transparency test for the two-layer antiplane strain model and incident plane harmonic SH
waves with different angles of incidence and frequencies using the full-space Green’s functions. Material
of the three domains are assumed to be the same, i.e., µ2 = µ1 = µ0 = 1, β2 = β1 = β0 = 1, ρ2 = ρ1 =
ρ0 = 1, ν2 = ν1 = ν0 = 1/3. N1 = N2 = P = 121, M = L11 = L21 = L12 = 55 , L22 = 22 , a = 0.5,
h1 = 1, h2 = 0.5, d1 = d2 = 0.2, 2ws = 2wc = 6 , dr = d0 = 0.1
2. Flat-Layer Test. In this test, the maximum deviation dj of the two cosine-shaped scatterers is assumed
to be zero. Using the same initial choice of the sources and collocation points used in transparency test,
the surface response for this zero-deviation case is shown in Figure 4.23. In this analysis, the material
properties are defined in equation (4.34). Apparently, the surface displacement is found to approach that
of the corresponding flat-layer model.
66
−5 0 5
7
8
9
Ω = 2
0 o
−5 0 5
6
7
8
30 o
−5 0 5
4
5
6
60 o
|v|
−5 0 5
0
1
2
85 o
x
1
−5 0 5
1
2
3
Ω = 4
0 o
−5 0 5
1
2
3
30 o
−5 0 5
1
2
3
60 o
|v|
−5 0 5
1
2
3
85 o
x
1
Figure 4.23: Flat-layer test for the two-layer antiplane strain model and incident harmonic SH waves with
different angles of incidence and frequencies using the full-space Green’s functions. N1 = N2 = P = 121,
M = L11 = L21 = L12 = 55 , L22 = 22 , a = 0.5, h1 = 1, h2 = 0.5, d1 = d2 = 0, 2ws = 2wc = 6 ,
dr = d0 = 0.1
3. Matrix Squeness Test. This test investigates the ratio of the number of rows versus the number of
columns in matrix A (equation (A.10)) which is defined as
rm =
2(N1 + N2) + P
(L11 + L12) + M + (L21 + L22)
(4.35)
In this test, the material properties are defined by equations (4.34) and the number of collocation points
is assumed to be fixed while the number of sources is varied according to
N1 = N2 = P = 231; L11 = L12 = L21 = M = 5m; L22 = 3m; m = 1, 2, ..., 48 (4.36)
67
while all other parameters remain the same. Corresponding error Er is calculated for a range of 1 ≤ m ≤
48. Figure 4.24 illustrates the results of this test for one frequency. It is found that parameters 19 ≤ m ≤ 35
result in minimum error which correspond to rm = 2.6 to 1.4.
The similar results have been obtained for different frequencies (Ω = 1 − 4) and different angles
of incidence (θ0 = 0o − 85o
). Therefore, rm ≃ 2.5 has been accepted throughout further study of the
two-layer problem.
0 10 20 30 40 50
10−8
10−6
10−4
10−2
100
Er
0
o
0 10 20 30 40 50
10−8
10−6
10−4
10−2
100
Er
30 o
0 10 20 30 40 50
10−8
10−6
10−4
10−2
100
m
Er
60 o
0 10 20 30 40 50
10−8
10−6
10−4
10−2
100
m
Er
85 o
Figure 4.24: Testing for the optimum matrix squeness rm. Error Er is shown as a function of m for the twolayer antiplane strain model subjected to an incident antiplane harmonic wave with a frequency Ω = 4
using the full-space Green’s functions. The angles of incidence are 0
o
, 30o
, 60o
and 85o
. N1 = N2 =
P = 231, M = L11 = L21 = L12 = 5m, L22 = 3m, m = 1, 2, ..., 48; a = 0.5, h1 = 1, h2 = 0.5,
d1 = d2 = 0.2, 2ws = 2wc = 4, dr = d0 = 3ds
4. Locations dr and d0 of the Auxiliary Surfaces C00, C11, C12, C21 and C22 Test. This test deals
with the estimate of the location dr of auxiliary surfaces C00, C11, C12, C21 relative to the interfaces C1,
68
C1, C2, C2, and the location d0 of auxiliary surface C22 relative to the top surface SF , respectively. To
perform this test, the location dr of auxiliary surfaces is varied according to the following rule
d0 = dr = 0.02m, m = 1, 2, ..., 48 (4.37)
while all other parameters remain the same. The relative error Er based on the surface response is then
computed for a range of auxiliary surfaces (0.02 ⩽ dr ⩽ 0.96).
Figure 4.25 demonstrates the results of this test for one frequency. The optimum results are found
to be 3ds ⩽ dr ⩽ 7ds for a frequency Ω = 4 which agrees with the relationship (3.29) proposed by J.
E. Luco (private communication) at dr = 0.1 since the spacing between two adjacent collocation points
ds = 0.0333. Like in the one-layer model, it is interesting to find that the optimum range of the dr reduces
following the reduction of the numbers of collocation points and sources. However, the optimum results
of dr always starts from dr = 3ds. Therefore, the Luco’s relationship (3.29) is accepted throughout this
two-layer analysis.
69
0 0.2 0.4 0.6 0.8 1
10−4
10−2
100
Er
0
o
0 0.2 0.4 0.6 0.8 1
10−4
10−2
100
Er
30 o
0 0.2 0.4 0.6 0.8 1
10−4
10−2
100
d
r
Er
60 o
0 0.2 0.4 0.6 0.8 1
10−4
10−2
100
d
r
Er
85 o
Figure 4.25: Estimate of auxiliary surface location dr. Error Er is shown as a function of the spacing dr
between the auxiliary surfaces C0, C11, C12, C21 and the interface C1, C1, C2, C2, respectively for the
two-layer antiplane strain model using the full-space Green’s function approach. The angles of incidence
are 0
o
, 30o
, 60o
and 85o
. The dimensionless frequency Ω = 4. N1 = N2 = P = 121, M = L11 = L21 =
L12 = 55, L22 = 33; a = 0.5, h1 = 1, h2 = 0.5, d1 = d2 = 0.2, 2ws = 2wc = 4 , d0 = dr. The dash line
represents the optimum location of auxiliary surfaces d0 = dr = 3ds proposed by J. E. Luco
5. Optimum Spacing ds of the Collocation Points Test. This test deals with the estimate of the
optimum spacing ds between two adjacent collocation points which is defined by
ds =
2wc
N1 − 1
(4.38)
For this purpose, the collocation points and sources vary according to
N1 = N2 = P = 11m; L11 = L12 = L21 = M = 5m; L22 = 3m; m = 1, 2, ..., 25 (4.39)
70
to maintain a size ratio of rm ≃ 2.5 while keeping the other parameters unchanged. The relative error Er
is calculated for different values of m. Typical results of this test are depicted in Figure 4.26. For different
cases, the results of the same test are summarized in Table 4.3. It can be seen that the higher the frequency,
the smaller the spacing (ds) between two adjacent collocation points.
Based on this test, the value m = 7, i.e., ds = 0.0263 is acceptable for all cases of Ω = 1 − 4 and
θ0 = 0o − 85o
.
For conservatism, it is reasonable to choose m = 21 and ds = 0.0087 throughout this two-layer
antiplane strain model for a wider frequency range of 0 < Ω ≤ 8.
0 5 10 15 20 25
10−10
10−5
100
Er
0
o
0 5 10 15 20 25
10−10
10−5
100
Er
30 o
0 5 10 15 20 25
10−8
10−6
10−4
10−2
100
m
Er
60 o
0 5 10 15 20 25
10−8
10−6
10−4
10−2
100
m
Er
85 o
Figure 4.26: Testing of the optimum spacing of collocation points ds. Error Er is shown as a function of m
for the two-layer antiplane strain model subjected to an incident antiplane harmonic wave using the fullspace Green’s functions. The angles of incidence are 0
o
, 30o
, 60o
and 85o
. Ω = 2. N1 = N2 = P = 11m,
M = L11 = L21 = L12 = 5m, L22 = 3m , m = 1, 2, ..., 25; a = 0.5, h1 = 1, h2 = 0.5, d1 = d2 = 0.2,
2ws = 2wc = 2 , dr = d0 = 3ds
71
Table 4.3: The optimum spacing ds between two adjacent collocation points based on the minimum error Er for the two-layer antiplane strain model using the full-space Green’s functions. The critical Er is
assumed to be 0.0001. The angles of incidence are 0
o
, 30o
, 60o
and 85o
. a = 0.5, h1 = 1, h2 = 0.5,
d1 = d2 = 0.2, dr = d0 = 3ds, 2ws = 2wc = 2
Ω m ds
1 ≥ 4 ≤ 0.0465
2 ≥ 5 ≤ 0.0370
4 ≥ 7 ≤ 0.0263
6. Optimum Length 2wc of the Interfaces/Top Surface (C1, C2 and SF ) and the length 2ws of
the auxiliary surfaces Test. The six test is employed to determine the optimum length 2wc of the
interfaces C1 and C2 and the top surface SF where collocation points are distributed as well as the optimum
length 2ws of the auxiliary surfaces. The numbers of collocation points and sources vary also according
to equation (4.39) while the spacing ds between two adjacent collocation points and all other parameters
are fixed. Here a small value ds = 0.0125, which is less than ds = 0.0263 determined by the fifth test, is
chosen for this test.
Figure 4.27 illustrates the relative error Er with respect to the parameter m while 2wc and 2ws can be
expressed by
2wc = 2ws = (N1 − 1)ds, N1 = 11m, m = 1, 2, ..., 25 (4.40)
72
0 5 10 15 20 25
10−6
10−5
10−4
10−3
Er
0
o
0 5 10 15 20 25
10−6
10−5
10−4
10−3
Er
30 o
0 5 10 15 20 25
10−5
10−4
10−3
m
Er
60 o
0 5 10 15 20 25
10−5
10−4
10−3
m
Er
85 o
Figure 4.27: Testing of the optimum length 2wc using full-space Green’s functions - Error Er as a function
of 2wc for the two-layer antiplane strain model subjected to an incident antiplane harmonic wave with a
dimensionless frequency Ω = 2. The angles of incidence are 0
o
, 30o
, 60o
and 85o
. a = 0.5, h1 = 1, h2 =
0.5, d1 = d2 = 0.2. ds = 0.0125; dr = d0 = 3ds, N1 = N2 = P = 11m, M = L11 = L21 = L12 = 5m,
L22 = 3m , m = 1, 2, ..., 25, 2wc = 2ws = (N1 − 1)ds.
For different cases, the results of the same test are summarized in Table 4.4.
Table 4.4: The optimum length 2wc of the interfaces and the top surface based on the minimum error Er
for the two-layer antiplane strain model using the full-space Green’s functions. The angles of incidence
are 0
o
, 30o
, 60o
and 85o
. a = 0.5, h1 = 1, h2 = 0.5, d1 = d2 = 0.2. ds = 0.0125, dr = d0 = 3ds, N1 =
N2 = P = 11m, M = L11 = L21 = L12 = 5m, L22 = 3m , m = 1, 2, ..., 25; 2wc = 2ws = (N1 − 1)ds
Ω m 2ws = 2wc
1 14 ≤ m ≤ 20 1.9 ≤ 2wc ≤ 2.7
2 14 ≤ m ≤ 16 1.9 ≤ 2wc ≤ 2.2
4 m ≥ 10 2wc ≥ 1.4
73
The accepted value based on this test is m = 15 or 2wc = 2ws = 2 for all combined cases of Ω = 1−4
and θ0 = 0o − 85o
.
By inspection of Table 4.4, it is reasonable to apply 2wc = 2ws = 2 to incident SH or Love waves with
dimensionless frequency up to Ω = 8.
Based on the above parametric tests, the steady-state surface responses for different angles of incidence
and different frequencies are presented next.
4.3.3.2 Steady-State Response
Based on the above convergence tests, the parameters used for the steady-state surface displacements for
a two-layer antiplane strain model are
a = 0.5, h1 = 1, h2 = 0.5, 2wc = 2ws = 2;
ds = 0.0087, dr = d0 = 3ds;
N1 = N2 = P = 231, M = L11 = L21 = L12 = 105, L22 = 63.
Figure 4.28 and Figure 4.29 show the surface displacements for different frequencies and angles of
incidence for SH incident waves. Surface displacements for the same model using the half-space Green’s
functions in Chapter 3 are also displayed in this figure to confirm the validity of this approach. It can be
seen that a good agreement is achieved between the two approaches.
Here, the parameters for the two-layer model using the half-space Green’s functions are identical to
those for the same model using the full-space Green’s functions except P = L22 = 0 to account for highfrequency SH incidences. The parameters provided in Figure 3.13 are not applicable to the two-layer model
using the half-space Green’s functions when Ω > 4.
74
Figure 4.28: SH wave surface displacements for the two-layer antiplane strain model. d1 = d2 = 0.2;
Ω = 1, 2, 4. Solid line: using full-space Green’s functions; Dash line: using half-space Green’s functions.
Figure 4.29: SH wave surface displacements for the two-layer antiplane strain model. d1 = d2 = 0.2;
Ω = 3, 5, 7. Solid line: using full-space Green’s functions; Dash line: using half-space Green’s functions.
75
To investigate the effect of the scatterer deviations (d1 and d2) of the irregular interfaces C1 and C2 on
the surface response, the surface displacements for vertical SH incidences are calculated for various d1 and
d2 values for three frequencies (Ω = 1, 2, 4) are displayed in Figure 4.30 to Figure 4.32. As in the one-layer
model, the surface response approaches the value of the free field away from the irregularity. It is also
found that the response is symmetric about the x3-axis for a vertical SH incidence. A small change of the
deviation of the irregular interfaces may cause a significant change for the surface response pattern. The
total wave field is found to be very sensitive to the incident SH wave (angle of incidence and frequency),
the site location, the geometry (thickness of each layer and the shape of each interface) and the material
properties of the layered medium.
In addition, the resonance at the site (x1, x3) = (0, 0) is also studied and the surface response at this
location is given in Figure 4.33 for a vertical incidence. The first three resonance frequencies at this site are
determined to be Ω1 = 0.85, Ω2 = 2.02 and Ω3 = 3.00. The resonance frequencies for the corresponding
flat-layer model are Ω1 = 0.88, Ω2 = 2 and Ω3 = 3.12. While the resonance frequencies for the irregularand flat-layer models are similar, the corresponding amplification at higher modes is not.
76
-4 -2 0 2 4
5
10
15
20
25
30
|v|
Case 1: d1
=0.1; d2
varies
-0.2
-0.1
0.0
0.1
0.2
-4 -2 0 2 4
5
10
15
20
25
30
|v|
Case 2: d2
=0.1; d1
varies
-0.2
-0.1
0.0
0.1
0.2
-4 -2 0 2 4
x
1
5
10
15
20
25
30
|v|
Case 3: d2
=d1
; d1
varies
-0.2
-0.1
0.0
0.1
0.2
-4 -2 0 2 4
x
1
5
10
15
20
25
30
|v|
Case 4: d2
=-d1
; d1
varies
-0.2
-0.1
0.0
0.1
0.2
Figure 4.30: Surface displacement for the two-layer model for a vertical SH incidence (θ0 = 0o
) with
different combinations of d1 and d2 values. Ω = 1.
-4 -2 0 2 4
0
5
10
15
20
25
|v|
Case 1: d1
=0.1; d2
varies
-0.2
-0.1
0.0
0.1
0.2
-4 -2 0 2 4
0
5
10
15
20
25
|v|
Case 2: d2
=0.1; d1
varies
-0.2
-0.1
0.0
0.1
0.2
-4 -2 0 2 4
x
1
0
5
10
15
20
25
|v|
Case 3: d2
=d1
; d1
varies
-0.2
-0.1
0.0
0.1
0.2
-4 -2 0 2 4
x
1
0
5
10
15
20
25
|v|
Case 4: d2
=-d1
; d1
varies
-0.2
-0.1
0.0
0.1
0.2
Figure 4.31: Surface displacement for the two-layer model for a vertical SH incidence (θ0 = 0o
) with
different combinations of d1 and d2 values. Ω = 2.
77
-4 -2 0 2 4
0
5
10
15
|v|
Case 1: d1
=0.1; d2
varies
-0.2
-0.1
0.0
0.1
0.2
-4 -2 0 2 4
0
2
4
6
8
|v|
Case 2: d2
=0.1; d1
varies
-0.2
-0.1
0.0
0.1
0.2
-4 -2 0 2 4
x
1
0
5
10
15
20
|v|
Case 3: d2
=d1
; d1
varies
-0.2
-0.1
0.0
0.1
0.2
-4 -2 0 2 4
x
1
0
5
10
|v|
Case 4: d2
=-d1
; d1
varies
-0.2
-0.1
0.0
0.1
0.2
Figure 4.32: Surface displacement for the two-layer model for a vertical SH incidence (θ0 = 0o
) with
different combinations of d1 and d2 values. Ω = 4.
0 0.5 1 1.5 2 2.5 3 3.5 4
0
5
10
15
20
25
30
35
Ω
|v|
With Scatterer
Flat−layer
Figure 4.33: Surface displacement at (x1, x3) = (0, 0) for the two-layer model for a vertical SH incidence
(θ0 = 0o
) as a function of dimensionless frequency Ω. d1 = d2 = 0.1
78
Figure 4.34 through Figure 4.36 display the Love wave surface displacement for the two-layer model for
dimensionless frequencies Ω = 1.008, 2, 4, respectively. Similar to the surface displacement for incident
SH waves, the amplitude of the Love wave surface displacement approaches that of the flat-layer model far
away from the scatterer. It is also found that the pattern of Love wave surface displacement is dominated
by the upper-interface scatterer (d2), see Case 2 in each of these figures. Variation of the lower-interface
scatterer (d1) has less influence on the surface response. This is due to the following two factors:
• Effect of Saint-Venant’s Principle.
• Rapid decay of the Love wave amplitude as x3 increases with depth, see Table 4.5, indicates a weak
influence of scattering from the lower irregular interface(s) to the top surface.
This interesting effect is confirmed by more tests with different fixed upper-interface scatterers (d2),
see Figure 4.37.
Figure 4.34: Love wave surface displacement for the two-layer model with different combinations of d1
and d2 values. Ω = 1.008.
79
Figure 4.35: Love wave surface displacement for the two-layer model with different combinations of d1
and d2 values. Ω = 2.
Figure 4.36: Love wave surface displacement for the two-layer model with different combinations of d1
and d2 values. Ω = 4.
80
Figure 4.37: Further Love wave surface displacement test for the two-layer model with various d1 values
for different fixed d2 values. Ω = 2.
Table 4.5: Reduction of Love wave amplitude (Alove) at x3 → ∞, 2-layer free field
ω(rad/sec) Ω Alove@x3 = 0 Alove@x3 = 0.5 Alove@x3 = 1
π 4 0.1945 0.0071 0.0000077
0.5π 2 3.0902 0.2479 0.0050
0.252π 1.008 8.2895 1.6454 0.2535
81
To further validate the accuracy of the two-layer antiplane strain formulation for the incident Love
wave, a transparency test is conducted by letting the material properties of the lower two domains, D1
and D0, be identical, i.e.,
D0 : µ0 = 1, β0 = 1, ρ0 = 1, ν0 = 1/3
D1 : µ1 = 1, β1 = 1, ρ1 = 1, ν1 = 1/3 (4.41)
D2 : µ2 = 0.125, β2 = 0.5, ρ2 = 0.5, ν2 = 1/3
The geometry and the parameters are then defined as follows to simulate the one-layer model shown
by Figure 4.2,
a = 0.5, h1 = 1.5, h2 = 1, 2wc = 2ws = 2;
ds = 0.0087, dr = d0 = 3ds;
N1 = N2 = P = 231, M = L11 = L21 = L12 = 105, L22 = 63.
Figure 4.38 through Figure 4.41 display the Love wave surface displacements for the transparency test
for Ω = 0.96, 2.96, 3.44, and 5.60, respectively.
By comparing the Love wave surface displacements for Cases 1 / 3 from Figure 4.38 through Figure 4.41
with the one-layer Love wave surface displacements displayed by Figure 4.15 through Figure 4.18, it can
be seen that a good agreement is achieved between the one-layer model and the transparency two-layer
model.
This transparency test further confirms the accuracy of the two-dimensional one- and two-layer Love
wave formulations.
82
Figure 4.38: Transparency test. Love wave surface displacement for the two-layer model with different
combinations of d1 and d2 values. Ω = 0.96.
Figure 4.39: Transparency test. Love wave surface displacement for the two-layer model with different
combinations of d1 and d2 values. Ω = 2.96.
83
Figure 4.40: Transparency test. Love wave surface displacement for the two-layer model with different
combinations of d1 and d2 values. Ω = 3.44.
Figure 4.41: Transparency test. Love wave surface displacement for the two-layer model with different
combinations of d1 and d2 values. Ω = 5.60.
84
The Love wave resonance at the site (x1, x3) = (0, 0) for the two-dimensional 2-layer model is also
studied and the surface response at this location for a model with scatterer amplitude d1 = 0.1 and d2 =
−0.1 is given in Figure 4.42. The only resonance frequency in the range 1.00776 ≤ Ω ≤ 8 at this site is
Ω = 1.07. The resonance frequency for the corresponding flat-layer model is Ω = 1.018.
Figure 4.42: Love wave surface displacement at (x1, x3) = (0, 0) versus dimensionless frequency Ω for the
two-layer model. d1 = 0.1, d2 = −0.1
This concludes the steady-state analysis of the two-layer antiplane strain model.
4.4 Conclusions
In this Chapter, an indirect boundary integral equation method using the full-space Green’s functions was
applied to investigate the scattering of elastic SH and Love waves in one- and two-layer models with irregular interfaces. Detailed parametric error analysis was performed for all parameters for the one- and
two-layer SH models to assess the convergence of the proposed method. Results from this SH approach for
85
a one-layer structure were compared with those of Aki and Larner [1] and a good agreement was achieved
between the two approaches. In addition, a comparison was also made between the results obtained using the half-space Green’s functions and those using the full-space Green’s functions. Good agreement
was found for both the one- and the two-layer SH models. Subsequently, the numerical results for the
steady-state response were given for the one- and two-layer antiplane strain models for the SH and Love
incidences, respectively.
The results demonstrate that the steady-state response strongly depends upon the nature of the incident waves (frequency, and angle of incidence for SH incidence), the site of observation, the geometry and
material properties of the layered media. Plane-symmetric models for SH incidence were chosen to verify
the formulation for scattering waves in the multilayered media in addition to the regular transparency test
and flat-layer test. The surface motion amplitude is found to be symmetric for vertically incident SH waves
thus further confirming the validity of the model. A small change of the deviation value of the irregular
interfaces may cause a significant change in the surface response pattern. Resonance frequencies were
determined for both the one- and two-layer models. It was found that fundamental resonance frequencies
for the irregular- and flat-layer models are very close. However, for incident SH waves, as the order of the
resonance frequency increases, the difference between the resonance frequencies for irregular- and flatlayer models becomes larger. Similar observation can be made about the amplification of surface motion
at the resonance frequencies for irregular- and flat-layer models, i.e., with more pronounced differences
at the higher resonance frequencies. Finally, it was shown that away from the irregularity, the scattered
wave field decreases in amplitude and the total wave field approaches the values of the corresponding free
field.
In particular, for Love waves in the two-layer model, the steady-state response on the top surface is
found to be dominated by the scatterer of the upper interface. This is due to the effect of Saint-Venant’s
86
Principle, and the rapid decrease of the Love wave amplitude with increase of x3 which may result in a
weak influence of the scattering from the lower-interface irregularity to the top surface.
87
Chapter 5
Plane Strain Model
5.1 Statement of Problem
The geometry of the problem is depicted by Figure 3.1 with the irregular interface varying only in the
x1x3−plane. The geometry parameters of the model such as the domains Dj , the interfaces Ci and the
corresponding reference surfaces Si
, the top surface SF and the corresponding flat-layer depth hi of each
layer Dj are defined in Chapter 3. The system is subjected to an incident plane harmonic P, SV or Rayleigh
wave.
For plane strain model, the motion is assumed to take place in x1x3−plane with only two non-zero
components, i.e., uj = (uj , 0, wj ), which can be expressed by the displacement potentials [2]
uj = ∇ϕj + ∇ × (0, ψj , 0), j = 0, 1, 2, ... , J (5.1)
where ϕj and ψj are P and SV wave potentials, and uj , wj denote the displacement components along
the x1 and the x3 axes of the j
th domain, respectively.
For steady-state solutions, the displacement potentials satisfy the following equations of motion [28]:
(∇2 + ζ
2
j
)ϕj (r, ω) = 0; r ∈ Dj (5.2)
88
(∇2 + η
2
j
)ψj (r, ω) = 0; r ∈ Dj (5.3)
∇2 =
∂
2
∂x2
1
+
∂
2
∂x2
2
+
∂
2
∂x2
3
, ζj =
ω
αj
; ηj =
ω
βj
, j = 0, 1, 2, ... , J (5.4)
where ζj and ηj are the wavenumbers associated with P and SV waves, respectively, and αj and βj denote
the P and SV wave velocities in the j
th domain, respectively.
The traction-free boundary conditions (2.3) along the top free surface SF are reduced to
σ33,J (r, ω) = 0; r ∈ SF (5.5)
σ13,J (r, ω) = 0; r ∈ SF (5.6)
The perfect bonding along the interfaces are expressed by
uj−1(r, ω) = uj (r, ω); r ∈ Cj , j = 1, 2, ... , J (5.7)
wj−1(r, ω) = wj (r, ω); r ∈ Cj , j = 1, 2, ... , J (5.8)
σnn,j−1(r, ω) = σnn,j (r, ω); r ∈ Cj , j = 1, 2, ... , J (5.9)
σnt,j−1(r, ω) = σnt,j (r, ω); r ∈ Cj , j = 1, 2, ... , J (5.10)
where σnn and σnt represent the normal and shear stresses along the interface Cj .
The incident P or SV waves from the bottom of the half-space D0 are assumed to be of the form
P wave: ϕ
inc = A0 exp(iωt − iζ0(x1 sin θ0 − x3 cos θ0)); r ∈ D0 (5.11)
89
SV wave: ψ
inc = B0 exp(iωt − iη0(x1 sin γ0 − x3 cos γ0)); r ∈ D0 (5.12)
where θ0 and γ0 are the angles of incidence for incident P and SV waves, respectively.
The form of the incident Rayleigh wave in the half-space is defined by [40]
ϕ
inc = A0 exp(i(ωt − κx1) − κ
r
1 − (
c
α0
)
2x3) (5.13)
ψ
inc = B0 exp(i(ωt − κx1) − κ
r
1 − (
c
β0
)
2x3) (5.14)
where c is the Rayleigh wave velocity and κ (=
ω
c
) represents the Rayleigh wavenumber.
Throughout, the amplitude of incident displacement potentials are chosen to be
P wave: A0 =
1
ω
, B0 = 0 (5.15)
SV wave: A0 = 0, B0 =
1
ω
(5.16)
Rayleigh wave: A0 ̸= 0, B0 = 1 (5.17)
where i =
√
−1, and h1 is the depth of the first flat reference interface (Figure 2.1). Therefore, the displacement amplitudes of the incident plane P and SV wave are 1/α0 and 1/β0, respectively. The factor
e
iωt will be omitted.
This completes the statement of the plane strain problem. Solution of the problem is considered next.
90
5.2 Steady-State Solution of the Problem
The total wave field in each domain can be expressed as a superposition of the free field and the scattered
field according to
uj = u
f f
j + u
s
j
; r ∈ Dj , j = 0, 1, 2, ... , J (5.18)
where the superscripts ff and s denote the free field of the corresponding flat-layer system and scattered
wave field caused by irregular interfaces, respectively.
5.2.1 Free Field
The displacement potentials for the free field for plane P or SV wave can be expressed by
ϕ
f f
j = Aj exp(−iζj (x1 sin θj − x3 cos θj )) + Ej exp(−iζj (x1 sin θj + x3 cos θj )) (5.19)
ψ
f f
j = Bj exp(−iηj (x1 sin γj − x3 cos γj )) + Fj exp(−iηj (x1 sin γj + x3 cos γj )) (5.20)
r ∈ Dj , j = 0, 1, 2, ... , J
while displacement potentials for the free field for Rayleigh incidence are given by
ϕ
f f
j = Aj exp(−iκx1 − κ
r
1 − (
c
αj
)
2x3) + Ej exp(−iκx1 + κ
r
1 − (
c
αj
)
2x3) (5.21)
ψ
f f
j = Bj exp(−iκx1 − κ
r
1 − (
c
βj
)
2x3) + Fj exp(−iκx1 + κ
r
1 − (
c
βj
)
2x3) (5.22)
91
r ∈ Dj , j = 0, 1, 2, ... , J
The parameters Aj , Bj , Ej , Fj and θj , γj are known [2] [94] [40]. For example, for a P wave incidence,
A0 and θ0 are given and B0 = 0 while E0, F0, A1, B1, E1, F1 and θ1 can be computed. For Rayleigh
incidence, E0 = F0 = 0 since the Rayleigh wave vanishes at x3 → ∞ in the bottom half-space D0.
Detailed solution for these parameters is presented in Appendix B for the one-layer model.
Thus the displacements u
f f
j = (u
f f
j
, 0, w
f f
j
) and stresses s
f f
j = (σ
f f
11,j , σ
f f
33,j , σ
f f
13,j ) corresponding to
the free field (ϕ
f f
j
, ψ
f f
j
) are also known [2] [65].
5.2.2 Scattered Wave Field
It is assumed that the displacement potentials for the scattered wave field can be expressed in terms of
single-layer potentials [54] [90]:
ϕ
s
0 =
Z
C00
p0(r0)G
P
0
(r, r0
)dr0; r ∈ D0 (5.23)
ϕ
s
j =
Z
Cj1
pj1(r0)G
P
j
(r, r0
)dr0 +
Z
Cj2
pj2(r0)G
P
j
(r, r0
)dr0; r ∈ Dj ; j = 1, 2, ..., J (5.24)
ψ
s
0 =
Z
C00
q0(r0)G
S
0
(r, r0
)dr0; r ∈ D0 (5.25)
ψ
s
j =
Z
Cj1
qj1(r0)G
S
j
(r, r0
)dr0 +
Z
Cj2
qj2(r0)G
S
j
(r, r0
)dr0; r ∈ Dj ; j = 1, 2, ..., J (5.26)
92
where C00, Cj1 and Cj2 are the auxiliary surfaces [27] defined in Figure 3.4, Figure 3.3 and Figure 4.1,
and p0, pj1 and pj2 and q0, qj1 and qj2 are the corresponding unknown density functions. The functions
GP
j
(r, r0
) and GS
j
(r, r0
) are the full-space Green’s functions for a P and SV type line load, respectively,
with the material properties of domain Dj . They satisfy the following equations [50]:
(∇2 + ζ
2
j
)G
P
j
(r, r0
) = −δ(|r − r0|); j = 0, 1, 2, ..., J (5.27)
(∇2 + η
2
j
)G
S
j
(r, r0
) = −δ(|r − r0|); j = 0, 1, 2, ..., J (5.28)
with δ(.) being the Dirac delta-function. The Green’s functions are given by [50] [93] [26]
G
P
j
(r, r0
) = i
4
H
(2)
0
(ζjσ); r ∈ Dj ; r0 ∈ Cjk; k = 1, 2; j = 0, 1, 2, ..., J (5.29)
G
S
j
(r, r0
) = i
4
H
(2)
0
(ηjσ); r ∈ Dj ; r0 ∈ Cjk; k = 1, 2; j = 0, 1, 2, ..., J (5.30)
σ =| r − r0 |
where H
(2)
0
(.) denotes the Hankel function of the second kind and order zero. In particular, if j = 0 in
expressions (5.29) - (5.30), then r0 ∈ C00 only.
If the scattered wave field is assumed in terms of discrete line sources, then
p0(r) = a
P
0,mδ(r − rm); r ∈ D0, rm ∈ C00; m = 1, 2, ..., M (5.31)
pj1(r) = a
P
j,lj1
δ(r − rlj1
); r ∈ Dj , rlj1 ∈ Cj1; lj1 = 1, 2, ..., Lj1 (5.32)
93
pj2(r) = a
P
j,lj2
δ(r − rlj2
); r ∈ Dj , rlj2 ∈ Cj2; lj2 = Lj1 + 1, Lj1 + 2, ..., Lj1 + Lj2 (5.33)
q0(r) = a
S
0,mδ(r − rm); r ∈ D0, rm ∈ C00; m = 1, 2, ..., M (5.34)
qj1(r) = a
S
j,lj1
δ(r − rlj1
); r ∈ Dj , rlj1 ∈ Cj1; lj1 = 1, 2, ..., Lj1 (5.35)
qj2(r) = a
S
j,lj2
δ(r − rlj2
); r ∈ Dj , rlj2 ∈ Cj2; lj2 = Lj1 + 1, Lj1 + 2, ..., Lj1 + Lj2 (5.36)
where a
P
0,m, a
P
j,lj1
, a
P
j,lj2
, a
S
0,m, a
S
j,lj1
, a
S
j,lj2
denote the unknown source intensities, and M, Lj1 and Lj2 are
the orders of approximation along the auxiliary surfaces C00, Cj1 and Cj2, respectively.
Substitution of expressions (5.31) - (5.36) into (5.23) - (5.26) yields the scattered wave field in the following form
ϕ
s
0 =
X
M
m=1
a
P
0,mG
P
0
(r, rm); r ∈ D0; rm ∈ C00 (5.37)
ϕ
s
j =
X
Lj1
lj1=1
a
P
j,lj1G
P
j
(r, rlj1
) +
Lj
X1+Lj2
lj2=Lj1+1
a
P
j,lj2G
P
j
(r, rlj2
); r ∈ Dj ; rlj1 ∈ Cj1; rlj2 ∈ Cj2 (5.38)
94
ψ
s
0 =
X
M
m=1
a
S
0,mG
S
0
(r, rm); r ∈ D0; rm ∈ C00 (5.39)
ψ
s
j =
X
Lj1
lj1=1
a
S
j,lj1G
S
j
(r, rlj1
) +
Lj
X1+Lj2
lj2=Lj1+1
a
S
j,lj2G
S
j
(r, rlj2
); r ∈ Dj ; rlj1 ∈ Cj1; rlj2 ∈ Cj2 (5.40)
Thus the scattered displacement field can be expressed by [2]
u
s
j = ∇ϕ
s
j + ∇ × (0, ψs
j
, 0); r ∈ Dj , j = 0, 1, 2, ... , J (5.41)
5.2.3 Source Intensities for the Scattered Wave Field
The unknown source intensities a
P
0,m, a
P
j,lj1
, a
P
j,lj2
and a
S
0,m, a
S
j,lj1
, a
S
j,lj2
can be determined by using the
traction-free boundary conditions (5.5) - (5.6) and the continuity conditions (5.7) - (5.10). By choosing
Nj collocation points along each interface Cj and P collocation points along the top free surface SF to
impose the boundary conditions and the continuity conditions, these unknown source intensities can be
determined by solving the following matrix equation
Aa = f (5.42)
in the least-square sense.
Here, the matrix A of the size (2P + 4PJ
j=1 Nj ) × (2M + 2PJ
j=1(Lj1 + Lj2)) consists of both
displacement and stress Green’s function elements, a is a vector of the order (2M + 2PJ
j=1(Lj1 + Lj2))
which includes all the unknown source intensities and f represents a vector of the order (2P +4PJ
j=1 Nj )
containing the free-field displacement and tractions along the interfaces Cj and the top free surface SF .
95
The detailed expressions of the displacement and stress Green’s functions of the P and SV line load and
the matrices A, a and f are given in the Appendix B.
5.3 Numerical Results
For the sake of reducing the number of figures, the numerical results using the full-space Green’s functions
are presented only for the one-layer plane strain model. As before, parametric error analysis is conducted
to determine the optimum parameters based on the error criteria of the surface response. Subsequently,
the steady-state surface response is determined based on the optimum parameters.
5.3.1 Error Criteria
Similar to the antiplane strain model, the surface response error is defined as the relative error of the two
consecutive iterations of the surface response according to
Er =
PNs
i=1[(|u
n
J
(ri) − u
n−1
J
(ri)|)
2
+ (|w
n
J
(ri) − w
n−1
J
(ri)|)
2
]
PNs
i=1[|u
n
J
(ri)|
2 + |wn
J
(ri)|
2
]
; ri ∈ SF (5.43)
where the subscript J indicates the top layer DJ , Ns is the number of observation points on the surface
SF , and the superscripts n and n − 1 represent two consecutive iterations of the surface response. When
the numerical calculations converge, the error Er should approach zero. In practice, the minimum of the
error Er is being sought.
Throughout the tests, the number of observation points used to compute the error Er is chosen to
be Ns = 151. The observation points are equally spaced over the range of |x1| ≤ 4. The dimensionless
frequency Ω is defined according to equation (3.27).
96
5.3.2 One-Layer Model
The geometry of the model refers to Figure 4.2. Definitions of the parameters are the same as those defined
in the one-layer model section of Chapter 4. The material properties used in this study are defined by
D0 : µ0 = 1, α0 = 2, β0 = 1, ρ0 = 1, ν0 = 1/3 (5.44)
D1 : µ1 = 0.125, α1 = 1, β1 = 0.5, ρ1 = 0.5, ν1 = 1/3 (5.45)
where µj , ρj , νj , αj and βj denote the shear modulus, mass density, Poisson’s ratio, P and SV wave
velocities in the j
th domain Dj (j = 0, 1), respectively.
The interface and the auxiliary surfaces are defined by equations (4.16) to (4.19).
5.3.2.1 Parametric Error Analysis
As in the case of the antiplane strain model, six convergence tests are performed to determine the values of
the various parameters of the problem. The relationships (4.22) - (4.23) assumed in antiplane strain model
are also accepted here. Since these tests follow the pattern discussed for the antiplane strain model, only
their summary is presented here as follows.
1. Transparency Test. Zero-scattering condition is tested for an initial choice of problem parameters
by assuming that the material properties in the top layer and bottom half-space are the same. As expected,
the contribution of the interface deviation is found to be negligible and the surface response of the plane
strain model tends to that of the corresponding half-space free field.
97
2. Flat-Layer Test. In this test, the interface deviation d of the scatterer is assumed to be zero, and the
same initial choice of parameters obtained in the transparency test is applied. It was found that the surface
response amplitude of the plane strain model approaches that of the corresponding flat-layer case.
3. Matrix Squeness Test. This test investigates the ratio rm of number of rows versus number of
columns in matrix A of equation (5.42) which is defined by
rm =
(4N + 2P)
(2L1 + 2L2 + 2M)
(5.46)
where N and P are the number of collocation points along interface C1 and the top surface SF , and L1,
L2 and M represent the number of sources along the auxiliary surfaces C11, C12 and C00, respectively.
In this test, the numbers of collocation points N and P and all other parameters are fixed while the
numbers of sources L1, L2 and M are varied according to
N = P = 165; L1 = M = 5m; L2 = 3m; m = 1, 2, ..., 17 (5.47)
The optimum values of the matrix squeness rm are found to be 2.4 − 2.9 which correspond to m =
13 − 16 for dimensionless frequency Ω = 3 − 5 and angle of incidence θ0 = 0o − 85o
. Therefore, a matrix
squeness value rm = 2.5 is accepted throughout the plane strain model.
4. Location dr and d0 of the Auxiliary Surfaces C00, C11 and C12 Test This test is conducted to
estimate the optimum location dr of the auxiliary surfaces C00 and C11 relative to the interface C1 as well
as the location d0 of the auxiliary surface C12 relative to the top surface SF , which are varied according
to expression (4.27) while all other parameters remain the same. The relationship (3.29) proposed by J. E.
Luco (private communication) is found to be satisfied in this test thus accepted throughout this analysis.
98
5. Optimum Spacing ds of the Collocation Points. Here, the spacing ds of the collocation points is
defined by expression (4.28) and the numbers of sources and collocation points are varied according to
relationship (4.29). The other parameters remain unchanged in the test.
6. Optimum Length 2wc and 2ws of the Interface C1, the Top Surface SF and the Auxiliary Surfaces (C00, C11 and C12) Test. In this test, the length 2wc of the interface/top surface (C1 and SF ) and
the length 2ws of the auxiliary surfaces are determined by minimizing the relative error Er based on the
surface response. The numbers of collocation points and sources vary according to equation (4.29). The
spacing ds of the collocation points is fixed which is obtained from the fifth parametric test. 2wc varies
according to equation (4.30). All other parameters are fixed.
Based on the above parametric error analyses, the optimum parameters which are accepted for the
steady-state response evaluation for the one-layer plane strain model using the full-space Green’s functions
are listed in Table 5.1.
Table 5.1: Parameters for the steady-state response for the one-layer plane strain model using the full-space
Green’s functions. Ω = 1 − 8
Wave type N, P L1, M L2 wc, ws dr, d0 ds
P 231 105 63 1 0.0261 0.0087
SV 231 105 63 1 0.0261 0.0087
Rayleigh 231 105 63 1 0.0261 0.0087
Based on the parameters in Table 5.1, the results of the steady-state response for the one-layer model
are presented next.
5.3.2.2 Steady-State Response
The surface displacement has been investigated for P, SV, and Rayleigh incidences for the one-layer plane
strain model with a cosine-shaped interface of various scatterer deviation values in the range −0.2 ≤
99
d ≤ 0.2. Here, d < 0 represents a cosine-shaped scatterer curved upward while d > 0 represents a
cosine-shaped scatterer curved downward.
Figure 5.1 - Figure 5.2 show the surface displacement for plane P waves, and Figure 5.3 - Figure 5.4 display the surface displacement for plane SV waves with various scatterer deviations, frequencies and angles
of incidence. It is noted that when the angle of incidence changes, the pattern of the surface response also
changes significantly. The results clearly show that for a vertical incidence, the predominant amplitude of
surface motion is in the vertical direction for P waves and the horizontal direction for SV waves. However,
this distinction becomes less pronounced for an oblique incidence. In addition, the amplification of surface
motion strongly depends on the frequency of the incident wave and the location of the observation station.
Lastly, the surface response is very sensitive to the magnitude of the irregularity along the interface.
This suggests that surface motion results from a complex interaction of elastic waves within the media.
Constructive (destructive) interaction will result in the amplification (reduction) of surface motion.
As expected, the surface motion amplitude for vertically incident P and SV waves possess certain
symmetry with respect to the origin.
Figure 5.5 shows the surface displacement for Rayleigh waves. Apparently both surface displacement
components have a similar range of amplification. In addition, it is found that a higher frequency results
in a more complicated pattern of the surface response for this incident wave. Again, the surface response
is very sensitive to the deviation magnitude of the cosine-shaped irregularity along the interface.
It should be noted that the reflected P wave becomes a surface wave if the angle of incident SV wave
γ0 is greater than arcsin(β0/α0).
100
-4 -2 0 2 4
0
0.2
0.4
|u|
P2D
0
= 0o
-4 -2 0 2 4
1.2
1.4
1.6
1.8
|w|
P2D
0
= 0o
-4 -2 0 2 4
0
2
4
|u|
0
= 30o
-4 -2 0 2 4
1
1.5
2
|w|
0
= 30o
-4 -2 0 2 4
0
2
4
|u|
0
= 60o
-4 -2 0 2 4
0.5
1
1.5
2
|w|
0
= 60o
-4 -2 0 2 4
x
1
0
0.5
|u|
0
= 85o
-4 -2 0 2 4
x
1
0.1
0.2
0.3
|w|
0
= 85o
Figure 5.1: Steady-state surface displacement for a one-layer plane strain model subjected to incident P
waves with different angles of incidence. Ω = 5. a = 0.5, h = 1. Solid: d = 0.2, Dash: d = 0.1, Points:
d = 0.0, Dash-Dot: d = −0.1, Dot: d = −0.2.
-4 -2 0 2 4
0
0.5
1
|u|
P2D
0
= 0o
-4 -2 0 2 4
2
4
6
|w|
P2D
0
= 0o
-4 -2 0 2 4
0
0.5
1
|u|
0
= 30o
-4 -2 0 2 4
1.5
2
2.5
3
3.5
|w|
0
= 30o
-4 -2 0 2 4
0.5
1
|u|
0
= 60o
-4 -2 0 2 4
0.6
0.8
1
1.2
|w|
0
= 60o
-4 -2 0 2 4
x
1
0.2
0.4
|u|
0
= 85o
-4 -2 0 2 4
x
1
0.2
0.4
|w|
0
= 85o
Figure 5.2: Steady-state surface displacement for a one-layer plane strain model subjected to incident P
waves with different angles of incidence. Ω = 6. a = 0.5, h = 1. Solid: d = 0.2, Dash: d = 0.1, Points:
d = 0.0, Dash-Dot: d = −0.1, Dot: d = −0.2.
101
Figure 5.3: Steady-state surface displacement for a one-layer plane strain model subjected to incident SV
waves with different angles of incidence. Ω = 4. a = 0.5, h = 1. Solid: d = 0.2, Dash: d = 0.1, Points:
d = 0.0, Dash-Dot: d = −0.1, Dot: d = −0.2.
Figure 5.4: Steady-state surface displacement for a one-layer plane strain model subjected to incident SV
waves with different angles of incidence. Ω = 5. a = 0.5, h = 1. Solid: d = 0.2, Dash: d = 0.1, Points:
d = 0.0, Dash-Dot: d = −0.1, Dot: d = −0.2.
102
-4 -2 0 2 4
0.26
0.28
0.3
0.32
|u|
R2D
= 4
-4 -2 0 2 4
0.14
0.16
0.18
|u|
= 5
-4 -2 0 2 4
x
1
0.08
0.085
0.09
|u|
= 6
-4 -2 0 2 4
0.42
0.44
0.46
0.48
|w|
R2D
= 4
-4 -2 0 2 4
0.23
0.24
0.25
0.26
0.27
|w|
= 5
-4 -2 0 2 4
x
1
0.125
0.13
0.135
|w|
= 6
Figure 5.5: Steady-state surface displacement for a one-layer plane strain model subjected to incident
Rayleigh waves for various frequencies. a = 0.5, h = 1. Solid: d = 0.2, Dash: d = 0.1, Points: d = 0.0,
Dash-Dot: d = −0.1, Dot: d = −0.2.
Figure 5.6, Figure 5.7 and Figure 5.8 display the surface displacement at the site x1 = 0 versus frequency
for the one-layer plane strain model with a scatterer deviation d = 0.1 subjected to vertically incident P,
SV, and Rayleigh surface waves, respectively. Surface displacement of the corresponding flat-layer model is
also shown in these three figures for comparison. Resonances are detected for all these three models [104].
For the vertically incident P wave, the first two resonance frequencies for the surface displacement
component |w| at the site x1 = 0 are found to be Ωi = 1.992 and 5.960, which are close to the resonance
frequencies Ωi = 2 and 6 for the corresponding flat-layer model, see Figure 5.6. The difference between
the model with a scatterer and the flat-layer model is attributed to the presence of irregularity.
For the vertically incident SV wave, the first three resonance frequencies for the surface displacement
component |u| at the site x1 = 0 are found to be Ωi = 1.001, 2.996 and 4.965, which are close to the
first three resonance frequencies Ωi = 1, 3 and 5 for the corresponding flat-layer model, see Figure 5.7.
103
The difference between the model with a scatterer and the flat-layer model is attributed to the presence of
irregularity.
It is also shown that, for the vertically incident P and SV waves, the difference of resonance surface displacement amplitude between the one-layer model with an irregularity and the flat-layer model increases
with increase of frequency.
For the incident Rayleigh wave, the first two sharp resonance frequencies for both the two surface
displacement component |u| and |w| at the site x1 = 0 are found to be Ωi = 5.12 and 5.92, which
are identical to the ones of the corresponding flat-layer model, see Figure 5.8. However, the resonance
displacement amplitude of the one-layer model with a scatterer is slightly higher than that of the flat-layer
model, see Table 5.2. This difference is attributed to the presence of irregularity.
Table 5.2: Peak displacement amplitude comparison between the one-layer model with scatterer and the
corresponding flat-layer model for incident Rayleigh waves
Resoance Frequency (Ω) |u| |u
f f | %, |u| |w| |w
f f | %, |w|
5.12 1.8338 1.8106 1.3% 2.8492 2.8384 0.4%
5.92 8.5420 8.4409 1.2% 13.2826 13.2399 0.3%
104
0 1 2 3 4 5 6 7 8
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
|w|
Flat Layer
With Scatterer
Figure 5.6: Steady-state surface displacement at x1 = 0 for the one-layer plane strain model subjected to
a vertically incident P wave as a function of dimensionless frequencies. a = 0.5, h = 1, d = 0.1
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
1
2
3
4
5
6
7
8
9
10
|u|
Flat Layer
With Scatterer
Figure 5.7: Steady-state surface displacement at x1 = 0 for the one-layer plane strain model subjected to
a vertically incident SV wave as a function of dimensionless frequencies. a = 0.5, h = 1, d = 0.1
105
2 3 4 5 6 7 8
0
5
10
|u|
d = 0.1
d = 0.0
2 3 4 5 6 7 8
0
5
10
15
|w|
d = 0.1
d = 0.0
Figure 5.8: Steady-state surface displacement at x1 = 0 for the one-layer plane strain model subjected to
an incident Rayleigh wave as a function of dimensionless frequencies. The first sharp resonance occurs at
Ω1 = 5.12 while the second one at Ω2 = 5.92. a = 0.5, h = 1, d = 0.1
To further validate the accuracy of the one-layer plane strain formulation for the incident Rayleigh
wave, a transparency test is conducted by letting the material properties of the two domains, D1 and D0,
be identical, i.e.,
D0 : µ0 = 1, α0 = 2, β0 = 1, ρ0 = 1, ν0 = 1/3 (5.48)
D1 : µ1 = 1, α1 = 2, β1 = 1, ρ1 = 1, ν1 = 1/3 (5.49)
The geometry of this one-layer model is the same as that shown by Figure 4.2, i.e., a = 0.5, h = 1.
The amplitude of the cosine-shaped scatterer is chosen to be d = 0.2.
In this transparency test, the parameters given in Table 5.1 are used to simulate the surface response
of the Rayleigh wave in a pure half-space for a constant Rayleigh wave speed derived in Appendix B at
circular frequencies ω = i2π/20, i = 1, 2, ..., 20.
106
The material properties of the pure half-space are identical to those of the transparency test, see (5.48).
As expected, the Rayleigh wave surface response of the transparency one-layer model remains constant
at a given frequency as the scattering from the scatterer vanishes due to the identical material properties
on both sides of the irregular interface. A good agreement has been achieved for the Rayleigh wave surface
displacements between the transparency one-layer model and the pure half-space model, see Table 5.3.
Table 5.3: Comparison of Rayleigh wave surface displacements between transparency one-layer and pure
half-space models
ω(rad/sec) |u
HS| |w
HS| |u
1L| |w
1L| %, |u| %, |w|
0.31415927 0.09358615 0.14648083 0.09358615 0.14648083 0.000004% 0.000000%
0.62831853 0.18717230 0.29296165 0.18717230 0.29296166 0.000000% 0.000003%
0.94247780 0.28075845 0.43944248 0.28075845 0.43944248 0.000000% 0.000000%
1.25663710 0.37434460 0.58592331 0.37434460 0.58592331 0.000000% 0.000000%
1.57079630 0.46793074 0.73240413 0.46793074 0.73240413 0.000000% 0.000000%
1.88495560 0.56151689 0.87888496 0.56151691 0.87888494 0.000004% −0.000002%
2.19911490 0.65510304 1.02536580 0.65510303 1.02536580 −0.000002% 0.000000%
2.51327410 0.74868919 1.17184660 0.74868921 1.17184660 0.000003% 0.000000%
2.82743340 0.84227534 1.31832740 0.84227534 1.31832740 0.000000% 0.000000%
3.14159270 0.93586149 1.46480830 0.93586150 1.46480830 0.000001% 0.000000%
3.45575190 1.02944760 1.61128910 1.02944760 1.61128910 0.000000% 0.000000%
3.76991120 1.12303380 1.75776990 1.12303380 1.75776990 0.000000% 0.000000%
4.08407040 1.21661990 1.90425070 1.21661990 1.90425070 0.000000% 0.000000%
4.39822970 1.31020610 2.05073160 1.31020610 2.05073150 0.000000% −0.000005%
4.71238900 1.40379220 2.19721240 1.40379220 2.19721240 0.000000% 0.000000%
5.02654820 1.49737840 2.34369320 1.49737840 2.34369320 0.000000% 0.000000%
5.34070750 1.59096450 2.49017410 1.59096460 2.49017410 0.000006% 0.000000%
5.65486680 1.68455070 2.63665490 1.68455070 2.63665490 0.000000% 0.000000%
5.96902600 1.77813680 2.78313570 1.77813690 2.78313570 0.000006% 0.000000%
6.28318530 1.87172300 2.92961650 1.87172300 2.92961650 0.000000% 0.000000%
Here, in Table 5.3, the superscript HS indicates the pure half-space model while 1L represents the
one-layer transparency model.
This transparency test further confirms the accuracy of the 2D one-layer Rayleigh wave formulation.
This concludes the steady-state analysis of the one-layer plane strain model.
107
5.4 Conclusions
The plane strain model formulation for the scattering of harmonic elastic waves in a media of multiple
layers with irregular interfaces has been derived using an indirect boundary integral equation approach.
The total wave field is assumed to be a superposition of the known free field and the unknown scattered
wave field. The unknown scattered wave field can be expressed in terms of single-layer potentials involving
line load full-space Green’s functions associated with unknown density functions. The unknown density
functions are determined in the least square sense.
Detailed numerical work has been performed for the one-layer plane strain model with one cosineshaped irregular interface subjected to incident plane harmonic P, SV, and Rayleigh waves. Convergence
analysis was conducted to estimate the optimum parameters of the model. Subsequently, using the parameters determined by the convergence tests, the steady-state surface response was calculated for the
one-layer model with an irregularity for different incident plane waves at various frequencies. The influence of the scatterer deviation amplitude on the surface response was considered for this model as well.
The presented numerical results show that the surface response near the irregularities is very sensitive
to the nature of the incident waves (such as wave type, angle of incidence, and frequency), the shape of
the irregular interfaces and the location of the observation point. It was shown that the presence of layer
irregularity may result in a significant increase in amplification of the surface motion as compared to the
flat-layer model. Furthermore, the surface motion amplitudes for vertically incident P and SV waves are
found to exhibit appropriate symmetry about the origin due to the symmetric geometry of the model which
provides additional support about the accuracy of the proposed solution.
For observation sites directly above the irregular interface, the surface response exhibits resonance
features [104]. The resonance frequencies are closely related to those of the corresponding flat layer.
The difference of resonance frequencies and amplitudes between the models with and without irregular
scatterers is attributed to the presence of irregularity.
108
Chapter 6
Three-Dimensional Model
In this chapter, the scattering of elastic plane harmonic P, SV, SH, Love and Rayleigh waves for the threedimensional multilayered model is investigated by using an indirect boundary integral equation method.
The scattered wave field is represented in terms of the single-layer potentials with a point load full-space
Green’s functions incorporated. A closed-form solution is available for the point load full-space Green’s
functions thus reducing errors in the evaluation of Green’s functions using a numerical integration approach. Therefore, the key issue is to determine the optimum parameters from which the numerical results
are obtained.
6.1 Statement of Problem
Statement of the problem for the three-dimensional model refers to Chapter 2.
6.2 Steady-State Solution of the Problem
The total wave field in each domain can be expressed as a superposition of the free field and the scattered
wave field according to
uj = u
f f
j + u
s
j
; r ∈ Dj, j = 0, 1, 2, ..., J (6.1)
109
where the superscripts ff and s denote the free field of the corresponding flat-layer system and the scattered wave field caused by irregular interfaces, respectively.
6.2.1 Free Field
The time factor associated with the three-dimensional Green’s functions is e
−iωt [50]. Therefore, for an
incident plane wave coming from below in the x1x3−plane of a global orthogonal Cartesian coordinate
system r = (x1, x2, x3) shown in Figure 2.1, the free field should be rewritten as follows:
SH waves
v
f f
j = Aj exp(ikj (x1 sin θj − x3 cos θj )) + Bj exp(ikj (x1 sin θj + x3 cos θj )) (6.2)
r ∈ Dj , j = 0, 1, 2, ... , J
Love waves, assuming that the SH- and Love-wave speed values satisfy β0 > β1,..., > c > βJ [40].
v
f f
J = AJ exp(iκx1 + iκx3
r
(
c
βJ
)
2 − 1) + BJ exp(iκx1 − iκx3
r
(
c
βJ
)
2 − 1); r ∈ DJ, (6.3)
v
f f
j = Aj exp(iκx1 − κx3
r
1 − (
c
βj
)
2)+Bj exp(iκx1 + κx3
r
1 − (
c
βj
)
2); r ∈ Dj; j = 0, 1, ..., J−1
(6.4)
Plane P or SV waves
ϕ
f f
j = Aj exp(iζj (x1 sin θj − x3 cos θj )) + Ej exp(iζj (x1 sin θj + x3 cos θj )) (6.5)
110
ψ
f f
j = Bj exp(iηj (x1 sin γj − x3 cos γj )) + Fj exp(iηj (x1 sin γj + x3 cos γj )) (6.6)
r ∈ Dj , j = 0, 1, 2, ... , J
Rayleigh waves
ϕ
f f
j = Aj exp(iκx1 − κ
r
1 − (
c
αj
)
2x3) + Ej exp(iκx1 + κ
r
1 − (
c
αj
)
2x3) (6.7)
ψ
f f
j = Bj exp(iκx1 − κ
r
1 − (
c
βj
)
2x3) + Fj exp(iκx1 + κ
r
1 − (
c
βj
)
2x3) (6.8)
r ∈ Dj , j = 0, 1, 2, ... , J
The parameters Aj , Bj , Ej , Fj and θj , γj are known [2] [94] [40].
The incident waves from the bottom of the half-space D0 are then assumed to be of the form
SH waves
v
inc = exp(−iωt + ik0(x1 sin θ0 − x3 cos θ0)) (6.9)
Love waves
v
inc = exp(−iωt + iκx1 − κx3
r
1 − (
c
β0
)
2); κ =
ω
c
(6.10)
P waves
ϕ
inc = A0 exp(−iωt + iζ0(x1 sin θ0 − x3 cos θ0)) (6.11)
111
SV waves
ψ
inc = B0 exp(−iωt + iη0(x1 sin γ0 − x3 cos γ0)) (6.12)
Rayleigh waves
ϕ
inc = A0 exp(−iωt + iκx1 − κ
r
1 − (
c
α0
)
2x3); κ =
ω
c
(6.13)
ψ
inc = B0 exp(−iωt + iκx1 − κ
r
1 − (
c
β0
)
2x3); κ =
ω
c
(6.14)
Here, all the above parameters are defined in Chapters 3 through 5.
However, if the incident wave propagates in a plane with an azimuthal angle φ0 with respect to the
x1−axis, a local Cartesian coordinate system r
′
= (x
′
1
, x
′
2
, x
′
3
) needs to be defined in such a way that the
plane wave propagates in the x
′
1x
′
3−plane (see Figure 2.1). The transformation relationship between the
two coordinate systems can be expressed as
r
′
= Lr; L =
cos φ0 sin φ0 0
− sin φ0 cos φ0 0
0 0 1
(6.15)
where L is the transformation matrix.
Therefore, the free-field displacements u
f f and tractions t
f f in the global coordinates r = (x1, x2, x3)
can then be obtained by multiplying those ((u
′
)
f f and (t
′
)
f f ) in the local coordinates r
′
= (x
′
1
, x
′
2
, x
′
3
) by
the transpose (L
T
) of the transformation matrix, i.e.,
u
f f = L
T
(u
′
)
f f
; t
f f = L
T
(t
′
)
f f (6.16)
112
This completes the analysis of the free field for the 3D model. The scattered wave field is considered
next.
6.2.2 Scattered Wave Field
It is assumed that the scattered wave field can be expressed in terms of single-layer potentials [54] [90]:
u
s(0)
p
(r,ω) = Z
C00
q
(0)
k
(r0)G
(0)
pk (r, r0
; ω)dr0; (6.17)
r ∈ D0; p = 1, 2, 3
u
s(j)
p
(r,ω) = Z
Cj1
q
(j)
k
(r0)G
(j)
pk (r, r0
; ω)dr0 +
Z
Cj2
q
(j)
k
(r0)G
(j)
pk (r, r0
; ω)dr0; (6.18)
r ∈ Dj ; j = 1, 2, ..., J; p = 1, 2, 3
where r is the position vector, C00, Cj1 and Cj2 are the auxiliary surfaces defined by Figure 3.4, Figure 3.3
and Figure 4.1 [27] [102], respectively, q
(j)
k
are the corresponding unknown density functions, and summation over the repeated subscript k is understood (but not with respect to the repeated superscript j).
The functions G
(j)
pk (r, r0
; ω) are the three-dimensional full-space displacement Green’s functions in the
domain Dj due to a point load at location r0 which satisfy the following equation of motion [50]:
µj
∂
2
∂xi∂xi
G
(j)
pk (r, r0
; ω) + (λj + µj )
∂
2
∂xp∂xi
G
(j)
ik (r, r0
; ω) + ρjω
2G
(j)
pk (r, r0
; ω) = −δpkδ(r − r0) (6.19)
r ∈ Dj ; j = 0, 1, 2, ..., J; p, k = 1, 2, 3
113
Here ρj denotes the mass density, λj and µj are the Lame’s constants, ω is circular frequency and
δ(r − r0) represents the Dirac delta-function. The explicit expressions for the 3D full-space Green’s functions are given in Appendix C.
If the scattered wave field is assumed in terms of discrete point sources, then it follows that the unknown density functions assume the following form
D0 :
q
(0)
1
(r) = a
(0)
m δ(r − rm)
q
(0)
2
(r) = b
(0)
m δ(r − rm) (6.20)
q
(0)
3
(r) = c
(0)
m δ(r − rm)
r ∈ D0, rm ∈ C00, m = 1, 2, ..., M
Dj :
q
(j)
1
(r) = d
(j)
lj
δ(r − rlj
)
q
(j)
2
(r) = e
(j)
lj
δ(r − rlj
) (6.21)
q
(j)
3
(r) = f
(j)
lj
δ(r − rlj
)
r ∈ Dj , rlj ∈ Cj1 + Cj2, lj = 1, 2, ..., Lj1 + Lj2
Consequently, the scattered wave fields (6.17) and (6.18) can further be expressed as
u
s
0 = a
(0)
m G
(0)
11 (r, rm; ω) + b
(0)
m G
(0)
12 (r, rm; ω) + c
(0)
m G
(0)
13 (r, rm; ω); r ∈ D0
v
s
0 = a
(0)
m G
(0)
21 (r, rm; ω) + b
(0)
m G
(0)
22 (r, rm; ω) + c
(0)
m G
(0)
23 (r, rm; ω); r ∈ D0 (6.22)
114
w
s
0 = a
(0)
m G
(0)
31 (r, rm; ω) + b
(0)
m G
(0)
32 (r, rm; ω) + c
(0)
m G
(0)
33 (r, rm; ω); r ∈ D0
u
s
j = d
(j)
lj
G
(j)
11 (r, rlj
; ω) + e
(j)
lj
G
(j)
12 (r, rlj
; ω) + f
(j)
lj
G
(j)
13 (r, rlj
; ω);
v
s
j = d
(j)
lj
G
(j)
21 (r, rlj
; ω) + e
(j)
lj
G
(j)
22 (r, rlj
; ω) + f
(j)
lj
G
(j)
23 (r, rlj
; ω); (6.23)
w
s
j = d
(j)
lj
G
(j)
31 (r, rlj
; ω) + e
(j)
lj
G
(j)
32 (r, rlj
; ω) + f
(j)
lj
G
(j)
33 (r, rlj
; ω);
r ∈ Dj ; j = 1, 2, ..., J
rm ∈ C00, m = 1, 2, ..., M; rlj ∈ Cj1 + Cj2, lj = 1, 2, ..., Lj1 + Lj2
where a
(0)
m , b
(0)
m , c
(0)
m and d
(j)
lj
, e
(j)
lj
, f
(j)
lj
denote the unknown source intensities, M , Lj1 and Lj2 are the
number of sources along the auxiliary surfaces C00, Cj1 and Cj2, respectively. Summation convention is
over the repeated subscripts m and lj is understood (but not with respect to the repeated superscript j).
6.2.3 Source Intensities for the Scattered Wave Field
The unknown source intensities can be determined by imposing the traction-free boundary conditions (2.3)
along the top surface SF and the continuity conditions (2.4) - (2.5) along each interface Cj . By choosing
Nj collocation points on each interface Cj to impose the continuity conditions and P collocation points
on the top free surface SF to impose the traction-free boundary conditions [102], the unknown source
intensities can be determined by solving a matrix equation of the form
Aa = f (6.24)
115
in the least-square sense. Here, the matrix A consists of both displacement and stress Green’s functions which is of the size (3P + 6PJ
j=1 Nj ) × (3M + 3PJ
j=1(Lj1 + Lj2)), a is a vector of the order
(3M + 3PJ
j=1(Lj1 +Lj2)) containing all the unknown source intensities and f represents a vector of the
order (3P + 6PJ
j=1 Nj ) that includes the free-field displacements along the interfaces Cj and the freefield tractions along the same interfaces and the surface SF . The detailed expressions for the point-load
displacement and stress Green’s functions and the matrices A, a and f are given in the Appendix C.
6.3 Numerical Results
For the sake of simplicity, only a one-layer three-dimensional model is investigated. As before, the optimum parameters are determined based on the parametric error analysis of the surface response. Subsequently, the numerical steady-state surface response is evaluated using the optimum parameters obtained
from the parametric error analysis along the two perpendicular line segments (see Figure 6.1).
Line Segment AA′
: |x1| ≤ 4, x2 = x3 = 0 (6.25)
Line Segment BB′
: |x2| ≤ 4, x3 = x1 = 0 (6.26)
116
A A’
B
B’
O
(-4,0,0) (4,0,0)
(0,4,0)
(0,-4,0)
X
X
X
1
2
3
Figure 6.1: Line segments AA′
and BB′
on which the three-dimensional surface response is evaluated
6.3.1 Error Criteria
Since there are no exact solutions available, the surface response error is defined as the relative error of
the two consecutive iterations of the surface response according to
Er = (E
(A)
r + E
(B)
r
)/2 (6.27)
where
E
(A)
r =
PNs
i=1[(|u
n
J
(ri) − u
n−1
J
(ri)|)
2
+ (|v
n
J
(ri) − v
n−1
J
(ri)|)
2
+ (|w
n
J
(ri) − w
n−1
J
(ri)|)
2
]
PNs
i=1[|u
n
J
(ri)|
2 + |v
n
J
(ri)|
2 + |wn
J
(ri)|
2
]
; (6.28)
117
ri ∈ AA′
E
(B)
r =
PNs
i=1[(|u
n
J
(ri) − u
n−1
J
(ri)|)
2
+ (|v
n
J
(ri) − v
n−1
J
(ri)|)
2
+ (|w
n
J
(ri) − w
n−1
J
(ri)|)
2
]
PNs
i=1[|u
n
J
(ri)|
2 + |v
n
J
(ri)|
2 + |wn
J
(ri)|
2
]
; (6.29)
ri ∈ BB′
Here, the subscript J indicates the top layer DJ , AA′
and BB′
are the line segments defined in (6.25)
- (6.26), also shown in Figure 6.1. Ns is the number of observation points along each line segment on the
surface SF , and the superscripts n and n − 1 represent two consecutive iterations of the surface response.
When the numerical calculations converge, the error Er should approach zero. In practice, the minimum
of the error Er is being sought.
Throughout the tests, the number of observation points used to compute the error Er along each line
segment is chosen to be Ns = 101. The observation points are equally spaced over the range of |x1| ≤ 4
for line segment AA′
and |x2| ≤ 4 for line segment BB′
, respectively.
The dimensionless frequency Ω is defined according to equation (3.27).
6.3.2 One-Layer Model
Only an axisymmetric one-layer three-dimensional model is investigated to reduce the amount of numerical work. This model is subjected to the incident plane harmonic P, SV, SH, Love or Rayleigh waves.
The cross-section view in the x1x3− plane of the one-layer model is depicted in Figure 4.2 with a
cosine-shaped scatterer. The three-dimensional interface C1, the top surface SF and the auxiliary surfaces
C00, C11 and C12 can be obtained by rotating the corresponding two-dimensional generator curves for
180o
around the x3−axis. Therefore, the interface C1 and the top surface SF are defined by
C1:
118
x3 =
h for a < rc ≤ wc
h +
d
2
(1 + cos πrc
a
) for rc ≤ a
; rc =
q
x
2
1 + x
2
2
(6.30)
SF :
x3 = 0; rc ≤ wc; rc =
q
x
2
1 + x
2
2
(6.31)
while the auxiliary surfaces C00, C11 and C12 can be expressed by
C00:
x3 =
h − dr for a < rs ≤ ws
h − dr +
d
2
(1 + cos πrs
a
) for rs ≤ a
; rs =
q
x
2
1 + x
2
2
(6.32)
C11:
x3 =
h + dr for a < rs ≤ ws
h + dr +
d
2
(1 + cos πrs
a
) for rs ≤ a
; rs =
q
x
2
1 + x
2
2
(6.33)
C12:
x3 = −d0; rs ≤ ws; rs =
q
x
2
1 + x
2
2
(6.34)
Here, definition of the parameters h, a, d, dr and d0 are the same as in the two-dimensional onelayer model discussed in Chapters 4 and 5 (see Figure 4.2), while wc and ws are the maximum radii of the
collocations and sources with respect to the x3−axis.
Based on the above definition, the location of the collocation points on the interface C1 is designed
according to
119
x
(i,j)
1 = rc cos
π
4
j
x
(i,j)
2 = rc sin
π
4
j; rc =
wc
Nc
i (6.35)
x
(i,j)
3 =
h for a < rc ≤ wc
h +
d
2
(1 + cos πrc
a
) for rc ≤ a
i = 0, 1, 2, ..., Nc; j = 0, 1, 2, 3, 4, 5, 6, 7;
N = 8Nc + 1; (x
(i,j)
1
, x
(i,j)
2
, x
(i,j)
3
) ∈ C1 (6.36)
where Nc is the number of circles with respect to x3−axis in the x1x2− plane with eight equally spaced
collocation points along each circle, N is the number of total collocation points on the interface C1.
The locations of the collocation points on the top traction-free surface SF are arranged according to
x
(i,j)
1 = rc cos
π
4
j
x
(i,j)
2 = rc sin
π
4
j; rc =
wc
Nc
i (6.37)
x
(i,j)
3 = 0
i = 0, 1, 2, ..., Nc; j = 0, 1, 2, 3, 4, 5, 6, 7;
P = 8Nc + 1; (x
(i,j)
1
, x
(i,j)
2
, x
(i,j)
3
) ∈ SF (6.38)
where P is the number of total collocation points on the surface SF .
The collocation points for the one-layer three-dimensional model are displayed in Figure 6.2.
120
Figure 6.2: A one-layer three-dimensional model with collocation points (circles) distributed on the interface C1 and the top surface SF . Azimuthal angle φ0 = 0o
Similarly, the locations of the sources on the auxiliary surfaces C00, C11 and C12 are defined as follows:
C00:
x
(i,j)
1 = rs cos
π
4
j
x
(i,j)
2 = rs sin
π
4
j; rs =
ws
Ns
i (6.39)
x
(i,j)
3 =
h − dr for a < rs ≤ ws
h − dr +
d
2
(1 + cos πrc
a
) for rs ≤ a
i = 0, 1, 2, ..., Ns; j = 0, 1, 2, 3, 4, 5, 6, 7;
M = 8Ns + 1; (x
(i,j)
1
, x
(i,j)
2
, x
(i,j)
3
) ∈ C00 (6.40)
121
C11:
x
(i,j)
1 = rs cos
π
4
j
x
(i,j)
2 = rs sin
π
4
j; rs =
ws
Ns
i (6.41)
x
(i,j)
3 =
h + dr for a < rs ≤ ws
h + dr +
d
2
(1 + cos πrc
a
) for rs ≤ a
i = 0, 1, 2, ..., Ns; j = 0, 1, 2, 3, 4, 5, 6, 7;
L11 = 8Ns + 1; (x
(i,j)
1
, x
(i,j)
2
, x
(i,j)
3
) ∈ C11 (6.42)
C12:
x
(i,j)
1 = r
′
s
cos
π
4
j
x
(i,j)
2 = r
′
s
sin
π
4
j; r
′
s =
ws
N
′
s
i (6.43)
x
(i,j)
3 = −d0
i = 0, 1, 2, ..., N′
s
; j = 0, 1, 2, 3, 4, 5, 6, 7;
L12 = 8N
′
s + 1; (x
(i,j)
1
, x
(i,j)
2
, x
(i,j)
3
) ∈ C12 (6.44)
where Ns and N
′
s
are the numbers of circles centered at the x3−axis in the x1x2−plane with eight equally
spaced sources along each circle, M, L11 and L12 are the numbers of total point sources on the auxiliary
surfaces C00, C11 and C12, respectively.
The point sources for the one-layer three-dimensional model are displayed in Figure 6.3.
122
Figure 6.3: A one-layer three-dimensional model with point sources (stars) distributed on the auxiliary
surfaces C00, C11, and C12
The material properties used in this study are defined by expressions (5.44)-(5.45). The optimum locations of the collocation points and sources are determined through parametric error analysis which is
presented next.
6.3.2.1 Parametric Error Analysis
For this model, six tests are designed to estimate the values of the various parameters for the threedimensional problem. The relationships (4.22) - (4.23) assumed in antiplane strain model are also accepted
here. Since the parametric error analysis of three-dimensional problem is very similar to the procedure for
the two-dimensional plane strain model, details of the parametric error analysis of the three-dimensional
problems are omitted.
123
1. Transparency Test. By letting the material properties in the top layer (D1) be the same as those of the
bottom half-space (D0) and the scatterer deviation d to be a non-zero value, the zero-scattering condition
is tested for an initial choice of the parameters. Then it was found that the surface response approaches
that of the free field for the half-space.
2. Flat-Layer Test. In this test, it is assumed that the scatterer deviation d is zero and the material
properties in the two domains are different. By using the same initial parameters of the transparency test,
the surface response amplitude is found to be the same as in the corresponding flat-layer model.
3. Matrix Squeness Test. The matrix A squeness in equation (6.24) for this model is defined by
rm =
3P + 6N
3M + 3(L11 + L12)
(6.45)
The optimum value of the matrix squeness for both the antiplane strain model and plane strain model
has been found to be rm ∼ 2.5. Therefore, this squeness value is accepted throughout the study of the
three-dimensional model as well.
4. Location dr and d0 of the Auxiliary Surfaces C00, C11 and C12. By estimating the surface response
error Er with respect to the locations dr and d0 while keeping all other parameters unchanged, an optimum
value of the locations dr and d0 of the auxiliary surfaces (C00, C11 and C12) can be determined.
5. Optimum Spacing ds of the Collocation Points. Here, the parameter ds is defined as the spacing
of the two adjacent circles for the collocation points.
By gradually increasing the number of circles Nc for the collocation points and the number of circles
Ns and N
′
s
for the sources, one tries to find an optimum ds by evaluating the surface response error Er.
124
Here, rm ∼ 2.5 for equation (6.45) and the optimum value of the locations dr and d0 of the auxiliary
surfaces (C00, C11 and C12) from the fourth parametric test are applied. All other parameters are fixed.
6. Optimum of the Maximum Radii wc and ws of the Collocations and Sources. In this test,
the maximum radii wc and ws are determined by minimizing the relative error Er based on the surface
response. Gradually increase the numbers of circles Nc for the collocation points and the numbers of
circles Ns and N
′
s
for the sources, while keeping the spacing ds of the collocation points, as obtained
from the fifth parametric test, unchanged. Here, rm ∼ 2.5 for equation (6.45) and the optimum value of
the locations dr and d0 of the auxiliary surfaces (C00, C11 and C12) from the fourth parametric test are
applied. All other parameters are fixed.
The accepted parameters for this one-layer three-dimensional model, utilizing the full-space Green’s
functions, have been determined through the parametric error analysis detailed above and are presented
in Table 6.1.
Table 6.1: Parameters for the steady-state response for the one-layer three-dimensional model using the
full-space Green’s functions. Ω = 4 − 5; Angles of incidence are 0
o − 85o
Wave type N, P L1, M L2 wc, ws ds dr, d0
P 265 121 73 1 0.0303 0.0909
SV 265 121 73 1 0.0303 0.0909
SH 265 121 73 1 0.0303 0.0909
Rayleigh 265 121 73 1 0.0303 0.0909
It is understood that the optimum parameters listed in Table 6.1 can be applied to the one-layer model
with frequencies lower than Ω = 5.
Results of the steady-state response of the one-layer three-dimensional model by using the parameters
in Table 6.1 are presented next.
Parametric error analysis is not performed for the Love wave. However, it is still reasonable to apply
the parameters given in Table 6.1 to calculate the steady-state response for the Love wave.
125
6.3.2.2 Steady-State Response
The surface displacement for the one-layer three-dimensional model with an axisymmetric cosine-shaped
interface has been calculated for incident P, SV, SH, Love, and Rayleigh waves. Figure 6.4 - Figure 6.45
depict the surface displacement for incident P, SV, SH, Love, and Rayleigh waves for different frequencies
and various angles of incidence. The azimuthal angle is φ0 = 0o
.
Because of the axisymmetric geometry of the model, some particular features should be expected from
the surface response along the observation line segments AA′
and BB′
for a vertically incident wave:
(1) For a vertically incident P wave, the surface response should be axisymmetric everywhere, i.e., the
displacement |u| and |w| along the x1−axis should be the same as|v| and |w| along the x2−axis. Therefore,
the following relationships must hold for vertically incident P waves:
|u
P
(x1, 0, 0)| = |v
P
(0, x2, 0)|
|w
P
(x1, 0, 0)| = |w
P
(0, x2, 0)| (6.46)
|u
P
(0, x2, 0)| = |v
P
(x1, 0, 0)| = 0
where the superscript P denotes the type of the incident wave.
This feature is displayed in Figure 6.4, Figure 6.18 and Figure 6.32.
(2) For a vertically incident plane SH wave, the motion of the media takes place along the x2−axis
only, i.e., u = (0, v, 0), until the incident wave hits the scatterer. For a vertically incident plane SV wave
on the other hand, the motion of the media takes place along the x1−axis only, i.e., u = (u, 0, 0), until
the propagating wave strikes the scatterer. Therefore, a wave that appears in section AA′
as a vertically
126
incident SH wave can be viewed in section BB′
as a vertically incident SV wave. Consequently, the
following relationships must hold for vertically incident SV and SH waves:
|u
SV (x1, 0, 0)| = |v
SH(0, x2, 0)|
|w
SV (x1, 0, 0)| = |w
SH(0, x2, 0)| (6.47)
|u
SV (0, x2, 0)| = |v
SH(x1, 0, 0)|
where the superscripts SV and SH denote the type of the incident wave.
The above properties can be easily seen from the results depicted by Figure 6.8 versus Figure 6.12,
Figure 6.22 versus Figure 6.26, and Figure 6.36 versus Figure 6.40.
(3) The surface response amplitudes along the line segment BB′
should be symmetric about the
x3−axis for all the five types of incidences, including the Love and Rayleigh surface waves, as well as
the vertically and obliquely incident P, SV, and SH waves in the x1x3−plane. This feature can be observed
by inspecting the surface displacement response due to different incident waves in Figure 6.4 - Figure 6.45.
Therefore, the symmetric features provide additional support to the validity of the results for the general three-dimensional model.
127
−4 −2 0 2 4
0
0.05
0.1
|u|
P3D θ
0
= 0 o
−4 −2 0 2 4
0
0.05
0.1
|v|
−4 −2 0 2 4
1.2
1.3
1.4
1.5
|w|
x
1
−4 −2 0 2 4
0
0.05
0.1
P3D θ
0
= 0 o
|u|
−4 −2 0 2 4
0
0.05
0.1
|v|
−4 −2 0 2 4
1.2
1.3
1.4
1.5
|w|
x
2
Figure 6.4: Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model subjected to incident P wave, θ0 = 0o
. Ω = 3. h = 1, a = 0.5, d = 0.2
−4 −2 0 2 4
1
1.5
2
|u|
P3D θ
0
= 30 o
−4 −2 0 2 4
0
0.05
0.1
|v|
−4 −2 0 2 4
1.2
1.4
1.6
|w|
x
1
−4 −2 0 2 4
1
1.5
2
P3D θ
0
= 30 o
|u|
−4 −2 0 2 4
0
0.05
0.1
|v|
−4 −2 0 2 4
1.2
1.4
1.6
|w|
x
2
Figure 6.5: Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model subjected to incident P wave, θ0 = 30o
. Ω = 3. h = 1, a = 0.5, d = 0.2
128
−4 −2 0 2 4
0.8
0.9
1
1.1
1.2
|u|
P3D θ
0
= 60 o
−4 −2 0 2 4
0
0.05
0.1
|v|
−4 −2 0 2 4
0.8
0.9
1
1.1
1.2
|w|
x
1
−4 −2 0 2 4
0.8
0.9
1
1.1
1.2
|u|
P3D θ
0
= 60 o
−4 −2 0 2 4
0
0.05
0.1
|v|
−4 −2 0 2 4
0.8
0.9
1
1.1
1.2
|w|
x
2
Figure 6.6: Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model subjected to incident P wave, θ0 = 60o
. Ω = 3. h = 1, a = 0.5, d = 0.2
−4 −2 0 2 4
0.3
0.35
0.4
|u|
P3D θ
0
= 85 o
−4 −2 0 2 4
0
0.01
0.02
0.03
0.04
|v|
−4 −2 0 2 4
0.35
0.4
0.45
|w|
x
1
−4 −2 0 2 4
0.3
0.35
0.4
|u|
P3D θ
0
= 85 o
−4 −2 0 2 4
0
0.01
0.02
0.03
0.04
|v|
−4 −2 0 2 4
0.35
0.4
0.45
|w|
x
2
Figure 6.7: Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model subjected to incident P wave, θ0 = 85o
. Ω = 3. h = 1, a = 0.5, d = 0.2
129
−4 −2 0 2 4
6
7
8
9
10
|u|
SV3D γ0
= 0 o
−4 −2 0 2 4
0
0.05
0.1
|v|
−4 −2 0 2 4
0
0.5
1
|w|
x
1
−4 −2 0 2 4
6
7
8
9
10
SV3D γ0
= 0 o
|u|
−4 −2 0 2 4
0
0.05
0.1
|v|
−4 −2 0 2 4
0
0.5
1
|w|
x
2
Figure 6.8: Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model subjected to incident SV wave, γ0 = 0o
. Ω = 3. h = 1, a = 0.5, d = 0.2
−4 −2 0 2 4
3
3.5
4
4.5
5
|u|
SV3D γ0
= 30 o
−4 −2 0 2 4
0
0.05
0.1
0.15
0.2
|v|
−4 −2 0 2 4
2
2.2
2.4
2.6
2.8
|w|
x
1
−4 −2 0 2 4
3
3.5
4
4.5
5
SV3D γ0
= 30 o
|u|
−4 −2 0 2 4
0
0.05
0.1
0.15
0.2
|v|
−4 −2 0 2 4
2
2.2
2.4
2.6
2.8
|w|
x
2
Figure 6.9: Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model subjected to incident SV wave, γ0 = 30o
. Ω = 3. h = 1, a = 0.5, d = 0.2
130
−4 −2 0 2 4
2.2
2.25
2.3
2.35
2.4
|u|
SV3D γ0
= 60 o
−4 −2 0 2 4
0
0.05
0.1
|v|
−4 −2 0 2 4
0.1
0.15
0.2
0.25
0.3
|w|
x
1
−4 −2 0 2 4
2.2
2.25
2.3
2.35
2.4
|u|
SV3D γ0
= 60 o
−4 −2 0 2 4
0
0.05
0.1
|v|
−4 −2 0 2 4
0.1
0.15
0.2
0.25
0.3
|w|
x
2
Figure 6.10: Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model subjected to incident SV wave, γ0 = 60o
. Ω = 3. h = 1, a = 0.5, d = 0.2
−4 −2 0 2 4
1.6
1.65
1.7
1.75
1.8
|u|
SV3D γ0
= 85 o
−4 −2 0 2 4
0
0.05
0.1
|v|
−4 −2 0 2 4
0.3
0.35
0.4
0.45
0.5
|w|
x
1
−4 −2 0 2 4
1.6
1.65
1.7
1.75
1.8
|u|
SV3D γ0
= 85 o
−4 −2 0 2 4
0
0.05
0.1
|v|
−4 −2 0 2 4
0.3
0.35
0.4
0.45
0.5
|w|
x
2
Figure 6.11: Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model subjected to incident SV wave, γ0 = 85o
. Ω = 3. h = 1, a = 0.5, d = 0.2
131
−4 −2 0 2 4
0
0.05
0.1
|u|
SH3D θ
0
= 0 o
−4 −2 0 2 4
6
7
8
9
10
|v|
−4 −2 0 2 4
0
0.5
1
|w|
x
1
−4 −2 0 2 4
0
0.05
0.1
SH3D θ
0
= 0 o
|u|
−4 −2 0 2 4
6
7
8
9
10
|v|
−4 −2 0 2 4
0
0.5
1
|w|
x
2
Figure 6.12: Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model subjected to incident SH wave, θ0 = 0o
. Ω = 3. h = 1, a = 0.5, d = 0.2
−4 −2 0 2 4
0
0.05
0.1
0.15
0.2
|u|
SH3D θ
0
= 30 o
−4 −2 0 2 4
5
6
7
8
|v|
−4 −2 0 2 4
0
0.5
1
|w|
x
1
−4 −2 0 2 4
0
0.05
0.1
0.15
0.2
SH3D θ
0
= 30 o
|u|
−4 −2 0 2 4
5
6
7
8
|v|
−4 −2 0 2 4
0
0.5
1
|w|
x
2
Figure 6.13: Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model subjected to incident SH wave, θ0 = 30o
. Ω = 3. h = 1, a = 0.5, d = 0.2
132
−4 −2 0 2 4
0
0.1
0.2
0.3
0.4
|u|
SH3D θ
0
= 60 o
−4 −2 0 2 4
2.5
3
3.5
4
|v|
−4 −2 0 2 4
0
0.1
0.2
0.3
0.4
|w|
x
1
−4 −2 0 2 4
0
0.1
0.2
0.3
0.4
|u|
SH3D θ
0
= 60 o
−4 −2 0 2 4
2.5
3
3.5
4
|v|
−4 −2 0 2 4
0
0.1
0.2
0.3
0.4
|w|
x
2
Figure 6.14: Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model subjected to incident SH wave, θ0 = 60o
. Ω = 3. h = 1, a = 0.5, d = 0.2
−4 −2 0 2 4
0
0.05
0.1
0.15
0.2
|u|
SH3D θ
0
= 85 o
−4 −2 0 2 4
0.8
0.9
1
1.1
|v|
−4 −2 0 2 4
0
0.05
0.1
|w|
x
1
−4 −2 0 2 4
0
0.05
0.1
0.15
0.2
|u|
SH3D θ
0
= 85 o
−4 −2 0 2 4
0.8
0.9
1
1.1
|v|
−4 −2 0 2 4
0
0.05
0.1
|w|
x
2
Figure 6.15: Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model subjected to incident SH wave, θ0 = 85o
. Ω = 3. h = 1, a = 0.5, d = 0.2
133
Figure 6.16: Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model subjected to incident Love wave. Ω = 3.04. h = 1, a = 0.5, d = 0.2
−4 −2 0 2 4
0.3
0.35
0.4
0.45
0.5
|u|
R3D
−4 −2 0 2 4
0
0.005
0.01
0.015
0.02
|v|
−4 −2 0 2 4
0.65
0.7
0.75
|w|
x
1
−4 −2 0 2 4
0.3
0.35
0.4
0.45
0.5
|u|
R3D
−4 −2 0 2 4
0
0.005
0.01
0.015
0.02
|v|
−4 −2 0 2 4
0.65
0.7
0.75
|w|
x
2
Figure 6.17: Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model subjected to incident Rayleigh wave. Ω = 3. h = 1, a = 0.5, d = 0.2
134
−4 −2 0 2 4
0
0.01
0.02
0.03
0.04
|u|
P3D θ
0
= 0 o
−4 −2 0 2 4
0
0.01
0.02
0.03
0.04
|v|
−4 −2 0 2 4
0.98
0.99
1
1.01
1.02
|w|
x
1
−4 −2 0 2 4
0
0.01
0.02
0.03
0.04
P3D θ
0
= 0 o
|u|
−4 −2 0 2 4
0
0.01
0.02
0.03
0.04
|v|
−4 −2 0 2 4
0.98
0.99
1
1.01
1.02
|w|
x
2
Figure 6.18: Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model subjected to incident P wave, θ0 = 0o
. Ω = 4. h = 1, a = 0.5, d = 0.2
−4 −2 0 2 4
0.3
0.35
0.4
0.45
0.5
|u|
P3D θ
0
= 30 o
−4 −2 0 2 4
0
0.01
0.02
0.03
0.04
|v|
−4 −2 0 2 4
0.9
0.95
1
|w|
x
1
−4 −2 0 2 4
0.3
0.35
0.4
0.45
0.5
P3D θ
0
= 30 o
|u|
−4 −2 0 2 4
0
0.01
0.02
0.03
0.04
|v|
−4 −2 0 2 4
0.9
0.95
1
|w|
x
2
Figure 6.19: Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model subjected to incident P wave, θ0 = 30o
. Ω = 4. h = 1, a = 0.5, d = 0.2
135
−4 −2 0 2 4
0.4
0.5
0.6
0.7
0.8
|u|
P3D θ
0
= 60 o
−4 −2 0 2 4
0
0.01
0.02
0.03
0.04
|v|
−4 −2 0 2 4
0.7
0.75
0.8
0.85
0.9
|w|
x
1
−4 −2 0 2 4
0.4
0.5
0.6
0.7
0.8
|u|
P3D θ
0
= 60 o
−4 −2 0 2 4
0
0.01
0.02
0.03
0.04
|v|
−4 −2 0 2 4
0.7
0.75
0.8
0.85
0.9
|w|
x
2
Figure 6.20: Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model subjected to incident P wave, θ0 = 60o
. Ω = 4. h = 1, a = 0.5, d = 0.2
−4 −2 0 2 4
0.12
0.14
0.16
0.18
0.2
|u|
P3D θ
0
= 85 o
−4 −2 0 2 4
0
0.005
0.01
|v|
−4 −2 0 2 4
0.18
0.19
0.2
0.21
0.22
|w|
x
1
−4 −2 0 2 4
0.12
0.14
0.16
0.18
0.2
|u|
P3D θ
0
= 85 o
−4 −2 0 2 4
0
0.005
0.01
|v|
−4 −2 0 2 4
0.18
0.19
0.2
0.21
0.22
|w|
x
2
Figure 6.21: Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model subjected to incident P wave, θ0 = 85o
. Ω = 4. h = 1, a = 0.5, d = 0.2
136
−4 −2 0 2 4
1.8
1.9
2
2.1
2.2
|u|
SV3D γ0
= 0 o
−4 −2 0 2 4
0
0.05
0.1
|v|
−4 −2 0 2 4
0
0.05
0.1
|w|
x
1
−4 −2 0 2 4
1.8
1.9
2
2.1
2.2
SV3D γ0
= 0 o
|u|
−4 −2 0 2 4
0
0.05
0.1
|v|
−4 −2 0 2 4
0
0.05
0.1
|w|
x
2
Figure 6.22: Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model subjected to incident SV wave, γ0 = 0o
. Ω = 4. h = 1, a = 0.5, d = 0.2
−4 −2 0 2 4
3.2
3.4
3.6
3.8
4
4.2
|u|
SV3D γ0
= 30 o
−4 −2 0 2 4
0
0.05
0.1
0.15
0.2
|v|
−4 −2 0 2 4
0
0.5
1
|w|
x
1
−4 −2 0 2 4
3.2
3.4
3.6
3.8
4
4.2
SV3D γ0
= 30 o
|u|
−4 −2 0 2 4
0
0.05
0.1
0.15
0.2
|v|
−4 −2 0 2 4
0
0.5
1
|w|
x
2
Figure 6.23: Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model subjected to incident SV wave, γ0 = 30o
. Ω = 4. h = 1, a = 0.5, d = 0.2
137
−4 −2 0 2 4
1.6
1.7
1.8
1.9
2
|u|
SV3D γ0
= 60 o
−4 −2 0 2 4
0
0.05
0.1
0.15
0.2
|v|
−4 −2 0 2 4
0.5
0.6
0.7
0.8
0.9
|w|
x
1
−4 −2 0 2 4
1.6
1.7
1.8
1.9
2
|u|
SV3D γ0
= 60 o
−4 −2 0 2 4
0
0.05
0.1
0.15
0.2
|v|
−4 −2 0 2 4
0.5
0.6
0.7
0.8
0.9
|w|
x
2
Figure 6.24: Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model subjected to incident SV wave, γ0 = 60o
. Ω = 4. h = 1, a = 0.5, d = 0.2
−4 −2 0 2 4
0.35
0.4
0.45
|u|
SV3D γ0
= 85 o
−4 −2 0 2 4
0
0.01
0.02
0.03
0.04
|v|
−4 −2 0 2 4
0.19
0.2
0.21
0.22
|w|
x
1
−4 −2 0 2 4
0.35
0.4
0.45
|u|
SV3D γ0
= 85 o
−4 −2 0 2 4
0
0.01
0.02
0.03
0.04
|v|
−4 −2 0 2 4
0.19
0.2
0.21
0.22
|w|
x
2
Figure 6.25: Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model subjected to incident SV wave, γ0 = 85o
. Ω = 4. h = 1, a = 0.5, d = 0.2
138
−4 −2 0 2 4
0
0.05
0.1
|u|
SH3D θ
0
= 0 o
−4 −2 0 2 4
1.8
1.9
2
2.1
2.2
|v|
−4 −2 0 2 4
0
0.05
0.1
|w|
x
1
−4 −2 0 2 4
0
0.05
0.1
SH3D θ
0
= 0 o
|u|
−4 −2 0 2 4
1.8
1.9
2
2.1
2.2
|v|
−4 −2 0 2 4
0
0.05
0.1
|w|
x
2
Figure 6.26: Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model subjected to incident SH wave, θ0 = 0o
. Ω = 4. h = 1, a = 0.5, d = 0.2
−4 −2 0 2 4
0
0.1
0.2
0.3
0.4
|u|
SH3D θ
0
= 30 o
−4 −2 0 2 4
1.8
2
2.2
2.4
|v|
−4 −2 0 2 4
0
0.01
0.02
0.03
0.04
|w|
x
1
−4 −2 0 2 4
0
0.1
0.2
0.3
0.4
SH3D θ
0
= 30 o
|u|
−4 −2 0 2 4
1.8
2
2.2
2.4
|v|
−4 −2 0 2 4
0
0.01
0.02
0.03
0.04
|w|
x
2
Figure 6.27: Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model subjected to incident SH wave, θ0 = 30o
. Ω = 4. h = 1, a = 0.5, d = 0.2
139
−4 −2 0 2 4
0
0.5
1
|u|
SH3D θ
0
= 60 o
−4 −2 0 2 4
2
2.5
3
|v|
−4 −2 0 2 4
0
0.5
1
|w|
x
1
−4 −2 0 2 4
0
0.5
1
|u|
SH3D θ
0
= 60 o
−4 −2 0 2 4
2
2.5
3
|v|
−4 −2 0 2 4
0
0.05
0.1
|w|
x
2
Figure 6.28: Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model subjected to incident SH wave, θ0 = 60o
. Ω = 4. h = 1, a = 0.5, d = 0.2
−4 −2 0 2 4
0
0.1
0.2
0.3
0.4
|u|
SH3D θ
0
= 85 o
−4 −2 0 2 4
0.8
0.9
1
1.1
1.2
|v|
−4 −2 0 2 4
0
0.05
0.1
|w|
x
1
−4 −2 0 2 4
0
0.1
0.2
0.3
0.4
|u|
SH3D θ
0
= 85 o
−4 −2 0 2 4
0.8
0.9
1
1.1
1.2
|v|
−4 −2 0 2 4
0
0.05
0.1
|w|
x
2
Figure 6.29: Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model subjected to incident SH wave, θ0 = 85o
. Ω = 4. h = 1, a = 0.5, d = 0.2
140
Figure 6.30: Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model subjected to incident Love wave. Ω = 4. h = 1, a = 0.5, d = 0.2
−4 −2 0 2 4
0.25
0.3
0.35
|u|
R3D
−4 −2 0 2 4
0
0.005
0.01
|v|
−4 −2 0 2 4
0.43
0.44
0.45
0.46
0.47
|w|
x
1
−4 −2 0 2 4
0.25
0.3
0.35
|u|
R3D
−4 −2 0 2 4
0
0.005
0.01
|v|
−4 −2 0 2 4
0.43
0.44
0.45
0.46
0.47
|w|
x
2
Figure 6.31: Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model subjected to incident Rayleigh wave. Ω = 4. h = 1, a = 0.5, d = 0.2
141
Following the symmetry analysis of the model response, the attention is turned to amplification of the
surface motion due to the presence of layer irregularity for different incident waves.
For a vertically incident P wave with Ω = 4 in the x1x3−plane shown by Figure 6.18, it is apparent
that in addition to the predominant component of motion w in the vertical direction, the model generates
one non-predominant component of motion (u for section AA′
and v for section BB′
) in the horizontal
direction. However, for an obliquely incident P wave in the x1x3−plane, in addition to the vertical predominant component of motion w, the model may generate both horizontal components of motion for
section BB′
(Figure 6.19). In other words, a mode conversion takes place for section BB′
see Figure 6.19
- Figure 6.21 for Ω = 4.
For a vertically incident SV wave with Ω = 4 in the x1x3−plane, the surface response is depicted
by Figure 6.22. The response consists of the predominant horizontal component of motion u for both
sections AA′
and BB′
, and one non-predominant vertical component w for section AA′
. However, for
an oblique SV incidence in the x1x3−plane, all three components of motion are present for section BB′
,
see Figure 6.23 - Figure 6.25 for Ω = 4. The additional horizontal component of motion v and vertical
component of motion w have been generated through a mode conversion.
A similar mode conversion phenomenon can be seen for an incident SH wave with Ω = 4 in the
x1x3−plane. Namely, for a vertical incidence, only the predominant horizontal component of motion v
for both sections AA′
and BB′
and one non-predominant vertical component of motion w for section
BB′ have been generated, see Figure 6.26. However, for an oblique incidence, all the three components of
motion are present for section BB′
, see Figure 6.27 through Figure 6.29.
The surface response due to an incident Love wave with Ω = 4 is displayed by Figure 6.30. The predominant component v is present in the surface response. All three components of motion are generated
for section BB′
.
142
Finally, Figure 6.31 depicts the surface response due to an incident Rayleigh waves with Ω = 4 in the
x1x3−plane. The two predominant components u and w are present in the surface response. All three
components of motion are generated for section BB′
.
Similar results have been obtained for various waves at different frequencies, see Figure 6.4 through
Figure 6.17 for Ω = 3, and Figure 6.32 through Figure 6.45 for Ω = 5.
143
−4 −2 0 2 4
0
0.1
0.2
0.3
0.4
|u|
P3D θ
0
= 0 o
−4 −2 0 2 4
0
0.1
0.2
0.3
0.4
|v|
−4 −2 0 2 4
1.3
1.35
1.4
1.45
1.5
|w|
x
1
−4 −2 0 2 4
0
0.1
0.2
0.3
0.4
P3D θ
0
= 0 o
|u|
−4 −2 0 2 4
0
0.1
0.2
0.3
0.4
|v|
−4 −2 0 2 4
1.3
1.35
1.4
1.45
1.5
|w|
x
2
Figure 6.32: Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model subjected to incident P wave, θ0 = 0o
. Ω = 5. h = 1, a = 0.5, d = 0.2
−4 −2 0 2 4
0
1
2
3
|u|
P3D θ
0
= 30 o
−4 −2 0 2 4
0
0.1
0.2
0.3
0.4
|v|
−4 −2 0 2 4
1
1.2
1.4
|w|
x
1
−4 −2 0 2 4
0
1
2
3
P3D θ
0
= 30 o
|u|
−4 −2 0 2 4
0
0.1
0.2
0.3
0.4
|v|
−4 −2 0 2 4
1
1.2
1.4
|w|
x
2
Figure 6.33: Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model subjected to incident P wave, θ0 = 30o
. Ω = 5. h = 1, a = 0.5, d = 0.2
144
−4 −2 0 2 4
0.5
1
1.5
2
|u|
P3D θ
0
= 60 o
−4 −2 0 2 4
0
0.1
0.2
0.3
0.4
|v|
−4 −2 0 2 4
0.5
1
1.5
|w|
x
1
−4 −2 0 2 4
0.5
1
1.5
2
|u|
P3D θ
0
= 60 o
−4 −2 0 2 4
0
0.1
0.2
0.3
0.4
|v|
−4 −2 0 2 4
0.5
1
1.5
|w|
x
2
Figure 6.34: Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model subjected to incident P wave, θ0 = 60o
. Ω = 5. h = 1, a = 0.5, d = 0.2
−4 −2 0 2 4
0
0.1
0.2
0.3
0.4
|u|
P3D θ
0
= 85 o
−4 −2 0 2 4
0
0.01
0.02
0.03
0.04
|v|
−4 −2 0 2 4
0.1
0.15
0.2
0.25
0.3
|w|
x
1
−4 −2 0 2 4
0
0.1
0.2
0.3
0.4
|u|
P3D θ
0
= 85 o
−4 −2 0 2 4
0
0.01
0.02
0.03
0.04
|v|
−4 −2 0 2 4
0.1
0.15
0.2
0.25
0.3
|w|
x
2
Figure 6.35: Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model subjected to incident P wave, θ0 = 85o
. Ω = 5. h = 1, a = 0.5, d = 0.2
145
−4 −2 0 2 4
4
6
8
10
|u|
SV3D γ0
= 0 o
−4 −2 0 2 4
0
0.05
0.1
|v|
−4 −2 0 2 4
0
0.5
1
1.5
2
|w|
x
1
−4 −2 0 2 4
4
6
8
10
SV3D γ0
= 0 o
|u|
−4 −2 0 2 4
0
0.05
0.1
|v|
−4 −2 0 2 4
0
0.5
1
1.5
2
|w|
x
2
Figure 6.36: Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model subjected to incident SV wave, γ0 = 0o
. Ω = 5. h = 1, a = 0.5, d = 0.2
−4 −2 0 2 4
3
4
5
6
|u|
SV3D γ0
= 30 o
−4 −2 0 2 4
0
0.1
0.2
0.3
0.4
|v|
−4 −2 0 2 4
0
0.5
1
|w|
x
1
−4 −2 0 2 4
3
4
5
6
SV3D γ0
= 30 o
|u|
−4 −2 0 2 4
0
0.1
0.2
0.3
0.4
|v|
−4 −2 0 2 4
0
0.5
1
|w|
x
2
Figure 6.37: Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model subjected to incident SV wave, γ0 = 30o
. Ω = 5. h = 1, a = 0.5, d = 0.2
146
−4 −2 0 2 4
6
7
8
9
10
|u|
SV3D γ0
= 60 o
−4 −2 0 2 4
0
0.5
1
|v|
−4 −2 0 2 4
0
1
2
3
4
|w|
x
1
−4 −2 0 2 4
6
7
8
9
10
|u|
SV3D γ0
= 60 o
−4 −2 0 2 4
0
0.5
1
|v|
−4 −2 0 2 4
0
1
2
3
4
|w|
x
2
Figure 6.38: Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model subjected to incident SV wave, γ0 = 60o
. Ω = 5. h = 1, a = 0.5, d = 0.2
−4 −2 0 2 4
0.3
0.35
0.4
0.45
0.5
|u|
SV3D γ0
= 85 o
−4 −2 0 2 4
0
0.01
0.02
0.03
0.04
|v|
−4 −2 0 2 4
0.2
0.25
0.3
0.35
|w|
x
1
−4 −2 0 2 4
0.3
0.35
0.4
0.45
0.5
|u|
SV3D γ0
= 85 o
−4 −2 0 2 4
0
0.01
0.02
0.03
0.04
|v|
−4 −2 0 2 4
0.2
0.25
0.3
0.35
|w|
x
2
Figure 6.39: Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model subjected to incident SV wave, γ0 = 85o
. Ω = 5. h = 1, a = 0.5, d = 0.2
147
−4 −2 0 2 4
0
0.05
0.1
|u|
SH3D θ
0
= 0 o
−4 −2 0 2 4
4
6
8
10
|v|
−4 −2 0 2 4
0
0.5
1
1.5
2
|w|
x
1
−4 −2 0 2 4
0
0.05
0.1
SH3D θ
0
= 0 o
|u|
−4 −2 0 2 4
4
6
8
10
|v|
−4 −2 0 2 4
0
0.5
1
1.5
2
|w|
x
2
Figure 6.40: Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model subjected to incident SH wave, θ0 = 0o
. Ω = 5. h = 1, a = 0.5, d = 0.2
−4 −2 0 2 4
0
0.5
1
|u|
SH3D θ
0
= 30 o
−4 −2 0 2 4
0
5
10
|v|
−4 −2 0 2 4
0
0.5
1
|w|
x
1
−4 −2 0 2 4
0
0.5
1
SH3D θ
0
= 30 o
|u|
−4 −2 0 2 4
0
5
10
|v|
−4 −2 0 2 4
0
0.5
1
|w|
x
2
Figure 6.41: Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model subjected to incident SH wave, θ0 = 30o
. Ω = 5. h = 1, a = 0.5, d = 0.2
148
−4 −2 0 2 4
0
0.5
1
|u|
SH3D θ
0
= 60 o
−4 −2 0 2 4
1
2
3
4
5
|v|
−4 −2 0 2 4
0
0.5
1
|w|
x
1
−4 −2 0 2 4
0
0.5
1
|u|
SH3D θ
0
= 60 o
−4 −2 0 2 4
1
2
3
4
5
|v|
−4 −2 0 2 4
0
0.5
1
|w|
x
2
Figure 6.42: Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model subjected to incident SH wave, θ0 = 60o
. Ω = 5. h = 1, a = 0.5, d = 0.2
−4 −2 0 2 4
0
0.5
1
|u|
SH3D θ
0
= 85 o
−4 −2 0 2 4
0
1
2
3
|v|
−4 −2 0 2 4
0
0.1
0.2
0.3
0.4
|w|
x
1
−4 −2 0 2 4
0
0.5
1
|u|
SH3D θ
0
= 85 o
−4 −2 0 2 4
0
1
2
3
|v|
−4 −2 0 2 4
0
0.1
0.2
0.3
0.4
|w|
x
2
Figure 6.43: Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model subjected to incident SH wave, θ0 = 85o
. Ω = 5. h = 1, a = 0.5, d = 0.2
149
Figure 6.44: Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model subjected to incident Love wave. Ω = 5.04. h = 1, a = 0.5, d = 0.2
−4 −2 0 2 4
0.15
0.16
0.17
0.18
|u|
R3D
−4 −2 0 2 4
0
0.005
0.01
|v|
−4 −2 0 2 4
0.24
0.245
0.25
0.255
0.26
|w|
x
1
−4 −2 0 2 4
0.15
0.16
0.17
0.18
|u|
R3D
−4 −2 0 2 4
0
0.005
0.01
|v|
−4 −2 0 2 4
0.24
0.245
0.25
0.255
0.26
|w|
x
2
Figure 6.45: Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model subjected to incident Rayleigh wave. Ω = 5. h = 1, a = 0.5, d = 0.2
150
As in the plane strain model, the influence of the deviation d of the rotating cosine-shaped scatterer
upon the steady-state surface response is investigated. The surface displacement is calculated for incident
plane harmonic P, SV, SH, Love and Rayleigh waves for various scatterer deviation values for a dimensionless frequency Ω = 4. Here, d < 0 represents a rotating cosine-shaped scatterer curved upward while
d > 0 represents a rotating cosine-shaped scatterer curved downward.
Figure 6.46 - Figure 6.50 demonstrate the surface displacement field for different incident waves. The
surface response is very sensitive to the magnitude of the irregularity on the interface.
The resonance features for the three-dimensional models are omitted for the sake of brevity.
This concludes the steady-state analysis of the one-layer three-dimensional model.
−4 −2 0 2 4
0
0.05
0.1
0.15
0.2
|u|
P3D θ
0
= 0 o
−4 −2 0 2 4
0
0.05
0.1
0.15
0.2
|v|
−4 −2 0 2 4
0.9
1
1.1
1.2
1.3
|w|
x
1
−4 −2 0 2 4
0
0.05
0.1
0.15
0.2
P3D θ
0
= 0 o
|u|
−4 −2 0 2 4
0
0.05
0.1
0.15
0.2
|v|
−4 −2 0 2 4
0.9
1
1.1
1.2
1.3
|w|
x
2
Figure 6.46: Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model with various scatterer deviations d subjected to a vertically incident P wave. Ω = 4.
h = 1, a = 0.5. Dash: d = −0.4, dash-dot: d = −0.2, solid: d = 0, plus: d = 0.2, dot: d = 0.4
151
−4 −2 0 2 4
1
1.5
2
2.5
3
|u|
SV3D γ0
= 0 o
−4 −2 0 2 4
0
0.05
0.1
|v|
−4 −2 0 2 4
0
0.5
1
|w|
x
1
−4 −2 0 2 4
1
1.5
2
2.5
3
SV3D γ0
= 0 o
|u|
−4 −2 0 2 4
0
0.05
0.1
|v|
−4 −2 0 2 4
0
0.5
1
|w|
x
2
Figure 6.47: Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model with various scatterer deviations d subjected to a vertically incident SV wave. Ω = 4.
h = 1, a = 0.5. Dash: d = −0.4, dash-dot: d = −0.2, solid: d = 0, plus: d = 0.2, dot: d = 0.4
−4 −2 0 2 4
0
0.05
0.1
|u|
SH3D θ
0
= 0 o
−4 −2 0 2 4
1
1.5
2
2.5
3
|v|
−4 −2 0 2 4
0
0.5
1
|w|
x
1
−4 −2 0 2 4
0
0.05
0.1
SH3D θ
0
= 0 o
|u|
−4 −2 0 2 4
1
1.5
2
2.5
3
|v|
−4 −2 0 2 4
0
0.5
1
|w|
x
2
Figure 6.48: Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model with various scatterer deviations d subjected to a vertically incident SH wave. Ω = 4.
h = 1, a = 0.5. Dash: d = −0.4, dash-dot: d = −0.2, solid: d = 0, plus: d = 0.2, dot: d = 0.4
152
Figure 6.49: Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model with various scatterer deviations d subjected to an incident Love surface wave. Ω = 4.
h = 1, a = 0.5. Dot: d = −0.4, dash-dot: d = −0.2, point: d = 0, dash: d = 0.2, solid: d = 0.4
−4 −2 0 2 4
0.2
0.25
0.3
0.35
0.4
|u|
R3D
−4 −2 0 2 4
0
0.01
0.02
0.03
0.04
|v|
−4 −2 0 2 4
0.4
0.45
0.5
|w|
x
1
−4 −2 0 2 4
0.2
0.25
0.3
0.35
0.4
|u|
R3D
−4 −2 0 2 4
0
0.01
0.02
0.03
0.04
|v|
−4 −2 0 2 4
0.4
0.45
0.5
|w|
x
2
Figure 6.50: Steady-state surface displacement along the x1−axis and x2−axis for a one-layer threedimensional model with various scatterer deviations d subjected to an incident Rayleigh surface wave.
Ω = 4. h = 1, a = 0.5. Dash: d = −0.4, dash-dot: d = −0.2, solid: d = 0, plus: d = 0.2, dot: d = 0.4
153
6.4 Conclusions
Scattering of elastic waves in a media of three-dimensional multiple layers with irregular interfaces has
been formulated using an indirect boundary integral equation approach. The total wave field is assumed to
be a superposition of the free field and scattered wave field. The unknown scattered wave field is expressed
in terms of single-layer potentials involving the point load full-space Green’s functions and the unknown
density functions. The densities are determined in the least-square sense.
An axisymmetric one-layer geometry was selected for detailed numerical investigation of the threedimensional model. Parametric error analysis was applied to estimate the optimum parameters of the
model, such as the location and number of sources on the auxiliary surfaces, and the location and number of collocation points on the interface and the top surface. Subsequently, the steady-state surface response was calculated using the optimum parameters for incident P, SV, SH, Love, and Rayleigh waves.
The presented numerical results show that, for vertically incident elastic waves, an axisymmetric threedimensional geometry will produce certain symmetric patterns for the surface response. The symmetry
analysis provided additional support for the validity of the three-dimensional results.
It was also shown that the surface response near the irregularities is very sensitive to the nature of
the incident waves and the location of observation points. In addition, it was shown that for oblique
incidence, mode conversion may take place resulting in all three displacement components to be present
in the surface response.
Finally, the surface responses are found to be very sensitive to the shape of the irregular interfaces.
154
Chapter 7
Summary and Conclusions
Scattering of elastic waves in multilayered media with irregular interfaces was formulated using an indirect
boundary integral equation approach for antiplane strain, plane strain, and three-dimensional models.
Both the half- and the full-space Green’s function approaches were applied to evaluate the scattered wave
field for the antiplane strain model respectively, while the full-space Green’s function approach was used
to simulate the scattered waves for the plane strain and three-dimensional models. The total wave field
is assumed to be a superposition of the scattered wave field and the free field of the corresponding flatlayer model. The unknown scattered wave field is expressed in terms of single-layer potentials involving
Green’s functions with unknown density functions, which are determined in the least square sense.
A systematic parametric error analysis was performed to investigate the role of different parameters
upon the surface response. These parameters, such as the location of auxiliary surfaces, the distribution
of sources along the auxiliary surfaces and collocation points along the interfaces, were determined by
minimizing an error on the surface response iterations. Based upon these parameters, numerical results
for steady-state surface response have been evaluated for the two- and three-dimensional models for the
incident P, SV, SH, Love, and Rayleigh waves for a range of frequencies.
155
The presented numerical results show that, for vertically incident elastic waves, a symmetric geometry
may result in certain symmetries for the surface response. This particular feature provides additional
support for the validity of the proposed solution.
It was found that the steady-state response near the scatterer strongly depends upon the nature of the
incident waves (the wave type, angle of incidence, and frequency), the location of observation points, and
the shape of the irregularities.
Another important feature is the resonance observed in the surface response at observation sites directly above an irregular interface. These resonance frequencies are closely related to those of the corresponding flat layer. The differences in resonance frequencies and amplitudes between models with and
without irregular scatterers arise from the presence of interface irregularities.
Finally, it was shown that a slight variation in the scatterer shape may significantly change the pattern
and amplification of the surface response.
The new contributions for each model in this study are as follows:
• For the antiplane strain model, transient response was also presented for the one- and two-layer
models incorporating the half-space Green’s functions in addition to the steady-state numerical results. Figure 3.11 shows that the transient surface response of a one-layer model with an irregular
interface, induced by a vertically incident SH Ricker wavelet, clearly reveals the location and amplitude of the irregular scatterer, and the thickness of the top layer away from the scatterer. Therefore,
the study of both steady-state and transient surface responses offers a solid theoretical foundation
for experimental earthquake engineering research, including investigations into ground motion amplification, subsurface structure, and other related topics [33] [34].
• For the antiplane strain model using full-space Green’s functions, it is found that in a two-layer
model, the Love wave surface response is dominated by the irregularity of the upper interface due
156
to the effect of Saint-Venant’s Principle and the rapid decay of the Love wave amplitude as x3 increases with depth, which results in a significant reduction in the scattering from the lower irregular
interface(s). A further transparency test was conducted for the two-layer antiplane strain model with
an incident Love wave by letting the material properties in the lower two domains be identical to
simulate the one-layer antiplane strain model with the same incidence. A comparison of the Love
wave surface displacements between the transparency two-layer model and the one-layer model reveals a good agreement, confirming the accuracy of the two-dimensional one- and two-layer Love
wave formulations.
• For the plane strain model, a further transparency test was conducted for the one-layer plane strain
model with an incident Rayleigh wave by letting the material properties in the two domains be
identical to simulate the pure half-space plane strain model with the same incidence. A comparison
of the Rayleigh wave surface displacements between the transparency one-layer model and the pure
half-space model shows a good agreement, providing further validity of the two-dimensional onelayer model and pure half-space model for the incident Rayleigh wave.
• For the three-dimensional model, the scattering of incident P, SV, SH, Love, and Rayleigh waves in
a multilayered media was formulated using the full-space Green’s functions. The optimum values
of the numerical parameters have been determined. Subsequently, the steady-state response was
obtained for a one-layer axisymmetric model. Furthermore, the three-dimensional results indicate
that certain mode conversions may occur in the surface response, which cannot be present in the
two-dimensional model. These findings demonstrate the need for three-dimensional modeling to
accurately describe motion amplification in irregular layered media.
157
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158
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169
Appendices
170
Appendix A
Antiplane Strain Model
A.1 Half-Space Green’s Function and Matrix System
The half-space Green’s function for the antiplane strain model is specified by equation (3.14).
The matrix A of size (2PJ
j=1 Nj ) × (M +
PJ
j=1 Lj1 +
PJ−1
j=1 Lj2) in equation (3.25) is defined by
171
A =
G
(1)
0 −G
(1)
1 0 ... 0
µ0
∂G
(1)
0
∂n −µ1
∂G
(1)
1
∂n 0 ... 0
0 G(2)
1 −G
(2)
2 0
0 µ1
∂G
(2)
1
∂n −µ2
∂G
(2)
2
∂n 0
.
.
.
.
.
.
.
.
.
.
.
.
0 G(j)
j−1 −G
(j)
j
0
0 µj−1
∂G
(j)
j−1
∂n −µj
∂G
(j)
j
∂n 0
.
.
.
.
.
.
.
.
.
.
.
.
0 G(J)
J−1 −G
(J)
J
0 µJ−1
∂G
(J)
J−1
∂n −µJ
∂G
(J)
J
∂n
(A.1)
where
G
(1)
0 = [G0(rn1
, rm)]; rn1 ∈ C1; n1 = 1, 2, ..., N1;
rm ∈ C00; m = 1, 2, ..., M (A.2)
G
(j)
i
(nj , li) = [Gi(rnj
, rli
)]; rnj ∈ Cj ; rli ∈ Ci1 ∪ Ci2; j = i, i + 1; (A.3)
i = 1, 2, ..., J; nj = 1, 2, ..., Nj ; li = 1, 2, ..., (Li1 + Li2) (A.4)
172
Here, the superscript (j) in the Green’s function matrix G
(j)
i
represents the j
th interface. In particular,
the number of sources LJ2 = 0 since the scattered wave field in the domain DJ is caused by the sources
along the auxiliary surface CJ1 only.
The forcing vector f in equation (3.25) is of the order (2PJ
j=1 Nj ) and specified by
f =
v
f f
11 − v
f f
10
µ1
∂v
ff
11
∂n − µ0
∂v
ff
10
∂n
v
f f
22 − v
f f
21
µ2
∂v
ff
22
∂n − µ1
∂v
ff
21
∂n
.
.
.
v
f f
jj − v
f f
j,j−1
µj
∂v
ff
jj
∂n − µj−1
∂v
ff
j,j−1
∂n
.
.
.
v
f f
JJ − v
f f
J,J−1
µJ
∂v
ff
JJ
∂n − µJ−1
∂v
ff
J,J−1
∂n
(A.5)
where the submatrices defined by
v
f f
ji = [v
f f
i
(rnj
)]; rnj ∈ Cj ; nj = 1, 2, ..., Nj ; j = 1, 2, ..., J; i = j − 1, j (A.6)
are of the order Nj . The first subscript j represents the j
th interface and the second subscript i represents
the i
th domain.
173
The unknown source intensity vector a of the order (M +
PJ
j=1(Lj1 + Lj2)) in equation (3.25) is
given by
a =
am
b1
b2
.
.
.
bj
.
.
.
bJ
(A.7)
where
am = [am]; m = 1, 2, ..., M (A.8)
bj = [blj
]; lj = 1, 2, ... , Lj1, Lj1 + 1, ... , Lj1 + Lj2; LJ2 = 0 (A.9)
174
A.2 Full-Space Green’s Function and Matrix System
The full-space Green’s function for the antiplane strain model is specified by equation (4.6).
The matrix A of the order (2PJ
j=1 Nj + P) × (M +
PJ
j=1(Lj1 + Lj2)) in equation (4.15) is defined
by
A =
G
(1)
0 −G
(1)
1 0 ... 0
µ0
∂G
(1)
0
∂n −µ1
∂G
(1)
1
∂n 0 ... 0
0 G(2)
1 −G
(2)
2
... 0
0 µ1
∂G
(2)
1
∂n −µ2
∂G
(2)
2
∂n ... 0
.
.
.
.
.
.
.
.
.
.
.
.
0 ... G
(j)
j−1 −G
(j)
j
0
0 µj−1
∂G
(j)
j−1
∂n −µj
∂G
(j)
j
∂n 0
.
.
.
.
.
.
.
.
.
.
.
.
0 ... 0 G(J)
J−1 −G
(J)
J
0 ... 0 µJ−1
∂G
(J)
J−1
∂n −µJ
∂G
(J)
J
∂n
0 ... 0 µJ
∂G
(J+1)
J
∂x3
(A.10)
where
175
G
(1)
0
(n1, m) = G0(rn1
, rm); rn1 ∈ C1; n1 = 1, 2, ..., N1; rm ∈ C00; m = 1, 2, ...M (A.11)
G
(j)
i
(nj , li) = Gi(rnj
, rli
); rnj ∈ Cj ; rli ∈ Ci1 ∪ Ci2; j = i, i + 1; i = 1, 2, ..., J;
nj = 1, 2, ..., Nj ; li = 1, 2, ...,(Li1 + Li2) (A.12)
Here, the superscript (j) in the Green’s function matrix G
(j)
i
represents the j
th interface. In particular,
the superscript (J + 1) represents the top free surface SF and NJ+1 is known as P.
The forcing vector f in equation (4.15) is of the order (2PJ
j=1 Nj + P) and is given by
176
f =
v
f f
11 − v
f f
10
µ1
∂v
ff
11
∂n − µ0
∂v
ff
10
∂n
v
f f
22 − v
f f
21
µ2
∂v
ff
22
∂n − µ1
∂v
ff
21
∂n
.
.
.
v
f f
jj − v
f f
j,j−1
µj
∂v
ff
jj
∂n − µj−1
∂v
ff
j,j−1
∂n
.
.
.
v
f f
JJ − v
f f
J,J−1
µJ
∂v
ff
JJ
∂n − µJ−1
∂v
ff
J,J−1
∂n
0
(A.13)
where
v
f f
ji = [v
f f
i
(rnj
)]; rnj ∈ Cj ; nj = 1, 2, ..., Nj ; j = 1, 2, ..., J; i = j − 1, j (A.14)
is of the order Nj . The first subscript j represents the j
th interface and the second subscript i represents
the i
th domain.
The unknown source intensity vector a of the order (M +
PJ
j=1(Lj1 + Lj2)) is specified by
177
a =
am
b1
b2
.
.
.
bj
.
.
.
bJ
(A.15)
where
am = [am]; m = 1, 2, ..., M (A.16)
bj = [blj
]; lj = 1, 2, ... , Lj1, Lj1 + 1, ..., Lj1 + Lj2 (A.17)
A.3 Free Field for Incident Love Surface Waves
A.3.1 One-Layer Model
By setting the scatterer deviation d to 0, the geometry depicted by Figure 4.2 becomes a one-flat-layer model
for which the free field is evaluated in this subsection. The material properties are given by expressions
(4.20)-(4.21).
178
Choose β0 > c > β1 [40] [56], where β0 and β1 represent the shear wave speed in the bottom half
space D0 and the top layer D1 respectively, and c is the Love wave speed. The Love wave displacement in
the two domains is then defined by
v
f f
0 = A0e
−iκx1−κx3
q
1−(
c
β0
)
2
; κ =
ω
c
; (x1, x3) ∈ D0,
(A.18)
v
f f
1 = A1e
−iκx1+iκx3
q
(
c
β1
)
2−1
+ B1e
−iκx1−iκx3
q
(
c
β1
)
2−1
; (x1, x3) ∈ D1,
(A.19)
The factor e
iωt is omitted throughout the free-field calculation.
Traction-free condition on the top surface yields
µ1
∂vf f
1
∂x3
= 0; x3 = 0, (A.20)
Or
e
−iκx1
(iκx3A1
r
(
c
β1
)
2 − 1 − iκx3B1
r
(
c
β1
)
2 − 1) = 0 (A.21)
Thus,
B1 = A1 (A.22)
Therefore the displacement field in the top layer (A.19) becomes
v
f f
1 = 2A1e
−iκx1 cos(κx3
r
(
c
β1
)
2 − 1) (A.23)
Displacement continuity condition at the flat interface (x3 = h) requires
A0e
−κhq
1−(
c
β0
)
2
= 2A1 cos(κhr
(
c
β1
)
2 − 1) (A.24)
179
Traction continuity condition at the flat interface requires
µ0
∂vf f
0
∂x3
= µ1
∂vf f
1
∂x3
; x3 = h (A.25)
i.e.,
A0e
−κhq
1−(
c
β0
)
2
µ0
r
1 − (
c
β0
)
2 = (2A1)µ1
r
(
c
β1
)
2 − 1 sin(κhr
(
c
β1
)
2 − 1) (A.26)
Dividing equation (A.26) by (A.24) yields
µ0
r
1 − (
c
β0
)
2 = µ1
r
(
c
β1
)
2 − 1 tan(κhr
(
c
β1
)
2 − 1) (A.27)
i.e.,
tan(κhr
(
c
β1
)
2 − 1) = µ0
µ1
q
1 − (
c
β0
)
2
q
(
c
β1
)
2 − 1
(A.28)
from which the Love wave speed c for the one-layer model can be obtained graphically as a function of
the circular frequency ω, see Figure A.1.
Letting A0 = 1, then the parameter A1 can be obtained by solving the equation (A.24).
A1 =
e
−κhq
1−(
c
β0
)
2
2 cos(κhq
(
c
β1
)
2 − 1)
(A.29)
180
Figure A.1: Love wave speed c versus circular frequency ω for the one-flat-layer model depicted by Figure 3.5 with d = 0 and material properties defined by (4.20)-(4.21). The left c-ω relationship is used for
surface displacement evaluation of the Love wave
Figure A.2 displays the relationship between wavelength λ (=
2πc
ω
) and Love wave speed c for the
one-layer model with c-ω that is used for this analysis, see Figure A.1.
Figure A.2: Wavelength λ versus wave speed c for the Love wave, one-layer model
181
A.3.2 Two-Layer Model
By setting the scatterer deviations d1 and d2 to 0, the geometry for the two-layer model depicted by
Figure 4.21 becomes a two-flat-layer model for which the free field is evaluated in this subsection. The
material properties are given by expression (4.34).
Choose β0 > β1 > c > β2 [40] [56], where β0, β1 and β2 represent the shear wave speed in the bottom
half space D0, the mid layer D1 and the top layer D2 respectively, and c is the Love wave speed. The Love
wave displacement in the three domains is then defined by
v
f f
0 = A0e
−iκx1−κx3
q
1−(
c
β0
)
2
; κ =
ω
c
; (x1, x3) ∈ D0 (A.30)
v
f f
1 = A1e
−iκx1−κx3
q
1−(
c
β1
)
2
+ B1e
−iκx1+κx3
q
1−(
c
β1
)
2
; (x1, x3) ∈ D1 (A.31)
v
f f
2 = A2e
−iκx1+iκx3
q
(
c
β2
)
2−1
+ B2e
−iκx1−iκx3
q
(
c
β2
)
2−1
; (x1, x3) ∈ D2 (A.32)
The factor e
iωt is omitted throughout the free-field calculation.
Traction-free condition on the top surface yields
µ2
∂vf f
2
∂x3
= 0; x3 = 0, (A.33)
Or
e
−iκx1
(iκx3A2
r
(
c
β2
)
2 − 1 − iκx3B2
r
(
c
β2
)
2 − 1) = 0 (A.34)
Thus,
B2 = A2 (A.35)
182
Therefore the displacement field in the top layer (A.32) becomes
v
f f
2 = 2A2e
−iκx1 cos(κx3
r
(
c
β2
)
2 − 1) (A.36)
Traction continuity condition at the flat interfaces requires
µ1
∂vf f
1
∂x3
= µ0
∂vf f
0
∂x3
; x3 = h1 (A.37)
µ1
∂vf f
1
∂x3
= µ2
∂vf f
2
∂x3
; x3 = h2 (A.38)
i.e.,
−µ1A1q
′
β1
e
−κh1q
′
β1 + µ1B1q
′
β1
e
κh1q
′
β1 = −µ0q
′
β0A0e
−κh1q
′
β0 (A.39)
−µ1A1q
′
β1
e
−κh2q
′
β1 + µ1B1q
′
β1
e
κh2q
′
β1 = −µ2qβ22A2 sin(κh2qβ2) (A.40)
Here, q
′
β1 =
q
1 − (
c
β1
)
2, q
′
β0 =
q
1 − (
c
β0
)
2 and qβ2 =
q
(
c
β1
)
2 − 1 are real numbers.
Displacement continuity condition at the flat interface requires
v
f f
1 = v
f f
0
; x3 = h1 (A.41)
v
f f
1 = v
f f
2
; x3 = h2 (A.42)
i.e.,
A1e
−κh1q
′
β1 + B1e
κh1q
′
β1 = A0e
−κh1q
′
β0 (A.43)
A1e
−κh2q
′
β1 + B1e
κh2q
′
β1 = 2A2 cos(κh2qβ2) (A.44)
183
Substitution of (A.43) into (A.39) yields
(1 −
µ1q
′
β1
µ0q
′
β0
)A1e
−κh1q
′
β1 + (1 +
µ1q
′
β1
µ0q
′
β0
)B1e
κh1q
′
β1 = 0 (A.45)
Substitution of (A.44) into (A.40) yields
(tan(κh2qβ2) −
µ1q
′
β1
µ2qβ2
)A1e
−κh2q
′
β1 + (tan(κh2qβ2) +
µ1q
′
β1
µ2qβ2
)B1e
κh2q
′
β1 = 0 (A.46)
Equations (A.45)-(A.46) can be rewritten in the following form:
Aa = 0 (A.47)
where
A =
(1 −
µ1q
′
β1
µ0q
′
β0
)e
−κh1q
′
β1 (1 + µ1q
′
β1
µ0q
′
β0
)e
κh1q
′
β1
(tan(κh2qβ2) −
µ1q
′
β1
µ2qβ2
)e
−κh2q
′
β1 (tan(κh2qβ2) + µ1q
′
β1
µ2qβ2
)e
κh2q
′
β1
(A.48)
while
a =
A1
B1
(A.49)
and
0 =
0
0
(A.50)
A non-trivial solution for equation (A.47) requires
det(A) =0 (A.51)
184
from which the Love wave speed c for the two-layer model can be solved graphically as a function of the
circular frequency ω, see Figure A.3.
Letting A0 = 1, then coefficients A1, B1 and A2 can be obtained by solving equations (A.43)-(A.45).
Figure A.3: Love wave speed c versus circular frequency ω for the two-flat-layer model depicted by Figure 4.21 with d1 = d2 = 0 and material properties defined by (4.34). The left c-ω relationship is used for
surface displacement evaluation of the Love wave
Figure A.4 displays the relationship between wavelength λ (=
2πc
ω
) and Love wave speed c for the
two-layer model with the c-ω that is used for this analysis, see Figure A.3.
Figure A.4: Wavelength λ versus wave speed c for the Love wave, two-layer model
185
Appendix B
Plane Strain Model
B.1 Full-Space Green’s Functions and Matrix System
The displacement field caused by the P- and SV-type line source at location r0 can be expressed by [93]
u
P
(r, r0; ζ) = −
i
4
ζH(2)
1
(ζr)t1 (B.1)
w
P
(r, r0; ζ) = −
i
4
ζH(2)
1
(ζr)t3 (B.2)
u
S
(r, r0; η) = −
i
4
ηH(2)
1
(ηr)t3 (B.3)
w
S
(r, r0; η) = i
4
ηH(2)
1
(ηr)t1 (B.4)
r = (x1, 0, x3); r0 = (x10, 0, x30); r = |r − r0|; t1 =
x1 − x10
r
; t3 =
x3 − x30
r
where u
P and w
P are the displacement in x1 and x3 direction, respectively, due to the P-type line load
while u
S
and w
S
are the displacement in x1 and x3 direction, respectively, due to the SV-type line load,
and ζ and η are the wavenumbers associated to P and SV waves, respectively.
186
The stress filed caused by the P- and SV-type line source at location r0 can be expressed by
σ
P
11(r, r0; µ, λ, ζ) = i
4
(
2µζ
r
H
(2)
1
(ζr) − (λ + 2µ)ζ
2H
(2)
0
(ζr))t
2
1
−
i
4
(λζ2H
(2)
0
(ζr) + 2µζ
r
H
(2)
1
(ζr))t
2
3
(B.5)
σ
P
33(r, r0; µ, λ, ζ) = i
4
(
2µζ
r
H
(2)
1
(ζr) − (λ + 2µ)ζ
2H
(2)
0
(ζr))t
2
3
−
i
4
(λζ2H
(2)
0
(ζr) + 2µζ
r
H
(2)
1
(ζr))t
2
1
(B.6)
σ
P
13(r, r0; µ, λ, ζ) = i
4
(2µζ)(2H
(2)
1
(ζr)
r
− ζH(2)
0
(ζr))t1t3 (B.7)
σ
S
11(r, r0; µ, λ, η) = i
4
(2µη)(2
r
H
(2)
1
(ηr) − ηH(2)
0
(ηr))t1t3 (B.8)
σ
S
33(r, r0; µ, λ, η) = −
i
4
(2µη)(2
r
H
(2)
1
(ηr) − ηH(2)
0
(ηr))t1t3 (B.9)
σ
S
13(r, r0; µ, λ, η) = −
i
4
µη(
2
r
H
(2)
1
(ηr) − ηH(2)
0
(ηr))(t
2
1 − t
2
3
) (B.10)
where σ
P
11, σ
P
33 and σ
P
13 denote the stresses due to the P-type line load while σ
S
11, σ
S
33 and σ
S
13 denote the
stresses due to the SV-type line load, respectively.
Therefore, the normal stress σ
P
nn, σ
S
nn and shear stress σ
P
nt, σ
S
nt on the interface Cj with a unit normal
nj (r) = (n1,j (r), 0, n3,j (r)) due to P and SV line source can be given by
σ
P
nn(r, r0; µ, λ, ζ) = σ
P
11(r, r0; µ, λ, ζ)n
2
1,j (r) + σ
P
33(r, r0; µ, λ, ζ)n
2
3,j (r)
+2σ
P
13(r, r0; µ, λ, ζ)n1,j (r)n3,j (r) (B.11)
187
σ
P
nt(r, r0; µ, λ, ζ) = (σ
P
33(r, r0; µ, λ, ζ) − σ
P
11(r, r0; µ, λ, ζ))n1,j (r)n3,j (r)
+σ
P
13(r, r0; µ, λ, ζ)(n
2
1,j (r) − n
2
3,j (r)) (B.12)
σ
S
nn(r, r0; µ, λ, η) = σ
S
11(r, r0; µ, λ, η)n
2
1,j (r) + σ
S
33(r, r0; µ, λ, η)n
2
3,j (r)
+2σ
S
13(r, r0; µ, λ, η)n1,j (r)n3,j (r) (B.13)
σ
S
nt(r, r0; µ, λ, η) = (σ
S
33(r, r0; µ, λ, η) − σ
S
11(r, r0; µ, λ, η))n1,j (r)n3,j (r)
+σ
S
13(r, r0; µ, λ, η)(n
2
1,j (r) − n
2
3,j (r)) (B.14)
where the superscript (j) on the left-hand side of equations (B.11)- (B.14) has been omitted.
For the free stress field, the normal stress σ
f f
nn and shear stress σ
f f
nt along the interface Cj can be
expressed by
σ
f f
nn(r) = σ
f f
11 (r; µ, λ, ζ)n
2
1,j (r) + σ
f f
33 (r; µ, λ, ζ)n
2
3,j (r)
+2σ
f f
13 (r; µ, λ, ζ)n1,j (r)n3,j (r) (B.15)
σ
f f
nt (r) = (σ
f f
33 (r; µ, λ, ζ) − σ
f f
11 (r; µ, λ, ζ))n1,j (r)n3,j (r)
+σ
f f
13 (r; µ, λ, ζ)(n
2
1,j (r) − n
2
3,j (r)) (B.16)
188
The matrix A in equation (5.42) is given as
A =
−U
(1)
0 U
(1)
1
−S
(1)
0 S
(1)
1
−U
(2)
1 U
(2)
2
−S
(2)
1 S
(2)
2
... ...
−U
(j)
j−1 U
(j)
j
−S
(j)
j−1 S
(j)
j
... ...
−U
(J)
J−1 U
(J)
J
−S
(J)
J−1 S
(J)
J
T
(B.17)
where
U
(j)
i =
u
P(j)
i u
S(j)
i
w
P(j)
i w
S(j)
i
(B.18)
S
(j)
i =
s
P(j)
nn,i s
S(j)
nn,i
s
P(j)
nt,i s
S(j)
nt,i
(B.19)
T =
t
P
33 t
S
33
t
P
13 t
S
13
(B.20)
while
u
P(j)
i =
u
P
i
(rnj
, rlj
; ζi)
Nj×(Lj1+Lj2)
(B.21)
u
S(j)
i =
u
S
i
(rnj
, rlj
; ηi)
Nj×(Lj1+Lj2)
(B.22)
189
w
P(j)
i =
w
P
i
(rnj
, rlj
; ζi)
Nj×(Lj1+Lj2)
(B.23)
w
S(j)
i =
w
S
i
(rnj
, rlj
; ηi)
Nj×(Lj1+Lj2)
(B.24)
s
P(j)
nn,i =
σ
P
nn,i(rnj
, rlj
; µi
, λi
, ζi)
Nj×(Lj1+Lj2)
(B.25)
s
S(j)
nn,i =
σ
S
nn,i(rnj
, rlj
; µi
, λi
, ηi)
Nj×(Lj1+Lj2)
(B.26)
s
P(j)
nt,i =
σ
P
nt,i(rnj
, rlj
; µi
, λi
, ζi)
Nj×(Lj1+Lj2)
(B.27)
s
S(j)
nt,i =
σ
S
nt,i(rnj
, rlj
; µi
, λi
, ηi)
Nj×(Lj1+Lj2)
(B.28)
rnj ∈ Cj , nj = 1, 2, ..., Nj ; rlj ∈ Cj1 ∪ Cj2, lj = 1, 2, ..., Lj1 + Lj2; i = j − 1, j
t
P
33 =
σ
P
33(rp, rlJ
; µJ , λJ , ζJ )
P ×(LJ1+LJ2)
(B.29)
t
S
33 =
σ
S
33(rp, rlJ
; µJ , λJ , ηJ )
P ×(LJ1+LJ2)
(B.30)
t
P
13 =
σ
P
13(rp, rlJ
; µJ , λJ , ζJ )
P ×(LJ1+LJ2)
(B.31)
t
S
13 =
σ
S
13(rp, rlJ
; µJ , λJ , ηJ )
P ×(LJ1+LJ2)
(B.32)
rp ∈ SF , p = 1, 2, ..., P; rlJ ∈ CJ1 ∪ CJ2, lJ = 1, 2, ..., LJ1 + LJ2
where the index j represents the j
th interface Cj , j = 1, 2, ..., J, while the subscript i denotes the i
th
domain Di
, i = j − 1, j.
190
In particular, for the case i = 0 and j = 1, rlj
should be replaced by rm, where rm ∈ C00, m =
1, 2, ..., M.
The vector a in equation (5.42) incorporates the unknown source intensities in the following form
a =
a
P
0
a
S
0
a
P
1
a
S
1
a
P
2
a
S
2
...
a
P
j
a
S
j
...
a
P
J
a
S
J
(B.33)
where
191
a
P
0
a
S
0
=
a
P
01
a
P
02
...
a
P
0,M
a
S
01
a
S
02
...
a
S
0,M
2M×1
(B.34)
a
P
j
a
S
j
=
a
P
j,1
a
P
j,2
...
a
P
j,Lj1+Lj2
a
S
j,1
a
S
j,2
...
a
S
j,Lj1+Lj2
2(Lj1+Lj2)×1
; j = 1, 2, ..., J (B.35)
Finally, the forcing vector f in matrix equation (5.42) is assumed to be in the form
192
f =
U
f f
10 − U
f f
11
S
f f
10 − S
f f
11
U
f f
21 − U
f f
22
S
f f
21 − S
f f
22
...
U
f f
j,j−1 − U
f f
jj
S
f f
j,j−1 − S
f f
jj
...
U
f f
J,J−1 − U
f f
JJ
S
f f
J,J−1 − S
f f
JJ
O
(2P +4PJ
j=1 Nj )×1
; j = 1, 2, ..., J (B.36)
where
U
f f
ij =
[u
f f
j
(rni
)]
[w
f f
j
(rni
)]
2Ni×1
(B.37)
rni ∈ Ci
, ni = 1, 2, ..., Ni
, i = 1, 2, ..., J; j = i − 1, i
S
f f
ij =
[σ
f f
nn,j (rni
)]
[σ
f f
nt,j (rni
)]
2Ni×1
(B.38)
rni ∈ Ci
, ni = 1, 2, ..., Ni
, i = 1, 2, ..., J; j = i − 1, i
193
O =
0
0
...
0
2P ×1
(B.39)
Here, the index j represents the j
th domain Dj while the index i denotes the i
th interface Ci
.
B.2 Free Field for Plane Strain Model
The parameters for the displacement potentials for plane harmonic P, SV (equations (5.19) - (5.20)) and
Rayleigh (equations (5.21) - (5.22)) waves are given for a one-layer model in this appendix.
Here, the free field for plane strain model is given due to the time factor of e
iωt
.
B.2.1 Plane Harmonic P or SV Wave
The wavenumbers associated to P and SV waves along the x1 and x3 directions are given by
ζj,1 = ζj sin(θj )
ζj,3 = ζj cos(θj )
ηj,1 = ηj sin(γj ) (B.40)
ηj,3 = ηj cos(γj )
ζ0,1 = ζ1,1 = η0,1 = η1,1; j = 0, 1
where the first subscript j for the wavenumbers ζj,i and ηj,i denotes the layer number and the second
subscript i represent the x1 or x3 direction.
194
By considering the traction-free boundary conditions (equations (5.5) - (5.6)) along the top free surface
SF and the perfect bonding (equations (5.7) - (5.10)) along the flat interface S1, the parameters E0, F0, A1,
B1, E1 and F1 can be solved by the following matrix equation:
Qp = q (B.41)
where
Q11 = 0
Q12 = 0
Q13 = λ1ζ
2
1 + 2µ1ζ
2
1,3
(B.42)
Q14 = λ1ζ
2
1 + 2µ1ζ
2
1,3
Q15 = −2µ1η1,1η1,3
Q16 = 2µ1η1,1η1,3
Q21 = 0
Q22 = 0
Q23 = 2ζ1,1
ζ1,3
(B.43)
Q24 = −2ζ1,1
ζ1,3
Q25 = η
2
1,3 − η
2
1,1
Q26 = η
2
1,3 − η
2
1,1
195
Q31 = −ζ0,1 exp(−iζ0,3h1)
Q32 = η0,3 exp(−iη0,3h1)
Q33 = ζ1,1 exp(iζ1,3h1) (B.44)
Q34 = ζ1,1 exp(−iζ1,3h1)
Q35 = η1,3 exp(iη1,3h1)
Q36 = −η1,3 exp(−iη1,3h1)
Q41 = −ζ0,3 exp(−iζ0,3h1)
Q42 = −η0,1 exp(−iη0,3h1)
Q43 = −ζ1,3 exp(iζ1,3h1) (B.45)
Q44 = ζ1,3 exp(−iζ1,3h1)
Q45 = η1,1 exp(iη1,3h1)
Q46 = η1,1 exp(−iη1,3h1)
196
Q51 = (λ0ζ
2
0 + 2µ0ζ
2
0,3
) exp(−iζ0,3h1)
Q52 = (2µ0η0,1η0,3) exp(−iη0,3h1)
Q53 = −(λ1ζ
2
1 + 2µ1ζ
2
1,3
) exp(iζ1,3h1) (B.46)
Q54 = −(λ1ζ
2
1 + 2µ1ζ
2
1,3
) exp(−iζ1,3h1)
Q55 = (2µ1η1,1η1,3) exp(iη1,3h1)
Q56 = −(2µ1η1,1η1,3) exp(−iη1,3h1)
Q61 = (2µ0ζ0,1ζ0,3) exp(−iζ0,3h1)
Q62 = −µ0(η
2
0,3 − η
2
0,1
) exp(−iη0,3h1)
Q63 = (2µ1ζ1,1ζ1,3) exp(iζ1,3h1) (B.47)
Q64 = −(2µ1ζ1,1ζ1,3) exp(−iζ1,3h1)
Q65 = µ1(η
2
1,3 − η
2
1,1
) exp(iη1,3h1)
Q66 = µ1(η
2
1,3 − η
2
1,1
) exp(−iη1,3h1)
197
q =
0
0
A0ζ0,1 exp(iζ0,3h1) + B0η0,3 exp(iη0,3h1)
−A0ζ0,3 exp(iζ0,3h1) + B0η0,1 exp(iη0,3h1)
−A0(λ0ζ
2
0 + 2µ0ζ
2
0,3
) exp(iζ0,3h1) + B0(2µ0η0,1η0,3) exp(iη0,3h1)
A0(2µ0ζ0,1ζ0,3) exp(iζ0,3h1) + B0µ0(η
2
0,3 − η
2
0,1
) exp(iη0,3h1)
(B.48)
p =
E0
F0
A1
E1
B1
F1
(B.49)
Here, h1 is the depth of the flat reference interface, and the coefficients A0 and B0 in expression (B.48)
are defined by (5.15) and (5.16) for incident P and SV waves, respectively.
B.2.2 Rayleigh Wave for One-layer Model
The wavenumbers associated to an incident plane Rayleigh surface wave (equations (5.21) - (5.22)) are
defined by
198
κ =
ω
c
υj,1 =
q
κ
2 − ζ
2
j
(B.50)
υj,2 =
q
κ
2 − η
2
j
; j = 0, 1
where κ is the Rayleigh wavenumber, c represents the Rayleigh wave speed, and the first subscript j of
the attenuation coefficients υj,i denotes the domain number while the second one i is the index associated
to P type wavenumber ζj or SV-type wavenumber ηj .
By using the traction-free boundary conditions (equations (5.5) - (5.6)) along the top free surface SF and
the perfect bonding (equations (5.7) - (5.10)) along the flat interface S1, a matrix equation can be obtained
as follows:
RpR= 0 (B.51)
where
R11 = 2κυ1,1
R12 = −2κυ1,1
R13 = (2κ
2 − η
2
1
) (B.52)
R14 = (2κ
2 − η
2
1
)
R15 = 0
R16 = 0
199
R21 = (2κ
2 − η
2
1
)
R22 = (2κ
2 − η
2
1
)
R23 = 2κυ1,2 (B.53)
R24 = −2κυ1,2
R25 = 0
R26 = 0
R31 = κ exp(−υ1,1h1)
R32 = κ exp(+υ1,1h1)
R33 = υ1,2 exp(−υ1,2h1) (B.54)
R34 = −υ1,2 exp(+υ1,2h1)
R35 = −κ exp(−υ0,1h1)
R36 = −υ0,2 exp(−υ0,2h1)
200
R41 = υ1,1 exp(−υ1,1h1)
R42 = −υ1,1 exp(+υ1,1h1)
R43 = κ exp(−υ1,2h1) (B.55)
R44 = κ exp(+υ1,2h1)
R45 = −υ0,1 exp(−υ0,1h1)
R46 = −κ exp(−υ0,2h1)
R51 = 2κυ1,1 exp(−υ1,1h1)
R52 = −2κυ1,1 exp(+υ1,1h1)
R53 = (2κ
2 − η
2
1
) exp(−υ1,2h1) (B.56)
R54 = (2κ
2 − η
2
1
) exp(+υ1,2h1)
R55 = −2κυ0,1
µ0
µ1
exp(−υ0,1h1)
R56 = −(2κ
2 − η
2
0
)
µ0
µ1
exp(−υ0,2h1)
201
R61 = (2κ
2 − η
2
1
) exp(−υ1,1h1)
R62 = (2κ
2 − η
2
1
) exp(+υ1,1h1)
R63 = 2κυ1,2 exp(−υ1,2h1) (B.57)
R64 = −2κυ1,2 exp(+υ1,2h1)
R65 = −(2κ
2 − η
2
0
)
µ0
µ1
exp(−υ0,1h1)
R66 = −2κυ0,2
µ0
µ1
exp(−υ0,2h1)
pR =
A1
E1
B1i
F1i
A0
B0i
(B.58)
A non-trivial solution for equation (B.51) requires
det(R) =0 (B.59)
from which the Rayleigh wave speed c can be obtained.
Letting B0 = 1, then the other parameters A0, A1, B1, E1 and F1 can be obtained by solving the
matrix equation (B.51).
202
B.2.3 Rayleigh Wave for Pure Half-Space Model
The form of the incident Rayleigh wave in the half-space refers to equation (5.13) for P-type wave potential
and equation (5.14) for SV-type wave potential.
The displacement components can be obtained as follows according to the displacement potentials
(5.1)
u0 = −iκA0 exp(−iκx1 − κ
r
1 − (
c
α0
)
2x3) + κ
r
1 − (
c
β0
)
2B0 exp(−iκx1 − κ
r
1 − (
c
β0
)
2x3) (B.60)
w0 = −κ
r
1 − (
c
α0
)
2A0 exp(−iκx1 − κ
r
1 − (
c
α0
)
2x3) − iκB0 exp(−iκx1 − κ
r
1 − (
c
β0
)
2x3) (B.61)
The strain and stress for the plane-strain model can then be expressed as
ϵ11 =
∂u0
∂x1
; ϵ33 =
∂w0
∂x3
; ϵ13 =
1
2
(
∂u0
∂x3
+
∂w0
∂x1
); e = ϵ11 + ϵ33 (B.62)
σ13 = 2µ0ϵ13; σ33 = λ0e + 2µ0ϵ33 (B.63)
Traction-free conditions on the top surface are
σ13 = 0; at x3 = 0 (B.64)
σ33 = 0; at x3 = 0 (B.65)
Therefore,
i
r
1 − (
c
α0
)
2A0 −
1
2
(2 − (
c
β0
)
2
)B0 = 0 (B.66)
(2 − (
c
β0
)
2
)A0 + 2i
r
1 − (
c
β0
)
2B0 = 0 (B.67)
203
Equations (B.66)-(B.67) can be rewritten in the following form:
Ra = 0 (B.68)
where
R =
i
q
1 − (
c
α0
)
2 −
1
2
(2 − (
c
β0
)
2
)
(2 − (
c
β0
)
2
) 2i
q
1 − (
c
β0
)
2
(B.69)
while
a =
A0
B0
(B.70)
and
0 =
0
0
(B.71)
A non-trivial solution for equation (B.68) requires
det(R) =0 (B.72)
i.e.,
4
r
1 − (
c
α0
)
2
r
1 − (
c
β0
)
2 = (2 − (
c
β0
)
2
)
2
(B.73)
By assuming α0 = 2 and β0 = 1, the solution of equation (B.73) can be obtained graphically as follows
c = 0.932525905931155 (B.74)
This is the Rayleigh wave speed, independent of the circular frequency, in this pure half-space model.
Letting B0 = 1, then A0 can be obtained by solving the equation (B.66) or (B.67).
204
B.3 Influence of the SV Incident Angle on the Reflected P wave
The free-field displacement potentials for the plane SV and P wave in the bottom half space domain D0
can be expressed by the following two equations respectively,
ψ
f f
0 = B0 exp(−iη0(x1 sin γ0 − x3 cos γ0)) + F0 exp(−iη0(x1 sin γ0 + x3 cos γ0)) (B.75)
ϕ
f f
0 = A0 exp(−iζ0(x1 sin θ0 − x3 cos θ0)) + E0 exp(−iζ0(x1 sin θ0 + x3 cos θ0)) (B.76)
For incident SV wave, A0 = 0 since there is no up-going P wave in domain D0.
The wavenumbers for all the P- and SV-wave displacement potentials along the x1 direction should be
identical thus
ζ0 sin θ0 = η0 sin γ0 (B.77)
Since ζ0 =
ω
α0
and η0 =
ω
β0
(5.4), the above equation (B.77) becomes
sin θ0 =
α0
β0
sin γ0 (B.78)
Therefore,
ζ0 cos θ0 = ζ0
p
1 − sin2
θ0 (B.79)
or
ζ0 cos θ0 = ζ0
r
1 − (
α0
β0
sin γ0)
2 (B.80)
Since the P-wave speed α0 is greater than the S-wave one β0 in a domain, it can be seen that
ζ0 cos θ0 =
ζ0
q
1 − (
α0
β0
sin γ0)
2 for sin γ0 ≤ β0/α0
+/ − iζ0
q
(
α0
β0
sin γ0)
2 − 1 for sin γ0 > β0/α0
(B.81)
205
When sin γ0 ≤ β0/α0 or γ0 ≤ arcsin(β0/α0), the reflected P-wave displacement potential in domian
D0 becomes
ϕ
f f
0 = E0 exp(−iζ0x1 sin θ0 − iζ0
r
1 − (
α0
β0
sin γ0)
2x3) (B.82)
which is a harmonic P wave.
However, when sin γ0 > β0/α0 or γ0 > arcsin(β0/α0), the reflected P-wave displacement potential
in domain D0 has two possible forms of surface wave
ϕ
f f
0 = E0 exp(−iζ0x1 sin θ0 + ζ0
r
(
α0
β0
sin γ0)
2 − 1x3) (B.83)
and
ϕ
f f
0 = E0 exp(−iζ0x1 sin θ0 − ζ0
r
(
α0
β0
sin γ0)
2 − 1x3) (B.84)
The amplitude of the surface wave expressed by equation (B.83) increases while the wave propagates
toward the depth of the domain D0. This is not real. Instead, the amplitude of the surface wave is expected
to vanish at x3 → ∞ in the bottom half-space D0. Therefore, the surface wave expressed by equation
(B.84) is reasonable and adopted in this research if γ0 > arcsin(β0/α0).
Finally, based on the above derivation, it can be concluded that the free-field displacement potentials
for both the P and SV wave in the top layer D1 are plane harmonic for an incident SV wave with any angle
of incidence (0
o ≤ γ0 < 90o
) since
α0 > β0 = α1 > β1 (B.85)
according to the material properties for the one-layer model defined by (5.44) and (5.45).
206
Appendix C
Three-Dimensional Model
C.1 Full-Space Green’s Functions for the 3D Model
The full-space three-dimensional displacement Green’s functions G
(j)
ik (r, r0; ω) for a point load at location
r0 are given by [50]
G
(j)
pk (r, r0; ω) = 1
4πµj
U
(j)
1
(r, r0; ω)δpk − U
(j)
2
(r, r0; ω)
∂r
∂xp
∂r
∂xk
(C.1)
in which
U
(j)
1
(r, r0; ω) = e
iηj r
r
+
i
ηjr
−
1
(ηjr)
2
e
iηj r
r
− (
ζj
ηj
)
2
[
i
ζjr
−
1
(ζjr)
2
]
e
iζj r
r
(C.2)
U
(j)
2
(r, r0; ω) =
1 +
3i
ηjr
−
3
(ηjr)
2
e
iηj r
r
− (
ζj
ηj
)
2
[1 + 3i
ζjr
−
3
(ζjr)
2
]
e
iζj r
r
(C.3)
ζj =
ω
αj
, ηj =
ω
βj
; r = |r − r0|, r ∈ Dj , j = 0, 1, 2, ..., J; p, k = 1, 2, 3
where ζj and ηj are the wavenumbers associated to P and SV waves, respectively, and αj and βj denote
the P and SV wave velocities in the j
th domain, respectively.
Corresponding three-dimensional stress Green’s functions are given by
207
h
(j)
pqk(r, r0; ω) = 2µjg
(j)
pkp(r, r0; ω) + λj [g
(j)
1k1
(r, r0; ω) + g
(j)
2k2
(r, r0; ω) + g
(j)
3k3
(r, r0; ω)] (C.4)
q = p; p, k = 1, 2, 3
h
(j)
pqk(r, r0; ω) = µj [g
(j)
pkq(r, r0; ω) + g
(j)
qkp(r, r0; ω)] (C.5)
p ̸= q; p, q, k = 1, 2, 3
Here, the functions g
(j)
pqk(r, r0; ω) are expressed by
g
(j)
pqk(r, r0; ω) = 1
4πµj
"
∂U(j)
1
∂xk
−
∂U(j)
2
∂xk
(
∂r
∂xp
)
2 − 2U
(j)
2
∂r
∂xp
∂
2
r
∂xp∂xk
#
(C.6)
q = p; p, k = 1, 2, 3
g
(j)
pqk(r, r0; ω) = g
(j)
qpk(r, r0; ω)
= −
1
4πµj
"
∂U(j)
2
∂xk
∂r
∂xp
∂r
∂xq
+ U
(j)
2
(
∂r
∂xp
∂
2
r
∂xq∂xk
+
∂r
∂xq
∂
2
r
∂xp∂xk
)
#
(C.7)
p ̸= q; p, q, k = 1, 2, 3
where the summation convention is not applied to p, q, k, and the partial differentials in the formulae (C.6)
- (C.7) are represented according to
∂U(j)
1
(r, r0; ω)
∂xk
=
xk
r
[(iηj −
1
r
+ ( 2
η
2
j
r
3
−
i
ηjr
2
) + ( i
ηjr
−
1
η
2
j
r
2
)(iηj −
1
r
))e
iηj r
r
−(
ζj
ηj
)
2
(
2
ζ
2
j
r
3
−
i
ζjr
2
+ ( i
ζjr
−
1
ζ
2
j
r
2
)(iζj −
1
r
))e
iζj r
r
] (C.8)
208
∂U(j)
2
(r, r0; ω)
∂xk
=
xk
r
[( 6
η
2
j
r
3
−
3i
ηjr
2
+ (1 + 3i
ηjr
−
3
η
2
j
r
2
)(iηj −
1
r
))e
iηj r
r
−(
ζj
ηj
)
2
(
6
ζ
2
j
r
3
−
i
ζjr
2
+ (1 + 3i
ζjr
−
3
ζ
2
j
r
2
)(iζj −
1
r
))e
iζj r
r
] (C.9)
∂r
∂xk
=
xk − x0k
r
(C.10)
∂
2
r
∂x2
k
=
1
r
(1 − (
xk − x0k
r
)
2
) (C.11)
∂
2
r
∂xp∂xk
= −
1
r
(
xp − x0p
r
)(xk − x0k
r
); p ̸= k (C.12)
p, k = 1, 2, 3
The displacements of the scattered wave field are shown in equations (6.22) - (6.23) and stresses of the
scattered wave field are given by
σ
s(0)
pk = a
(0)
m h
(0)
pk1
(r, rm; ω) + b
(0)
m h
(0)
pk2
(r, rm; ω) + c
(0)
m h
(0)
pk2
(r, rm; ω); r ∈ D0; p = 1, 2, 3 (C.13)
σ
s(j)
pk = d
(j)
lj
h
(j)
pk1
(r, rlj
; ω) + e
(j)
lj
h
(j)
pk2
(r, rlj
; ω) + f
(j)
lj
h
(j)
pk2
(r, rlj
; ω);
r ∈ Dj ; j = 1, 2, ..., J; p = 1, 2, 3 (C.14)
rm ∈ C00, m = 1, 2, ..., M; rlj ∈ Cj1 + Cj2, lj = 1, 2, ..., Lj1 + Lj2
209
C.2 Matrix System of the 3D Model
The matrix A in equation (6.24) is of the order (3P + 6PJ
i=1 Ni)×(3M + 3PJ
i=1(Li1+Li2)) and defined
by
A =
−G
(1)
0 G
(1)
1
−S
(1)
0 S
(1)
1
−G
(2)
1 G
(2)
2
−S
(2)
1 S
(2)
2
... ...
−G
(j)
j−1 G
(j)
j
−S
(j)
j−1 S
(j)
j
... ...
−G
(J)
J−1 G
(J)
J
−S
(J)
J−1 S
(J)
J
T
(C.15)
where the submatrices G
(i)
j
are given by
G
(i)
j =
[G
(j)
11 (rni
, rm; ω)] [G
(j)
12 (rni
, rm; ω)] [G
(j)
13 (rni
, rm; ω)]
[G
(j)
12 (rni
, rm; ω)] [G
(j)
22 (rni
, rm; ω)] [G
(j)
23 (rni
, rm; ω)]
[G
(j)
13 (rni
, rm; ω)] [G
(j)
32 (rni
, rm; ω)] [G
(j)
33 (rni
, rm; ω)]
(C.16)
where rni ∈ Ci
, n = 1, 2, ..., Ni
; rm ∈ C00, m = 1, 2, ..., M for j = 0; rm ∈ Cj1 + Cj2 , m = 1, 2, ...,
Lj1 + Lj2 for other j’s. Here the index j corresponds to the j
th domain Dj while the index i to the i
th
interface Ci
. Cj1 and Cj2 are the auxiliary surfaces below and above the layer Dj .
210
The submatrices S
(i)
j
in expression (C.15) are defined by
S
(i)
j =
[X
(j)
a (rni
, rm; ω)] [X
(j)
b
(rni
, rm; ω)] [X
(j)
c (rni
, rm; ω)]
[Y
(j)
a (rni
, rm; ω)] [Y
(j)
b
(rni
, rm; ω)] [Y
(j)
c (rni
, rm; ω)]
[Z
(j)
a (rni
, rm; ω)] [Z
(j)
b
(rni
, rm; ω)] [Z
(j)
c (rni
, rm; ω)]
(C.17)
where
[X(j)
a
(rni
, rm; ω)] = h
(j)
111(rni
, rm; ω)n1(rni
) + h
(j)
121(rni
, rm; ω)n2(rni
) + h
(j)
131(rni
, rm; ω)n3(rni
)
[X
(j)
b
(rni
, rm; ω)] = h
(j)
112(rni
, rm; ω)n1(rni
)+h
(j)
122(rni
, rm; ω)n2(rni
)+h
(j)
132(rni
, rm; ω)n3(rni
) (C.18)
[X(j)
c
(rni
, rm; ω)] = h
(j)
113(rni
, rm; ω)n1(rni
) + h
(j)
123(rni
, rm; ω)n2(rni
) + h
(j)
133(rni
, rm; ω)n3(rni
)
[Y
(j)
a
(rni
, rm; ω)] = h
(j)
211(rni
, rm; ω)n1(rni
) + h
(j)
221(rni
, rm; ω)n2(rni
) + h
(j)
231(rni
, rm; ω)n3(rni
)
[Y
(j)
b
(rni
, rm; ω)] = h
(j)
212(rni
, rm; ω)n1(rni
)+h
(j)
222(rni
, rm; ω)n2(rni
)+h
(j)
232(rni
, rm; ω)n3(rni
) (C.19)
[Y
(j)
c
(rni
, rm; ω)] = h
(j)
213(rni
, rm; ω)n1(rni
) + h
(j)
223(rni
, rm; ω)n2(rni
) + h
(j)
233(rni
, rm; ω)n3(rni
)
[Z
(j)
a
(rni
, rm; ω)] = h
(j)
311(rni
, rm; ω)n1(rni
) + h
(j)
321(rni
, rm; ω)n2(rni
) + h
(j)
331(rni
, rm; ω)n3(rni
)
[Z
(j)
b
(rni
, rm; ω)] = h
(j)
312(rni
, rm; ω)n1(rni
)+h
(j)
322(rni
, rm; ω)n2(rni
)+h
(j)
332(rni
, rm; ω)n3(rni
) (C.20)
[Z
(j)
c
(rni
, rm; ω)] = h
(j)
313(rni
, rm; ω)n1(rni
) + h
(j)
323(rni
, rm; ω)n2(rni
) + h
(j)
333(rni
, rm; ω)n3(rni
)
211
Here, the definitions of rni
, rm and the indexes i, j are the same of those in expression (C.16). And
n(rni
) = (n1(rni
), n2(rni
), n3(rni
)) is the unit normal vector at location rni
on the interface Ci
.
The submatrix T in expression (C.15) is defined by
T =
[h
(J)
131(rnp
, rm; ω)] [h
(J)
132(rnp
, rm; ω)] [h
(J)
133(rnp
, rm; ω)]
[h
(J)
231(rnp
, rm; ω)] [h
(J)
232(rnp
, rm; ω)] [h
(J)
233(rnp
, rm; ω)]
[h
(J)
331(rnp
, rm; ω)] [h
(J)
332(rnp
, rm; ω)] [h
(J)
333(rnp
, rm; ω)]
3P ×3(LJ1+LJ2)
(C.21)
where rnp ∈ SF , np = 1, 2, ..., P; rm ∈ CJ1 + CJ2 , m = 1, 2, ..., LJ1 + LJ2.
The forcing vector f in equation (6.24) is defined by
f = −
U
f f
10 − U
f f
11
S
f f
10 − S
f f
11
U
f f
21 − U
f f
22
S
f f
21 − S
f f
22
...
U
f f
j,j−1 − U
f f
jj
S
f f
j,j−1 − S
f f
jj
...
U
f f
J,J−1 − U
f f
JJ
S
f f
J,J−1 − S
f f
JJ
O
(3P +6PJ
j=1 Nj )×1
; j = 1, 2, ..., J (C.22)
212
where
U
f f
ij =
[u
f f
j
(rni
)]
[v
f f
j
(rni
)]
[w
f f
j
(rni
)]
3Ni×1
(C.23)
rni ∈ Ci
, ni = 1, 2, ..., Ni
, i = 1, 2, ..., J; j = i − 1, i
S
f f
ij =
[σ
f f
11,j (rni
)n1(rni
) + σ
f f
12,j (rni
)n2(rni
) + σ
f f
13,j (rni
)n3(rni
)]
[σ
f f
21,j (rni
)n1(rni
) + σ
f f
22,j (rni
)n2(rni
) + σ
f f
33,j (rni
)n3(rni
)]
[σ
f f
31,j (rni
)n1(rni
) + σ
f f
32,j (rni
)n2(rni
) + σ
f f
33,j (rni
)n3(rni
)]
3Ni×1
(C.24)
rni ∈ Ci
, ni = 1, 2, ..., Ni
, i = 1, 2, ..., J; j = i − 1, i
O =
0
0
...
0
3P ×1
(C.25)
The vector a including all the unknown source densities in equation (6.24) is defined by
213
a
=
a
0
b
0
c
0
d
1
e
1
f1...dj ej fj... dJ eJfJ
(3
P +6
P
Jj=1
Nj
)
×
1
;
j = 1
,
2, ..., J (C.26)
where
a
0
=
h
a
(0) m
i
M
×
1
b
0
=
h
b
(0) m
i
M
×
1
(C.27)
c
0
=
h
c
(0) m
i
M
×
1
214
dj =
h
d
(j)
lj
i
(Lj1+Lj2)×1
ej =
h
e
(j)
lj
i
(Lj1+Lj2)×1
(C.28)
fj =
h
f
(j)
lj
i
(Lj1+Lj2)×1
215
Abstract (if available)
Abstract
Steady-state scattering of elastic waves in multilayered media with irregular interfaces is investigated by using an indirect boundary integral equation method for antiplane strain, plane strain, and three-dimensional models. The material is assumed to be elastic, homogeneous, and isotropic. The wave field in each domain is expressed as a superposition of the free field of the corresponding flat-layer model and the scattered wave field caused by irregularities. The scattered wave field is generated by a series of sources distributed near the irregular interfaces. The half-space Green’s functions are first applied to model the scattered wave field of the antiplane strain model while later the full-space Green’s functions are applied to represent the sources of all three different models. The source intensities are determined by satisfying the boundary and continuity conditions in the least square sense.
For each of the three models mentioned above, systematic parametric error analysis is conducted to determine the optimum parameters for the steady-state response. Numerical results are then presented based on the parametric error analysis. Transient response is studied in detail in the antiplane strain model using the half-space Green’s function approach. The resonant feature of the multilayered structure is investigated in the antiplane strain and plane strain models, respectively.
It is found that the steady-state response strongly depends on the type, frequency, and angle of the incident wave, the shape of the irregularity, and the site location. The pattern of the surface response amplitude of the three-dimensional model is more complicated than those of the two-dimensional models since the transverse variation of the model structure plays an important role in the steady-state response.
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Asset Metadata
Creator
Ding, Gang
(author)
Core Title
Scattering of elastic waves in multilayered media with irregular interfaces
School
Viterbi School of Engineering
Degree
Doctor of Philosophy
Degree Program
Mechanical Engineering
Degree Conferral Date
2024-12
Publication Date
11/27/2024
Defense Date
10/31/2024
Publisher
Los Angeles, California
(original),
University of Southern California
(original),
University of Southern California. Libraries
(digital)
Tag
anti-plane strain model,boundary conditions,continuity conditions,elastic waves,free field,green’s function,indirect boundary integral equation method,irregular interfaces,least square,mode conversion,multilayered media,parametric error analysis,plane strain model,resonance,scattered wave field,scattering,source intensities,steady-state response,three-dimensional model,transient response
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Sadhal, Satwindar (
committee chair
), Flashner, Henryk (
committee member
), Lee, Vincent (
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)
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Tags
anti-plane strain model
boundary conditions
continuity conditions
elastic waves
free field
green’s function
indirect boundary integral equation method
irregular interfaces
least square
mode conversion
multilayered media
parametric error analysis
plane strain model
resonance
scattered wave field
scattering
source intensities
steady-state response
three-dimensional model
transient response