Scattering of a plane harmonic wave by a completely embedded corrugated scatterer

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Content

SCATTERING OF A PLANE HARMONIC WAVE BY A COMPLETELY
EMBEDDED CORRUGATED SCATTERER

by

Chih-Wei Yu

A Dissertation Presented to the
FACULTY OF THE GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(MECHANICAL ENGINEERING)

December 2008

Copyright 2008 Chih-Wei Yu

ii
Acknowledgements
I would like to express my deepest appreciation to my advisor M.
Dravinski for his constant guidance and advice throughout this study. I am
also grateful to Professors H. Flashner, S. Sadhal, P. Vashishta and W.
Proskurowski for their willingness to serve on my dissertation committee. In
addition, I would like to thank Professor W. Proskurowski for his constructive
criticism of the dissertation. My thanks go to many professors in the
Department of Aerospace and Mechanical Engineering with whom it was my
privilege to study.
I would like to express my gratitude to my wife, parents, parents-in-law
and every member of my family for their support and encouragement and to
whom I dedicate this dissertation.
The teaching assistantship from the Department of Aerospace and
Mechanical Engineering at University of Southern California is greatly
appreciated.

iii
Table of Contents
Acknowledgements ii

List of Tables v

List of Figures vii

Abstract xiii

Chapter 1: Introduction 1

Chapter 2: Statement of Problem 11
2.1. Common features for the problems 12
2.2. Half-space problems 15
2.3. Full-space problems 18

Chapter 3: Anti-Plane Strain Model Solutions 20
3.1. Half-space inclusion problem solution 20
3.2. Half-space cavity problem solution 27
3.3. Full-space inclusion problem solution 29
3.4. Full-space cavity problem solution 33

Chapter 4: Anti-Plane Strain Model Numerical Results 36
4.1. Half-space inclusion problem results 38
4.2. Half-space cavity problem results 50
4.3. Full-space inclusion problem results 57
4.4. Full-space cavity problem results 64

Chapter 5: Plane Strain Model Solutions 69
5.1. Half-space inclusion problem solution 69
5.2. Half-space cavity problem solution 75
5.3. Full-space inclusion problem solution 78
5.4. Full-space cavity problem solution 81

iv
Chapter 6: Plane Strain Model Numerical Results 84
6.1. Half-space inclusion problem results 84
6.2. Half-space cavity problem results 93
6.3. Full-space inclusion problem results 103
6.4. Full-space cavity problem results 107

Chapter 7: Summary and Conclusions 115

References 119

Appendix: Evaluation of Computer Programs 126

v
List of Tables
Table 4.1: The corrugation parameters for the anti-plane strain model. 36

Table 4.2: The relative error ) , (
0 0
S Err
HS
β
η along the half-space
surface S
0
as a function of frequency η
β
and the length of S
0
for a
transparency test ( 1 = = b a , 1
2 1
= =μ μ , 1
2 1
= =
β β
v v and 0 = ε )
subjected to a vertical plane harmonic SH wave. 39

Table 4.3: The relative error ) , (
0 1
S Err
HS
β
η as a function of frequency
η
β
and the length of the half-space surface S
0
for a semi-circular soft
valley ( 6 / 1
2
= μ , 2 / 1
2
=
β
v ) subjected to a vertical plane harmonic
SH wave. 41

Table 4.4: The relative error ) , (
2
K Err
HS
β
η based on the numerical
results with different numbers of elements K at equivalent nodes
along the corrugated interface S and the half-space surface S
0

subjected to a vertical plane harmonic SH wave as 2 . 0 = ε ,
1 = = b a , 8 = m , 6 / 1
2
= μ and 2 / 1
2
=
β
v . 42

Table 4.5: The relative error ) , (
0 1
S Err
HS
β
η as a function of frequency
η
β
and the length of the half-space surface S
0
for a semi-circular
canyon subjected to a vertical plane harmonic SH wave. 51

Table 4.6: The relative error ) , (
2
K Err
HS
β
η based on the numerical
results with different numbers of elements K at equivalent nodes
along the corrugated surface S and the half-space surface S
0

subjected to a vertical plane harmonic SH wave as 2 . 0 = ε , 1 = = b a
and 8 = m . 52

Table 4.7: The relative error ) , (
1
K Err
FS
β
η as a function of the
frequency η
β
and the number of elements K for a circular soft
inclusion ( 6 / 1
2
= μ , 2 / 1
2
=
β
v ) embedded in a full space subjected to
a plane harmonic SH wave. 58

vi
Table 4.8: The relative error ) , (
1
K Err
FS
β
η as a function of the
frequency η
β
and the number of elements K for a circular stiff
inclusion ( 8
2
= μ , 2
2
=
β
v ) embedded in a full space subjected to a
plane harmonic SH wave. 58

Table 4.9: The relative error ) , (
2
K Err
FS
β
η based on the numerical
results with different numbers of elements K at equivalent nodes
along the corrugated interface S as 2 . 0 = ε , 1 = = b a , 8 = m ,
6 / 1
2
= μ and 2 / 1
2
=
β
v . 59

Table 4.10: The relative error ) , (
1
K Err
FS
β
η as a function of the
frequency η
β
and the number of elements K for a circular cavity
embedded in a full space subjected to a plane harmonic SH wave. 65

Table 4.11: The relative error ) , (
2
K Err
FS
β
η based on the numerical
results with different numbers of elements K at equivalent nodes
along the corrugated surface S as 2 . 0 = ε , 1 = = b a and 8 = m . 65

Table 6.1: The corrugation parameters for the plane strain model. 84

Table A.1: The average CPU times for executing MATLAB programs
with an Intel Core 2 CPU Q6600 at 2.4 GHz and 3 GB RAM. 126

vii
List of Figures
Figure 2.1: A half-space model for a scatterer B
2
with a corrugated
surface S embedded at depth h within domain B
1
. The surface of the
half-space is denoted by S
0
, and S
1
is the boundary at infinity. S’
represents the elliptic smooth interface limit when the corrugation
amplitude 0 → ε , while a and b are the principal axes of S’.
Furthermore, u
inc
, u
sc(1)
and u
sc(2)
denote the incident wave and
scattered waves in domains B
1
and B
2
, respectively, and n is the
outward unit normal vector on
0 1
S S S ∪ ∪ . Finally, γ is the off-vertical
angle of the incidence. 12

Figure 3.1: Discretized boundaries S ∂ and
0
S ∂ for the half-space
surface problem, where
∑
=
∂ = ∂
P
k
k
S S
1
and
∑
+ =
∂ = ∂
K
P k
k
S S
1
0
. Here P
denotes the number of elements along the corrugation and
k
S ∂
denotes the element k, and K is the total number of elements. The
length of each element
k
S ∂ is L
k
, and n
k
denotes the corresponding
unit outward normal vector. 25

Figure 3.2: A discretized corrugated boundary
∑
=
∂ = ∂
K
k
k
S S
1
in a full
space using linear elements. Here, K denotes both the number of
nodes and the number of elements. The length of each element is
denoted by L
k
, k=1:K, and n
k
represents outward unit normal on the
element
k
S ∂ . 31

Figure 4.1: The geometry of the half-space problem where the dash
line denotes the smooth surface limit S’ and the solid line denotes the
corrugation S and the half-space surface S
0
. Two period coefficients
are considered: 2 = m (left) and 8 = m (right). In addition, 1 = = b a ,
2 . 0 = ε and a h 2 = . 43

Figure 4.2: Half-space inclusion anti-plane strain model displacement
amplitude along the half-space surface S
0
subjected to a vertical
plane harmonic SH wave for different corrugation peak amplitude ε
as a function of location x
1
(Fig. 2.1) when 2 =
β
η , 8 = m , 1 = = b a ,
1
1
= μ , 6 / 1
2
= μ , 1
1
=
β
v , 2 / 1
2
=
β
v , a h 2 = , 128 = P and 928 = K . 45

viii
Figure 4.3: Half-space inclusion anti-plane strain model displacement
amplitude along the half-space surface S
0
subjected to a grazing
plane harmonic SH wave for different corrugation peak amplitude ε
as a function of location x
1
(Fig. 2.1) when 1 =
β
η , 8 = m , 1 = = b a ,
1
1
= μ , 6 / 1
2
= μ , 1
1
=
β
v , 2 / 1
2
=
β
v , a h 2 = , 128 = P and 928 = K . 46

Figure 4.4: Half-space inclusion anti-plane strain model displacement
amplitude along the half-space surface S
0
subjected to a vertical
plane harmonic SH wave for different corrugation peak amplitude ε
as a function of location x
1
(Fig. 2.1) when 2 =
β
η , 2 = m , 1 = = b a ,
a h 2 = , 128 = P and 928 = K . Stiff inclusion case: 1
1
= μ , 8
2
= μ ,
1
1
=
β
v and 2
2
=
β
v . 47

Figure 4.5: Half-space inclusion anti-plane strain model displacement
amplitude along the half-space surface S
0
subjected to a grazing
plane harmonic SH wave for different corrugation peak amplitude ε
as a function of location x
1
(Fig. 2.1) when 1 =
β
η , 2 = m , 1 = = b a ,
a h 2 = , 128 = P and 928 = K . Stiff inclusion case: 1
1
= μ , 8
2
= μ ,
1
1
=
β
v and 2
2
=
β
v . 48

Figure 4.6: Half-space cavity anti-plane strain model displacement
amplitude along the half-space surface S
0
subjected to a vertical
plane harmonic SH wave for different corrugation peak amplitude ε
as a function of location x
1
(Fig. 2.1) when 2 =
β
η , 8 = m , 1 = = b a ,
1
1
= μ , 1
1
=
β
v , a h 2 = , 128 = P and 928 = K . 54

Figure 4.7: Half-space cavity anti-plane strain model displacement
amplitude along the half-space surface S
0
subjected to a grazing
plane harmonic SH wave for different corrugation peak amplitude ε
as a function of location x
1
(Fig. 2.1) when 1 =
β
η , 8 = m , 1 = = b a ,
1
1
= μ , 1
1
=
β
v , a h 2 = , 128 = P and 928 = K . 55

Figure 4.8: Full-space problem geometry with a smooth circular
cavity S’ (dash line, the principal axes 1 = = b a ) and the
corresponding corrugated cavities (solid lines) of amplitude 2 . 0 = ε
and period coefficients 2 = m (left) and 8 = m (right), respectively. 61

ix
Figure 4.9: Full-space inclusion anti-plane strain model displacement
amplitude along the corrugated interface S for different corrugation
peak amplitude ε as a function of angle θ (Fig. 4.8) when 2 =
β
η ,
8 = m , 1 = = b a , 1
1
= μ , 6 / 1
2
= μ , 1
1
=
β
v , 2 / 1
2
=
β
v and 128 = K . 61

Figure 4.10: Full-space inclusion anti-plane strain model
displacement amplitude along the corrugated interface S for different
corrugation peak amplitude ε as a function of angle θ (Fig. 4.8)
when 2 =
β
η , 8 = m , 1 = = b a and 128 = K . Stiff inclusion case:
1
1
= μ , 8
2
= μ , 1
1
=
β
v and 2
2
=
β
v . 63

Figure 4.11: Full-space cavity anti-plane strain model displacement
amplitude along the corrugated surface S for different corrugation
peak amplitude ε as a function of angle θ (Fig. 4.8) when 2 =
β
η ,
8 = m , 1 = = b a , 1
1
= μ , 1
1
=
β
v and 128 = K . 67

Figure 6.1: Half-space inclusion plane strain model displacement
amplitudes along the half-space surface S
0
due to a vertical P
incidence for different corrugation peak amplitude ε as a function of
location x
1
(Fig. 2.1) when 1 =
β
η , 2 = m , 1 = = b a , 2
1
= λ , 1
1
= μ ,
1
1
= ρ , 1
2
= λ , 6 / 1
2
= μ , 3 / 2
2
= ρ , 0 = γ , a h 2 = , 128 = P and
928 = K . 86

Figure 6.2: Half-space inclusion plane strain model displacement
amplitudes along the half-space surface S
0
due to an oblique P
incidence for different corrugation peak amplitude ε as a function of
location x
1
(Fig. 2.1) when 1 =
β
η , 2 = m , 1 = = b a , 2
1
= λ , 1
1
= μ ,
1
1
= ρ , 1
2
= λ , 6 / 1
2
= μ , 3 / 2
2
= ρ ,
ο
80 = γ , a h 2 = , 128 = P and
928 = K . 87

Figure 6.3: Half-space inclusion plane strain model displacement
amplitudes along the half-space surface S
0
due to a vertical SV
incidence for different corrugation peak amplitude ε as a function of
location x
1
(Fig. 2.1) when 1 =
β
η , 2 = m , 1 = = b a , 2
1
= λ , 1
1
= μ ,
1
1
= ρ , 1
2
= λ , 6 / 1
2
= μ , 3 / 2
2
= ρ , 0 = γ , a h 2 = , 128 = P and
928 = K . 88

x
Figure 6.4: Half-space inclusion plane strain model displacement
amplitudes along the half-space surface S
0
due to an oblique SV
incidence for different corrugation peak amplitude ε as a function of
location x
1
(Fig. 2.1) when 1 =
β
η , 8 = m , 1 = = b a , 2
1
= λ , 1
1
= μ ,
1
1
= ρ , 1
2
= λ , 6 / 1
2
= μ , 3 / 2
2
= ρ ,
ο
80 = γ , a h 2 = , 128 = P and
928 = K . 89

Figure 6.5: Half-space inclusion plane strain model displacement
amplitudes along the half-space surface S
0
due to a Rayleigh
incidence for different corrugation peak amplitude ε as a function of
location x
1
(Fig. 2.1) when 1 =
β
η , 8 = m , 1 = = b a , 2
1
= λ , 1
1
= μ ,
1
1
= ρ , 1
2
= λ , 6 / 1
2
= μ , 3 / 2
2
= ρ , a h 2 = , 128 = P and 928 = K . 91

Figure 6.6: Half-space cavity plane strain model displacement
amplitude along the half-space surface S
0
due to a vertical P
incidence as a function of location x
1
(Fig. 2.1) when 5 . 0 =
β
η ,
1 = = b a , 2
1
= λ , 1
1 1
= =ρ μ , 0 = γ , a h 5 . 1 = , 128 = P a n d
928 = K . Solid lines denote the results of this study while the open
circles denote those of Luco and Barros (1994). 94

Figure 6.7: Half-space cavity plane strain model displacement
amplitude along the half-space surface S
0
due to a vertical SV
incidence as a function of location x
1
(Fig. 2.1) when 5 . 0 =
β
η ,
1 = = b a , 2
1
= λ , 1
1 1
= =ρ μ , 0 = γ , a h 5 . 1 = , 128 = P a n d
928 = K . Solid lines denote the results of this study while the open
circles denote those of Luco and Barros (1994). 94

Figure 6.8: Half-space cavity plane strain model displacement
amplitude along the half-space surface S
0
due to a Rayleigh
incidence as a function of location x
1
(Fig. 2.1) when 5 . 0 =
β
η ,
1 = = b a , 2
1
= λ , 1
1 1
= =ρ μ , 0 = γ , a h 5 . 1 = , 128 = P a n d
928 = K . Solid lines denote the results of this study while the open
circles denote those of Luco and Barros (1994). 95

Figure 6.9: Half-space cavity plane strain model displacement
amplitude along the half-space surface S
0
due to a vertical P
incidence for different corrugation peak amplitude ε as a function of
location x
1
(Fig. 2.1) when 1 =
β
η , 8 = m , 1 = = b a , 2
1
= λ ,
1
1 1
= =ρ μ , 0 = γ , a h 2 = , 128 = P and 928 = K . 97

xi
Figure 6.10: Half-space cavity plane strain model displacement
amplitude along the half-space surface S
0
due to an oblique P
incidence for different corrugation peak amplitude ε as a function of
location x
1
(Fig. 2.1) when 1 =
β
η , 8 = m , 1 = = b a , 2
1
= λ ,
1
1 1
= =ρ μ ,
ο
80 = γ , a h 2 = , 128 = P and 928 = K . 98

Figure 6.11: Half-space cavity plane strain model displacement
amplitude along the half-space surface S
0
due to a vertical SV
incidence for different corrugation peak amplitude ε as a function of
location x
1
(Fig. 2.1) when 1 =
β
η , 8 = m , 1 = = b a , 2
1
= λ ,
1
1 1
= =ρ μ , 0 = γ , a h 2 = , 128 = P and 928 = K . 98

Figure 6.12: Half-space cavity plane strain model displacement
amplitude along the half-space surface S
0
due to an oblique SV
incidence for different corrugation peak amplitude ε as a function of
location x
1
(Fig. 2.1) when 1 =
β
η , 8 = m , 1 = = b a , 2
1
= λ ,
1
1 1
= =ρ μ ,
ο
80 = γ , a h 2 = , 128 = P and 928 = K . 100

Figure 6.13: Half-space cavity plane strain model displacement
amplitude along the half-space surface S
0
due to a Rayleigh
incidence for different corrugation peak amplitude ε as a function of
location x
1
(Fig. 2.1) when 1 =
β
η , 8 = m , 1 = = b a , 2
1
= λ ,
1
1 1
= =ρ μ , a h 2 = , 128 = P and 928 = K . 101

Figure 6.14: Radial and tangential displacement amplitudes along the
corrugated interface S for full-space plane strain inclusion models
with different corrugation amplitudes ε and a P incidence as a
function of angle θ (Fig. 4.8) when 1 =
β
η , 8 = m , 1 = = b a ,
2
1
= λ , 1
1
= μ , 1
1
= ρ , 1
2
= λ , 6 / 1
2
= μ , 3 / 2
2
= ρ and 128 = K . 105

Figure 6.15: Radial and tangential displacement amplitudes along the
corrugated interface S for full-space plane strain inclusion models
with different corrugation amplitudes ε and an SV incidence as a
function of angle θ (Fig. 4.8) when 1 =
β
η , 8 = m , 1 = = b a ,
2
1
= λ , 1
1
= μ , 1
1
= ρ , 1
2
= λ , 6 / 1
2
= μ , 3 / 2
2
= ρ and 128 = K . 106

xii
Figure 6.16: Full-space plane strain cavity model displacement
amplitude along the smooth surface S’ as a function of angle θ (Fig.
4.8) when 25 . 0 =
β
η , 1 = = b a , 2
1
= λ , 1
1 1
= =ρ μ and 128 = K .
Solid lines denote the results of this study while solid circles are those
of Mow and Mente (1963). 109

Figure 6.17: Full-space plane strain cavity model displacement
amplitude along the smooth surface S’ as a function of angle θ (Fig.
4.8) when 1 =
β
η , 1 = = b a , 2
1
= λ , 1
1 1
= =ρ μ and 128 = K . Solid
lines denote the results of this study while solid circles are those of
Mow and Mente (1963). 109

Figure 6.18: Radial and tangential displacement amplitude along the
corrugated surface S for full-space plane strain cavity models with
different corrugation amplitudes ε and a P incidence as a function
of angle θ (Fig. 4.8) when 1 =
β
η , 2 = m , 1 = = b a , 2
1
= λ ,
1
1 1
= =ρ μ and 128 = K . 111

Figure 6.19: Radial and tangential displacement amplitude along the
corrugated surface S for full-space plane strain cavity models with
different corrugation amplitudes ε and a P incidence as a function
of angle θ (Fig. 4.8) when 1 =
β
η , 8 = m , 1 = = b a , 2
1
= λ ,
1
1 1
= =ρ μ and 128 = K . 112

Figure 6.20: Radial and tangential displacement amplitude along the
corrugated surface S for full-space plane strain cavity models with
different corrugation amplitudes ε and an SV incidence as a
function of angle θ (Fig. 4.8) when 1 =
β
η , 8 = m , 1 = = b a ,
2
1
= λ , 1
1 1
= =ρ μ and 128 = K . 113

xiii
Abstract
Anti-plane strain and plane strain models for steady-state scattering of
elastic waves by a rough scatterer embedded either in a full space or in a half
space is considered by using a direct boundary integral equation method.
Both cavity and inclusion problems are investigated. The roughness of the
scatterer is assumed to be periodic with arbitrary amplitude and period.
Detailed testing of the numerical results is presented. The motion along
the surface of the scatterer and the half-space surface is evaluated for
different corrugations, frequencies and impedance contrasts of the materials.
The importance of the scatterer roughness upon the displacement field is
clearly demonstrated. Larger corrugation amplitudes, shorter corrugation
periods and higher frequencies may produce significant changes in the
displacement field when compared with the corresponding smooth scatterer
results. These effects strongly depend upon the frequency, corrugation shape
and/or the impedance contrast of the materials.

1
Chapter 1: Introduction
Scattering of elastic waves by obstacles of arbitrary shapes is important
in many fields of science and engineering, e.g., geophysical prospecting, site
effects in strong ground motion seismology, and nondestructive evaluation of
materials. In geophysics, detection of underground mineral resources or
estimation of the site effects on seismic motion can be done by investigating
the scattering of waves by underground scatterers or topography near the
Earth surface (Aki, 1988). Similarly, in nondestructive evaluation,
irregularities and defects inside composite materials can be investigated by
analyzing the scattering of waves by sub-surface flaws and cracks (Miklowitz,
1983; Beskos, 1987; Beskos, 1997).
Scattering of elastic waves by a smooth scatterer has been studied using
both analytical and numerical methods. The analytical solutions are
applicable to problems involving simple geometry and material properties.
For example, Mow and Mente (1963) used the wave function expansion
method to investigate scattering of a plane harmonic SV incidence by a
two-dimensional cylindrical cavity in a full space. Later, Trifunac (1971) used

2
the same technique to compute the half-space surface motion by a
semi-circular alluvial valley and a semi-circular canyon (Trifunac, 1973)
subjected to a plane harmonic SH wave. Pao and Mow (1973) used the same
method to solve two-dimensional steady-state problems for scatterers
embedded in a full space and subjected to an incident SH wave. The above
work provides the analytical solutions which can be used for testing the
numerical results.
Numerical methods, on the other hand, can be used to investigate
models with complex geometry and materials. They include the wave
function expansion method, the Aki-Larner technique, the ray methods, the
finite difference method, the finite element method, the hybrid methods, the
indirect boundary integral equation method, and the direct boundary integral
equation method.
Using the wave function expansion method, Lee (1977) investigated
scattering of SH waves by a cylindrical cavity in a two-dimensional half space.
Datta and El-Akily (1978) adopted the matched asymptotic expansions to
examine the diffraction of SH waves by an elliptic cavity embedded in a half

3
space. Sanchez-Sesma (1983) used the wave function expansion method to
investigate scattering of plane harmonic P waves by a semi-spherical cavity
and an alluvial deposit in a half space. Similarly, Eshraghi and Dravinski
(1989, 1991) used the same technique to examine the scattering of SH, SV, P
and Rayleigh waves by non-axisymmetric dipping layers of arbitrary shapes
embedded in a three-dimensional half space. The spherical wave functions
used in Eshraghi and Dravinski’s papers did not satisfy the stress-free
boundary conditions at the half-space surface.
Meanwhile, Bard and Bouchon (1980, 1985) modeled the geometry of a
two-dimensional half-space valley of arbitrary shape using the Aki-Larner
technique (Aki and Larner, 1970). The results showed the presence of the
resonance patterns. However, the method did not work at high frequencies
and for large interface slopes (Bard and Bouchon, 1980).
Hong and Helmberger (1977) applied the ray expansion to investigate
high-frequency scattering of an SH line source in a two-dimensional
wedge-shaped medium with a free surface and an elastic lower boundary.
Later, by using a ray technique, Lee and Langston (1983) examined

4
scattering of P and SH waves by a three-dimensional circular basin in a half
space. However, the ray technique is not accurate at low frequencies.
Using the finite difference method, Alterman and Karal (1968)
investigated the scattering of a compressional point source in a
two-dimensional half-space layered model. Boore (1972, 1973) adopted the
same method to examine the effect of a ridge and a canyon subjected to a
SH wave in a two-dimensional half space. Smith and Bolt (1976) used the
finite element method to calculate Rayleigh waves in a layered model. It was
shown that the finite element method is suitable for modeling inhomogeneous
structures. However, the finite difference method and the finite element
method require establishing a computational grid throughout the domain of
the model. If a fine mesh is needed, the effectiveness of these methods will
be reduced. In addition, the artificial reflections from the edge of the model
reduce the accuracy of the results. Clayton and Engquist (1977) showed how
to minimize the artificial reflections from the boundary of the computational
domain.

5
Wong (1982) used an indirect boundary integral equation method to
solve for the motions in a two-dimensional half-space semi-elliptical canyon
problem subjected to plane harmonic P, SV and Rayleigh waves. The
maximum amplification for each incident wave was determined in the paper.
Dravinski (1982) used the same technique to investigate the scattering of an
SH, P, SV or Rayleigh wave by an alluvial valley of arbitrary shape in a half
space. The results showed the effects of the interface depth upon the
half-space surface displacement field. Later, Dravinski (1983) examined an
anti-plane strain model for dipping layers in a half space by using the same
technique. It was shown that the existence of dipping layers may generate
great amplification in the surface motion. Using an indirect boundary integral
equation technique similar conclusions were obtained by Dravinski and
Mossessian (1987) who investigated scattering of plane P, SV and Rayleigh
waves by dipping layers of arbitrary shape. Using the same approach Zheng
and Dravinski (1999) considered the amplification of elastic waves by an
anisotropic basin in a half space. The results demonstrated that the surface
response strongly depends upon the anisotropic material properties of the

6
basin and the nature of the incident wave. It should be pointed out that the
indirect boundary integral equation methods have difficulties in modeling
motion in very shallow basins.
Mossessian and Dravinski (1987) adopted a hybrid method which
combines the finite element method in the near field with an indirect boundary
integral equation method in the far field to solve for scattering of plane
harmonic P, SV and Rayleigh waves by an alluvial basin in a two-dimensional
half space. It was shown that the hybrid method is more effective than
boundary integral equation methods.
As for the direct boundary integral equation methods, Pao and Mow
(1973) investigated scattering in a two-dimensional and a three-dimensional
full-space cavity models, while Wong and Jennings (1975) considered a
two-dimensional arbitrary-shaped canyon subjected to plane SH waves.
Rizzo et al. (1985) used the same approach to investigate scattering of
elastic waves by a scatterer of arbitrary shape in a three-dimensional full
space, and showed that the singularity in the integrals can be removed.
Using a direct boundary integral equation formulation, Bouchon et al. (1996)

7
investigated scattering of S waves by a hill in a three-dimensional half space.
They found that the maximum amplification was atop the hill. Niu and
Dravinski (2003a, 2003b) discussed problems for a cavity in a
three-dimensional anisotropic full or half space using the same method. It
was shown that the resulting motion strongly depends upon the material
anisotropy.
All the above investigations on scattering of elastic waves by embedded
obstacles dealt with smooth scatterers. Since actual problems in various
scientific and engineering applications often involve scatterers with rough
surfaces, there is a need to investigate these problems in more detail.
Using Rayleigh’s method Asano (1960, 1961 and 1966) computed
reflection and refraction of plane SH, SV and P waves by a periodically
corrugated interface between two elastic half spaces. It was shown that, in
addition to regularly reflected and refracted waves, there exist the interface
waves as well. The reflected waves were found to decrease while the
refracted waves increase with greater corrugation amplitude. Abubakar
(1962a, 1962b) used a perturbation method to investigate reflection and

8
refraction of plane harmonic SH waves at a periodic interface between two
semi-infinite media as well. Abubakar’s work confirmed the existence of the
interface waves. By using the Rayleigh’s method, Kuo and Nafe (1962)
examined the propagation of Rayleigh waves in an elastic layer separated
from a half space by a sinusoidal interface. The results showed the local
phases and local group velocities of Rayleigh waves on the surface are
independent of the wave propagation direction. In addition, the phase and the
group velocity were found to be related to the frequency and the distance.
For scattering of electromagnetic waves by random rough surfaces
detailed review of literature can be found in Warnick and Chew (2001) and
Elfouhaily and Guérin (2004).
Fu et al. (2002) used the boundary element technique to evaluate the
attenuation of the wave energy in a homogeneous area between a rough
half-space surface and an irregular layer interface. The regional wave
amplitude was found to be strongly affected by the topography. Later, Fu
(2005) presented a detailed literature review about numerical techniques
used for this class of problems.

9
Recently, Dravinski (2007) used an indirect boundary integral equation
method to investigate scattering of plane harmonic SH, P, SV and Rayleigh
waves by a two-dimensional basin with a corrugated interface embedded in a
half space. The results showed that the peak surface motion atop the basin
may be significantly reduced due to the presence of the interface corrugation.
The purpose of the present study is to investigate scattering of elastic
waves by completely embedded scatterers with rough surfaces. For that end,
scattering of a plane harmonic P, SV, SH or Rayleigh wave by a corrugated
scatterer in a two-dimensional full or half space is investigated using a direct
boundary integral equation method. This study considers in detail the role of
the corrugation in the resulting motion caused by the scattered waves. The
corrugation is assumed to be of arbitrary amplitude and period.
The dissertation is organized as follows. The statement of the problems
is introduced in Chapter 2. This includes the geometry of the problems,
description of the corrugation, equations of motion, incident waves, radiation
conditions and boundary conditions.

10
The direct boundary integral equations for the anti-plane strain model
are introduced in Chapter 3 together with the solution of the full/half space
cavity/inclusion problems. The corresponding numerical results are
presented in Chapter 4 with the analytical solutions of the testing problems
which are presented as well. In Chapter 5 the integral equations for plane
strain models and the corresponding solutions are stated. The corresponding
numerical results and extensive analyses of the accuracy of the method are
presented in Chapter 6.
The dissertation concludes with summary and conclusions in Chapter 7.

11
Chapter 2: Statement of Problem
In this study both full-space and half-space problems for scattering of
elastic waves by a corrugated scatterer are considered. The two-dimensional
geometry of the half-space problem is depicted by Fig. 2.1. The problem
incorporates a half space B
1
and a scatterer B
2
with a corrugated interface S
embedded at depth h. The half-space boundary consists of the flat surface S
0

and an infinite surface S
1
. The outward unit normal, incident wave, and
scattered waves in domains B
1
and B
2
are denoted by n, u
inc
, u
sc(1)
and u
sc(2)
,
respectively. The origin of the right-handed Cartesian coordinate system
{x
1
,x
2
,x
3
} is placed at the half-space surface S
0
.
The corresponding full-space model can be obtained from Fig. 2.1 by
removing the half-space surface S
0
and by placing the origin of the Cartesian
coordinate system {x
1
,x
2
,x
3
} at the center of the scatterer. For convenience,
the x
2
-axis for the full-space problem is assumed to point upwards.
The incident wave u
inc
is assumed to be a plane harmonic SH, P, SV or
Rayleigh wave. The incident displacement vector is described as u=(0,0,u
3
)
for the anti-plane strain model and as u=(u
1
,u
2
,0) for the plane strain model.

12
The unknown scattered wave fields, u
sc(1)
and u
sc(2)
for the inclusion problems
and u
sc(1)
for the cavity problems, are to be determined throughout the elastic
media.

Figure 2.1: A half-space model for a scatterer B
2
with a corrugated surface S
embedded at depth h within domain B
1
. The surface of the half-space is
denoted by S
0
, and S
1
is the boundary at infinity. S’ represents the elliptic
smooth interface limit when the corrugation amplitude 0 → ε , while a and b
are the principal axes of S’. Furthermore, u
inc
, u
sc(1)
and u
sc(2)
denote the
incident wave and scattered waves in domains B
1
and B
2
, respectively, and n
is the outward unit normal vector on
0 1
S S S ∪ ∪ . Finally, γ is the off-vertical
angle of the incidence.

2.1. Common features for the problems
For the half-space model the corrugated surface S can be defined as
(Dravinski, 2007)
) , ( ) , ( ' ) , (
2 1 2 1 2 1
x x S x x S x x S
ε
+ = (2.1)

13
where S’ denotes a smooth elliptical surface defined by
( )






=
−
+ 1 : ) , ( '
2
2
2
2
2
1
2 1
b
h x
a
x
x x S (2.2)
in which a and b are the principal axes (Fig. 2.1) and S
ε
denotes the
perturbation to the smooth surface defined by
{ } θ θ ε θ θ ε
ε
sin ) sin( ; cos ) sin( : ) , (
2 1 2 1
m x m x x x S = = ; 1 0 ≤ ≤ε ; π θ π ≤ < −
(2.3)
Here ε is the corrugation amplitude while m is the period coefficient. The
corrugated scatterer for the full-space problem is defined in a similar manner.
It is evident from the above equations that the zero corrugation
amplitude ε implies that the corrugated surface S reduces to the smooth limit
S’.
The equations of motion for steady-state SH waves are given by (Mal
and Singh, 1991)
0
) (
3
2 ) (
3
2
= + ∇
j
j
j
u k u
β
;
j
j
v
k
β
β
ω
= ;
j
j
j
v
ρ
μ
β
= ; 2 : 1 = j (2.4)
and for steady-state plane-strain waves

14
2 : 1 ;
2
; ; 0
) ( 2 ) ( 2
=
+
= = = Φ + Φ ∇ j v
v
k k
j
j j
j
j
j
j
j
j
ρ
μ λ
ω
α
α
α α
(2.5)
2 : 1 ; ; ; 0
) ( 2 ) ( 2
= = = = Ψ + Ψ ∇ j v
v
k k
j
j
j
j
j
j
j
j
ρ
μ
ω
β
β
β β
(2.6)
where
) (
3
j
u represent the displacement fields and Φ
(j)
and Ψ
(j)
are the
displacement potentials in domain B
j
. In addition, k
α
and k
β
denote the
wavenumbers of the dilatational waves and shear waves, respectively, ω is
the circular frequency, v
α
and v
β
are the speeds of the dilatational waves and
shear waves, respectively, λ and μ are the Lame constants, and ρ is the
density. Throughout the paper, the material properties corresponding to
domain B
j
; j=1:2 are labeled with the subscript j=1:2 (e.g., λ
1
, μ
1
, ρ
1
, … etc.).
Finally, the Laplacian is defined by
( ) ( )
ii ,
2
≡ ∇ ; 2 : 1 = i (2.7)
The summation convention over repeated indices is understood, and unless
stated otherwise the range of the indices is from 1 to 2. For the plane strain
model the displacement vectors u
(j)
in domain B
j
can be obtained from the
corresponding potentials by

15






Ψ + Φ
Ψ − Φ
=






=
) (
1 ,
) (
2 ,
) (
2 ,
) (
1 ,
) (
2
) (
1 ) (
j j
j j
j
j
j
u
u
u ; 2 : 1 = j (2.8)
where ,i (i=1:2) indicate partial differentiation with respect to x
i
. Since for the
plane strain model there is no displacement in the x
3
direction, the
component u
3
is suppressed throughout.
This completes the description of the scatterers and the equations of
motion. The incident waves and the boundary conditions are considered
next.
2.2. Half-space problems
For the anti-plane strain model an oblique incident SH wave is assumed
to be of the form
) cos sin (
3
2 1 1
) (
γ γ
β
x x ik
inc
e u
−
= x ;
1
B ∈ x (2.9)
where γ represents the off-vertical angle of incidence.
For plane strain model an oblique incident P wave is defined as
0 ) (
1
) (
) cos sin (
1
2 1 1
= Ψ
−
= Φ
−
x
x
inc
x x ik inc
e
ik
γ γ
α
α
;
1
B ∈ x (2.10)





−
=
−
−
) cos sin (
) cos sin (
2 1 1
2 1 1
cos
sin
) (
γ γ
γ γ
α
α
γ
γ
x x ik
x x ik
inc
e
e
x u ;
1
B ∈ x (2.11)

16
while an oblique incident SV wave is given by
) cos sin (
1
2 1 1
1
) (
0 ) (
γ γ
β
β
x x ik
inc
inc
e
ik
−
= Ψ
= Φ
x
x
;
1
B ∈ x (2.12)






=
−
−
) cos sin (
) cos sin (
2 1 1
2 1 1
sin
cos
) (
γ γ
γ γ
β
β
γ
γ
x x ik
x x ik
inc
e
e
x u ;
1
B ∈ x (2.13)
An incident Rayleigh wave is specified by
2
1
2
2 1
2
1
2
2 1
2
1
2 2
2
1
2
2
2
) (
1
) (
β
α
β
β
k k x x ik
R R
R inc
k k x x ik
R
inc
R R
R R
e
k k k
k k
e
ik
− −
− −
−
−
− = Ψ
= Φ
x
x
;
1
B ∈ x (2.14)














− −
−
−
−
−
− −
=
− − − −
− − − −
1
2
1
2
2
2
1
2
2
1
2
1
2
2
2
1
2
2
]
2
2
[
] )
2
1 ( [
) (
2
1
2 2
1
2
2
1
2 2
1
2
2
2
1
x ik
k k x
R R
R k k x
R
R
x ik
k k x
R
k k x
inc
R
R R
R
R R
e e
k k k k
k k
e
ik
k k
e e
k
k
e
β α
β α
β α
β α
β
x u ;
1
B ∈ x
(2.15)
Here k
R
denotes the wavenumber of the Rayleigh waves. For convenience,
the factor
t i
e
ω −
and the ω -dependence of all the variables are suppressed.
For the inclusion problem, when the incident wave strikes the scatterer, it
generates the scattered waves u
sc(1)
and u
sc(2)
within the half space and the
elastic inclusion, respectively. They must satisfy the equations of motion, and

17
u
sc(1)
must satisfy the radiation condition in the far field (Kupradze, 1963).
Therefore, the motion in the domains B
1
and B
2
can be described as
2
) 2 ( ) 2 (
1
) 1 ( ) 1 (
); ( ) (
); ( ) ( ) (
B
B
sc
sc ff
∈ =
∈ + =
x x u x u
x x u x u x u
(2.16)
where u
ff
denotes the free-field displacement field which is generated by the
incident wave in absence of the scatterer, and u
sc(1)
and u
sc(2)
represent the
unknown scattered waves.
For the cavity problem the motion in the half space can be described by
the above equation with only the scattered field u
sc(1)
to be determined.
The boundary and the continuity conditions for the inclusion problem are
given by
0 ) , (
) 1 (
= n x f ;
0
S ∈ x (2.17)
and
S
S
∈ =
∈ =
x n x f n x f
x x u x u
); , ( ) , (
); ( ) (
) 2 ( ) 1 (
) 2 ( ) 1 (
(2.18)
while the traction-free boundary conditions for the cavity problem are given
by
0 ) , (
) 1 (
= n x f ;
0
S S∪ ∈ x (2.19)

18
where f
(j)
denote the traction field in domain B
j
, and S, S
0
and n represent the
corrugation, half-space surface and outward unit normal, respectively.
This completes the formulation of the half-space problem. The full-space
model is considered next.
2.3. Full-space problems
For this problem the incident waves are assumed to propagate
horizontally through the full space in the positive x
1
direction. Thus, the
incident waves for SH, P and SV waves can be obtained by substituting
ο
90 = γ into equations (2.9) to (2.13). Therefore, the total wave field in the full
space consists of the incident and scattered waves
) ( ) ( ) (
) 1 ( ) 1 (
x u x u x u
sc inc
+ = ;
1
B ∈ x (2.20)
while for the inclusion problem the total wave field in the elastic inclusion
consists of the scattered wave only
) ( ) (
) 2 ( ) 2 (
x u x u
sc
= ;
2
B ∈ x (2.21)
In addition to the equations of motion, the scattered waves in the full space
must satisfy appropriate radiation conditions at infinity (Kupradze, 1963).

19
For the inclusion problem, the displacement and traction continuity
conditions along the interface S are specified by equations (2.18) while for
the cavity problem, the traction-free boundary condition at the surface S is
specified by
0 ) , (
) 1 (
= n x f ; S ∈ x (2.22)
where n represents the outward unit normal at the corrugated surface S.
This completes the formulation of all the problems. The corresponding
solutions for the anti-plane strain model are considered first.

20
Chapter 3: Anti-Plane Strain Model Solutions
The direct boundary integral equation method is used to solve for the
unknown scattered waves. The half-space inclusion problem is considered
first.
3.1. Half-space inclusion problem solution
For this problem, the boundary integral equation for SH waves in domain
B
1
becomes (Niu and Dravinski, 2003b)
∫ ∫
+
− =
0
) ( ) , , ( ) , ( ) , ( ) ( ) (
) 1 (
3
) 1 ( ) 1 (
3
) 1 ( ) 1 (
3
) 1 (
S S
x
sc
S
x
sc sc sc
dS u t PV dS f g u c x n y x n x y x y y ;
0
S S∪ ∈ y (3.1)
while for domain B
2
, one has
∫ ∫
− = −
S
x
S
x
sc
dS u t PV dS f g u c ) ( ) , , ( ) , ( ) , ( ) ( ) (
) 2 (
3
) 2 ( ) 2 (
3
) 2 ( ) 2 (
3
) 2 (
x n y x n x y x y y ; S ∈ y
(3.2)
Here g
(j)
(x,y) denote the full-space displacement Green’s functions at the
observation point x due to an anti-plane strain harmonic source at y, and
t
(j)
(x,y,n) are the corresponding traction Green’s functions for domain B
j
, j=1:2
(Kobayashi, 1987). In addition,
) 1 (
3
sc
u ,
) 2 (
3
u ,
) 1 (
3
sc
f and
) 2 (
3
f are the unknown

21
displacement and traction fields, c
sc(j)
represent the free terms and PV
denotes the principal value integrals (Paris and Caňas, 1997).
It should be noted that the presence of the half-space surface S
0
in
equation (3.1) occurs due to the fact that the full-space Green’s functions
adopted in the study do not satisfy the traction-free boundary condition along
the half-space surface S
0
(Kobayashi, 1987). If the half-space Green’s
functions are used for this problem, then the half-space surface S
0
is not
present in equation (3.1). However, the purpose of this study is to facilitate
future research of three-dimensional models for both isotropic and
anisotropic half spaces. Since for these general models the corresponding
full-space Green’s functions are much easier to evaluate numerically than the
equivalent half-space Green’s functions (Dravinski and Mossessian, 1988;
Dravinski and Zheng, 2000; Dravinski and Niu, 2002; Chen and Dravinski,
2007a and 2007b), the full-space Green’s functions are chosen to be used for
this study.
Using the boundary and continuity conditions, the last two equations
become

22
) ( ) ( ) ( ) , , ( ) , ( ) , (
) , ( ) , ( ) ( ) , , ( ) ( ) (
3
) 1 (
3
) 1 (
3
) 1 (
3
) 1 ( ) 1 (
3
) 1 ( ) 1 (
3
) 1 (
0
0
y y x n y x n x y x
n x y x x n y x y y
ff sc
S S
x
ff
S
x
ff
S
x
S S
x
sc
u c dS u t PV dS f g
dS f g dS u t PV u c
+ + − =
− +
∫ ∫
∫ ∫
+
+
;
0
S S∪ ∈ y (3.3)
0 ) , ( ) , ( ) ( ) , , ( ) ( ) (
3
) 2 (
3
) 2 (
3
) 2 (
= − + −
∫ ∫
S
x
S
x
sc
dS f g dS u t PV u c n x y x x n y x y y ; S ∈ y
(3.4)
where
) , ( ) , ( ) , (
) ( ) ( ) (
) 2 (
3
) 1 (
3 3
) 2 (
3
) 1 (
3 3
n x n x n x
x x x
f f f
u u u
= =
= =
; S ∈ x ;
0
S ∉ x (3.5)
The above integral equations must be solved for the unknown displacements
u
3
(x) and tractions f
3
(x,n) along the corrugated interface S, and for the
displacements ) (
) 1 (
3
x u along the half-space surface S
0
. Then the
displacement fields ) (
) 1 (
3
x u and ) (
) 2 (
3
x u can be evaluated throughout the
half space by using the integral representations
∫ ∫
+
− + =
0
) ( ) , , ( ) , ( ) , ( ) ( ) (
) 1 (
3
) 1 ( ) 1 (
3
) 1 (
3
) 1 (
3
S S
x
sc
S
x
sc ff
dS u t PV dS f g u u x n y x n x y x y y ;
1
B ∈ y ;
0
S S∪ ∉ y (3.6)
∫ ∫
+ − =
S
x
S
x
dS u t PV dS f g u ) ( ) , , ( ) , ( ) , ( ) (
3
) 2 (
3
) 2 ( ) 2 (
3
x n y x n x y x y ;
2
B ∈ y ; S ∉ y
(3.7)

23
In order to solve the integral equations (3.3) and (3.4), the boundaries S
and S
0
are discretized using linear elements, i.e.,
0 0
S S S S ∂ + ∂ ≈ + , where
S ∂ and
0
S ∂ denote the discretized boundaries (Fig. 3.1). At a single
collocation point, the integral equations (3.3) and (3.4) become
[ ] [ ] ) (
) 1 (
) (
) 1 (
) (
) ( ) (
1 3
3 ) 1 (
2
) 1 (
1
1
) 1 (
3
) 1 (
3 ) 1 (
2
) 1 (
1
) 1 (
3
) 1 (
l F
k f
k f
B B
k u
k u
A A l u l c
P
k
lk lk
K
k
lk lk
sc
=






+
−






+
+
∑ ∑
= =
;
0
S S
l
∂ ∪ ∂ ∈ y ; ) 1 ( : 1 + = K l (3.8)
[ ] [ ] 0
) 1 (
) (
) 1 (
) (
) ( ) (
1 3
3 ) 2 (
2
) 2 (
1
1 3
3 ) 2 (
2
) 2 (
1 3
) 2 (
=






+
−






+
+ −
∑ ∑
= =
P
k
lk lk
P
k
lk lk
sc
k f
k f
B B
k u
k u
A A l u l c ;
S
l
∂ ∈ y ; P l : 1 = (3.9)
where
[ ] [ ] ) ( ) (
) 1 (
) (
) 1 (
) (
) (
3
) 1 (
1
3
3 ) 1 (
2
) 1 (
1
1
3
3 ) 1 (
2
) 1 (
1
l u l c
k u
k u
A A
k f
k f
B B l F
ff sc
K
k
ff
ff
lk lk
P
k
ff
ff
lk lk
+






+
+






+
− =
∑ ∑
= =

(3.10)
Here l is the collocation point, u
3
(k) and f
3
(k) are the unknown displacement
and traction at node k, P denotes the number of elements used to discretize
the inclusion boundary S, and K is the total number of elements. The
integration constants
) ( j
ilk
A and
) ( j
ilk
B are specified by

24
∫
∫
−
−
=
=
1
1
) ( ) (
1
1
) ( ) (
) ( ) , (
) ( ) , , (
ξ ξ ξ
ξ ξ ξ
d J N l g B
d J N l t A
k
i
j j
ilk
k
i
k j j
ilk
n
; 2 : 1 , = j i (3.11)
where ξ, N
i
, and J
k
are the local coordinate, the shape functions and the
Jacobians, respectively, and n
k
is the unit normal on the element
k
S ∂ . It
should be pointed out that the integration constants
) ( j
ilk
A in the last equation
are computed in the sense of the principal value integrals (Paris and Caňas,
1997). For linear elements, the shape functions and Jacobians are defined as
(Paris and Caňas, 1997)
2
1
2
1
2
1
ξ
ξ
+
=
−
=
N
N
(3.12)
2
k k
L
J = (3.13)
Here L
k
denotes the length of the element
k
S ∂ . Finally, the full-space
displacement and traction Green’s functions for SH waves are given by
(Kobayashi, 1987)
) (
4
) , (
) 1 (
0
r k H
i
g
β
μ
= y x (3.14)
n
r
r k H
ik
t
∂
∂
−
= ) (
4
) , , (
) 1 (
1 β
β
n y x ; y x− = r (3.15)

25
Here H
0
(1)
and H
1
(1)
are the Hankel functions of the first kind of orders zero
and one, respectively, r is the distance between the source y and observation
point x, while n denotes the unit normal vector on a surface where the
traction is being evaluated.

Figure 3.1: Discretized boundaries S ∂ and
0
S ∂ for the half-space surface
problem, where
∑
=
∂ = ∂
P
k
k
S S
1
and
∑
+ =
∂ = ∂
K
P k
k
S S
1
0
. Here P denotes the number
of elements along the corrugation and
k
S ∂ denotes the element k, and K is
the total number of elements. The length of each element
k
S ∂ is L
k
, and n
k

denotes the corresponding unit outward normal vector.

After placing the collocation point y
l
at each node along the interface S
and the half-space surface S
0
, results in K+1 equations of the type (3.8) and
P equations of the type (3.9). Based on these equations, the unknown
displacement field u
3
and traction field f
3
along the corrugated interface, and

26
the half-space surface displacement field
) 1 (
) 1 , 2 ( 3
u can be computed from the
following linear system
HSI HSI HSI
3 3 3
F U A = ;
) 1 ( ) 1 (
3
P K P K HSI
C
+ + × + +
∈ A ;
1 ) 1 (
3
× + +
∈
P K HSI
C U ;
1 ) 1 (
3
× + +
∈
P K HSI
C F
(3.16)
where










−
−
−
=
HS HS
HS HS HS
HS HS HS
HSI
) 2 ( ) 2 (
) 1 (
) 1 , 2 (
) 1 (
) 2 , 2 (
) 1 (
) 1 , 2 (
) 1 (
) 1 , 1 (
) 1 (
) 2 , 1 (
) 1 (
) 1 , 1 (
3
B 0 A
B A A
B A A
A ;
P P HS
C
×
∈
) 1 (
) 1 , 1 (
A ;
) 1 ( ) 1 (
) 2 , 1 (
P K P HS
C
− + ×
∈ A ;
P P K HS
C
× − +
∈
) 1 ( ) 1 (
) 1 , 2 (
A ;
) 1 ( ) 1 ( ) 1 (
) 2 , 2 (
P K P K HS
C
− + × − +
∈ A ;
P P HS
C
×
∈
) 1 (
) 1 , 1 (
B ;
P P K HS
C
× − +
∈
) 1 ( ) 1 (
) 1 , 2 (
B ;
P P HS
C
×
∈
) 2 (
A ;
P P HS
C
×
∈
) 2 (
B (3.17)










=
0
F
F
F
) 1 , 2 (
) 1 , 1 (
3
HSI
;
1
) 1 , 1 (
×
∈
P
C F ;
1 ) 1 (
) 1 , 2 (
× − +
∈
P K
C F (3.18)










=
3
) 1 (
) 1 , 2 ( 3
3
3
f
u
u
U
HSI
;
1
3
×
∈
P
C u ;
1 ) 1 ( ) 1 (
) 1 , 2 ( 3
× − +
∈
P K
C u ;
1
3
×
∈
P
C f (3.19)
and
K K
C
×
denotes a K K× dimensional complex vector space. The matrix
HSI
3
A is expressed in terms of the integration constants
) ( j
ilk
A and
) ( j
ilk
B and
the free terms while the vector
HSI
3
F contains the free-field results.

27
The key feature of the linear system (3.16) is that is can be solved by
using standard Gauss elimination procedure.
Once the corrugation displacements u
3
and tractions f
3
, and the
half-space surface displacements
) 1 (
) 1 , 2 ( 3
u are known, the total displacement
fields throughout the half space can be evaluated by using the integral
representations (3.6) and (3.7).
This completes solution of the half-space inclusion problem. The
half-space cavity problem is considered next.
3.2. Half-space cavity problem solution
For this problem, the boundary integral equation for the half space is
given by equation (3.1). Using the traction-free boundary conditions, equation
(3.1) becomes
∫ ∫
− = +
+ S
x
ff
S S
x
sc sc sc
dS f g dS u t PV u c ) , ( ) , ( ) ( ) , , ( ) ( ) (
3
) 1 ( ) 1 (
3
) 1 ( ) 1 (
3
) 1 (
0
n x y x x n y x y y ;
0
S S∪ ∈ y (3.20)
where
ff
f
3
are the known free-field tractions along the corrugated surface S.
The above integral equation must be solved for the unknown displacements
along the corrugated surface S and the half-space surface S
0
. Then the

28
scattered displacement field ) (
) 1 (
3
x u can be evaluated throughout the half
space by using the first integral representation (3.6) for the inclusion problem.
When the boundaries S and S
0
are discretized using one-dimensional
linear elements, equation (3.20) at a single collocation point becomes
[ ] ) (
) 1 (
) (
) ( ) (
1
) 1 (
3
) 1 (
3 ) 1 (
2
) 1 (
1
) 1 (
3
) 1 (
l F
k u
k u
A A l u l c
K
k
sc
sc
lk lk
sc sc
=






+
+
∑
=
;
0
S S
l
∂ ∪ ∂ ∈ y ;
) 1 ( : 1 + = K l (3.21)
where
[ ]
∑
=






+
− =
P
k
ff
ff
lk lk
k f
k f
B B l F
1
3
3 ) 1 (
2
) 1 (
1
) 1 (
) (
) ( (3.22)
Here the collocation point y
l
is located at node l , and all the other terms in
the last two equations have been already defined following equation (3.10).
After placing the collocation point y
l
at each node of the surfaces S and
S
0
, there will be K+1 equations of the type (3.21) and they can be combined
as
HSC sc HSC
3 3 3
F U A = ;
) 1 ( ) 1 (
3
+ × +
∈
K K HSC
C A ;
1 ) 1 (
3
× +
∈
K sc
C U ;
1 ) 1 (
3
× +
∈
K HSC
C F (3.23)
where the matrix
HSC
3
A and the vector
HSC
3
F are known while
sc
3
U
represents the vector of the unknown scattered displacements along the

29
cavity surface S and the half-space surface S
0
. As before,
HSC
3
A is
expressed in terms of the integration constants and the free terms while
HSC
3
F involves the values of the integration constants and the free-field
tractions along the cavity surface.
Based on the last equation, the unknown displacements
sc
3
U along the
corrugated surface S and the half-space surface S
0
can be computed using
the Gauss elimination procedure. Then the total displacement field
throughout the half space can be evaluated through the equation (3.6).
This completes solution of the half-space cavity problem. The full-space
inclusion problem solution is considered next.
3.3. Full-space inclusion problem solution
For this problem, the general boundary integral equations (Niu and
Dravinski, 2003a) can be simplified for SH waves in domain B
1
as
∫ ∫
− + =
S
x
S
x
inc sc
dS u t PV dS f g u u c ) ( ) , , ( ) , ( ) , ( ) ( ) ( ) (
) 1 (
3
) 1 ( ) 1 (
3
) 1 (
3
) 1 (
3
) 1 (
x n y x n x y x y y y ;
S ∈ y (3.24)
while for domain B
2
one gets equation (3.2). Here S denotes the corrugation

30
surface, and all the other terms have been defined following equations (3.1)
and (3.2).
By substituting the continuity conditions into equations (3.24) and (3.2)
results in the following equation
) ( ) , ( ) , ( ) ( ) , , ( ) ( ) (
3 3
) 1 (
3
) 1 (
3
) 1 (
y n x y x x n y x y y
inc
S
x
S
x
sc
u dS f g dS u t PV u c = − +
∫ ∫
; S ∈ y
(3.25)
together with equation (3.4) obtained earlier. Furthermore,
) , ( ) , ( ) , (
) ( ) ( ) (
) 2 (
3
) 1 (
3 3
) 2 (
3
) 1 (
3 3
n x n x n x
x x x
f f f
u u u
= =
= =
; S ∈ x (3.26)
The integral equations (3.25) and (3.4) have to be solved for the unknown
displacement and traction fields u
3
(x) and f
3
(x,n) along the corrugated
interface S. Subsequently, the displacement fields
) 1 (
3
u and
) 2 (
3
u can be
evaluated throughout the full space using the integral representation
∫ ∫
− + =
S
x
S
x
inc
dS u t PV dS f g u u ) ( ) , , ( ) , ( ) , ( ) ( ) (
3
) 1 (
3
) 1 (
3
) 1 (
3
x n y x n x y x y y ;
1
B ∈ y ;
S ∉ y (3.27)
together with equation (3.7).

31

Figure 3.2: A discretized corrugated boundary
∑
=
∂ = ∂
K
k
k
S S
1
in a full space
using linear elements. Here, K denotes both the number of nodes and the
number of elements. The length of each element is denoted by L
k
, k=1:K, and
n
k
represents outward unit normal on the element
k
S ∂ .

When the boundary S is discretized using one-dimensional linear
elements, as shown by Fig. 3.2, at a single collocation point y
l
the integral
equations become
[ ] [ ] ) (
) 1 (
) (
) 1 (
) (
) ( ) (
3
1 3
3 ) 1 (
2
) 1 (
1
1 3
3 ) 1 (
2
) 1 (
1 3
) 1 (
l u
k f
k f
B B
k u
k u
A A l u l c
inc
K
k
lk lk
K
k
lk lk
sc
=






+
−






+
+
∑ ∑
= =
;
S
l
∂ ∈ y ; K l : 1 = ;
∑
=
∂ = ∂
K
k
k
S S
1
(3.28)
[ ] [ ] 0
) 1 (
) (
) 1 (
) (
) ( ) (
1 3
3 ) 2 (
2
) 2 (
1
1 3
3 ) 2 (
2
) 2 (
1 3
) 2 (
=






+
−






+
+ −
∑ ∑
= =
K
k
lk lk
K
k
lk lk
sc
k f
k f
B B
k u
k u
A A l u l c
(3.29)

32
Here the collocation point y
l
is located at node l . All the other terms in the
last two equations have been already defined following equations (3.8) and
(3.9).
Based on the last two equations, the unknown displacement vector u
3

and traction vector f
3
along the corrugated interface S can be computed from
the linear system
inc FSI FSI
3 3 3
U U A = ;
K K FSI
C
2 2
3
×
∈ A ;
1 2
3
×
∈
K FSI
C U ;
1 2
3
×
∈
K inc
C U (3.30)
where






−
−
=
FS FS
FS FS
FSI
) 2 ( ) 2 (
) 1 ( ) 1 (
3
B A
B A
A ;
K K FS
C
×
∈
) 1 (
A ;
K K FS
C
×
∈
) 1 (
B ;
K K FS
C
×
∈
) 2 (
A ;
K K FS
C
×
∈
) 2 (
B (3.31)






=
0
u
U
inc
inc 3
3
;
1
3
×
∈
K inc
C u (3.32)






=
3
3
3
f
u
U
FSI
;
1
3
×
∈
K
C u ;
1
3
×
∈
K
C f (3.33)
The matrices A
(1)FS
and A
(2)FS
incorporate the free terms and the integration
constants
) ( j
ilk
A while the matrices B
(1)FS
and B
(2)FS
include the integration
constants
) ( j
ilk
B . The vector
inc
3
u , on the other hand, contains the incident

33
wave field values. The unknown displacement and traction fields can be
found by using Gauss elimination.
Once the displacement and traction fields along the corrugation are
known, the total displacements throughout the full space and the elastic
inclusion can be evaluated by using the integral representations (3.7) and
(3.27).
This completes solution of the full-space inclusion problem. The
full-space cavity problem is considered next.
3.4. Full-space cavity problem solution
For this problem, the boundary integral equation for the full space is
given by equation (3.24). By substituting the traction-free boundary condition
into equation (3.24) results in the following
) ( ) ( ) , , ( ) ( ) (
3
) 1 (
3
) 1 ( ) 1 (
3
) 1 (
y x n y x y y
inc
S
x
sc
u dS u t PV u c = +
∫
; S ∈ y (3.34)
The last equation has to be solved first for the unknown displacement field
) 1 (
3
u along the corrugation surface S. Subsequently, the displacement field
) 1 (
3
u can be evaluated throughout the full space using the integral
representation

34
∫
− =
S
x
inc
dS u t PV u u ) ( ) , , ( ) ( ) (
) 1 (
3
) 1 (
3
) 1 (
3
x n y x y y ;
1
B ∈ y ; S ∉ y (3.35)
In order to solve the integral equation (3.34), the corrugated boundary S
is discretized using linear elements. Consequently, at a single collocation
point, the integral equation becomes
[ ] ) (
) 1 (
) (
) ( ) (
3
1
) 1 (
3
) 1 (
3 ) 1 (
2
) 1 (
1
) 1 (
3
) 1 (
l u
k u
k u
A A l u l c
inc
K
k
lk lk
sc
=






+
+
∑
=
; S
l
∂ ∈ y ; K l : 1 = ;
∑
=
∂ = ∂
K
k
k
S S
1
(3.36)
where l denotes the collocation point y
l
, and all the other terms have been
defined following equation (3.10).
When applied to all collocation points K l : 1 = , the above equation leads
to the following result
inc FSC FSC
3 3 3
u U A = ;
K K FSC
C
×
∈
3
A ;
1
3
×
∈
K FSC
C U ;
1
3
×
∈
K inc
C u (3.37)
The matrix
FSC
3
A and the vector
inc
3
u are known while
FSC
3
U represents the
vector of the unknown displacements at the nodes along the cavity surface S.
The elements of matrix
FSC
3
A incorporate the integration constants and the
free terms while the vector
inc
3
u consists of the incident wave field evaluated
along the cavity surface.

35
Based on the last result, the unknown displacements
FSC
3
U along the
corrugated surface S can be computed by using standard Gauss elimination
procedure. Once these displacements are known, the total displacements
throughout domain B
1
can be evaluated by using the integral representation
(3.35).
This completes the solution of the full-space cavity problem. Numerical
results for the anti-plane strain model are considered next.

36
Chapter 4: Anti-Plane Strain Model Numerical Results
For convenience, a dimensionless frequency
β
η is introduced as a ratio
of the smooth scatterer width and the wavelength of the shear waves
β
β
λ
η
a 2
= (4.1)
For the problems at hand, the corrugation parameters are tabulated in
Table 4.1. In addition, two types of elastic inclusions are considered for the
inclusion problems: a soft inclusion ( 6 / 1
2
= μ , 2 / 1
2
=
β
v ) and a stiff inclusion
( 8
2
= μ , 2
2
=
β
v ). The corresponding impedance contrasts for the two types of
the inclusions are 3
2 2
1 1
= =
β
β
β
ρ
ρ
v
v
I
soft
and
4
1
=
stiff
I
β
.

Table 4.1: The corrugation parameters for the anti-plane strain model.
Medium Scatterer Parameter
Half Space
Inclusion
2 . 0 0 ≤ ≤ε ,
8 2 ≤ ≤ m , 1 = a ,
75 . 1 ; 1 ; 75 . 0 = b ,
1
1
= μ , 1
1
=
β
v ,
2 25 . 0 ≤ ≤
β
η
8 ; 6 / 1
2
= μ ,
2 ; 2 / 1
2
=
β
v
928 = K ,
128 = P ,
a h 2 = ,
2 / ; 0π γ =
Cavity -
Full Space
Inclusion
8 ; 6 / 1
2
= μ ,
2 ; 2 / 1
2
=
β
v
128 = K
Cavity -
It should be noted that the integral equations for the exterior problems
(e.g., equations 3.3, 3.20, 3.25 and 3.34) have non-unique solutions at

37
certain frequencies (Burton and Miller, 1971; Shaw, 1979; Kobayashi and
Nishimura, 1982a and 1982b). Burton and Miller (1971) showed that for the
full-space cavity problem these fictitious frequencies can be evaluated by
considering the resonant frequencies of the corresponding interior problem
within the domain B
2
with the zero-displacement boundary conditions. In
other words, the fictitious frequencies for the exterior integral equation can be
assessed by investigating the eigenvalues of the corresponding interior
problem with an altered boundary condition. That is the Dirichlet/Neumann
boundary condition for the exterior problem needs to be modified to the
Neumann/Dirichlet boundary condition for the corresponding interior problem
(Burton and Miller, 1971; Shaw, 1979).
For all problems at hand with the form F AU= , the solution is
non-unique at eigenfrequencies η
β
if the determinant of ) (
β
η A is zero.
Since the purpose of the study is to determine the importance of the surface
corrugation, the details of finding the η
β
for 0 ) ( ≈
β
η A are omitted here.
Those eigenfrequencies η
β
for which 0 ) ( ≈
β
η A have been identified in

38
order to present the results only at the frequencies different from the
eigenfrequencies.
4.1. Half-space inclusion problem results
Initially, a transparency test is performed. Namely, when the domains B
1

and B
2
have the same material properties, the displacements along the
half-space surface S
0
should agree with the free-field result. Therefore, the
surface length S
0
has been increased to match the free-field response at
each frequency. The difference between the free-field and numerical results
along the half-space surface S
0
can be evaluated by computing the following
error
∑
∑
+
+ =
+
+ =
−
=
1
1
2
3
1
1
2
3 3
0 0
) (
) ( ) (
) , (
K
P i
i
ff
K
P i
i i
ff
HS
u
u u
S Err
x
x x
β
η ;
0
S
i
∈ x (4.2)
where
ff
u
3
and
3
u are the half-space surface displacements corresponding
to the free-field and the smooth interface inclusion limit, respectively and S
0

denotes the half-space surface. Table 4.2 shows the results for
HS
Err
0

computed for a vertical incidence as a function of the frequency and the

39
length of the half-space surface. Apparently, the relative error
HS
Err
0
varies
consistently with frequency η
β
and the surface length S
0
. That is, the error
increases with the increase of the frequency, and decreases with the
increase of the surface length. Similar calculations have been performed for a
grazing incidence as well. These results show similar consistency observed
for a vertical incidence and thus they are omitted.

Table 4.2: The relative error ) , (
0 0
S Err
HS
β
η along the half-space surface S
0

as a function of frequency η
β
and the length of S
0
for a transparency test
( 1 = = b a , 1
2 1
= =μ μ , 1
2 1
= =
β β
v v and 0 = ε ) subjected to a vertical plane
harmonic SH wave.
0
\ S
β
η 10
0
= S 20
0
= S 40
0
= S 80
0
= S
25 . 0 =
β
η
0.0050% 0.0036% 0.0026% 0.0018%
1 =
β
η
0.0412% 0.0390% 0.0344% 0.0285%
2 =
β
η
0.1277% 0.0954% 0.0737% 0.0568%
In addition, the length of the half-space surface S
0
can be confirmed
based on the smooth interface response of a half-space semi-cylindrical
valley model for which the exact solution is available (Trifunac, 1971). As the
length of the half-space surface S
0
increases, the smooth interface limit
response of the valley problem should agree with the exact solution.

40
Therefore, at each frequency, the length of the half-space surface S
0
has
been increased to match the exact solution. Table 4.3 illustrates this for a
vertical incidence in terms of an error computed as a function of frequency
and the length of the half-space surface. The error
HS
Err
1
is defined as
∑
∑
+
=
+
=
−
=
1
1
2
1
1
2
3
0 1
) (
) ( ) (
) , (
K
i
i exact
K
i
i i exact
HS
u
u u
S Err
x
x x
β
η ;
0
S
i
∈ x (4.3)
where
exact
u and
3
u are displacements corresponding to the exact solution
and the smooth surface valley limit, respectively while S
0
is the half-space
surface.
It is apparent from Table 4.3 that the relative error uniformly increases
with the increase of the frequency. Based on the results of Tables 4.2 and 4.3,
the computational range of the half-space surface is chosen as { } a x S 20 :
0
≤ .
Any further increase of the range of S
0
produced a negligible effect upon the
resulting motion.
The relative errors for a grazing incidence provided similar conclusions
obtained for the vertical incidence, so they are omitted.

41
Table 4.3: The relative error ) , (
0 1
S Err
HS
β
η as a function of frequency η
β
and
the length of the half-space surface S
0
for a semi-circular soft valley ( 6 / 1
2
= μ ,
2 / 1
2
=
β
v ) subjected to a vertical plane harmonic SH wave.
0
\ S
β
η 10
0
= S 20
0
= S 40
0
= S 80
0
= S
25 . 0 =
β
η
0.0685% 0.0377% 0.0278% 0.0201%
1 =
β
η
0.2293% 0.2073% 0.1723% 0.1346%
2 =
β
η
0.4321% 0.3735% 0.3028% 0.2372%
Finally, a convergence test is considered as well. For a nonzero
corrugation amplitude ε, the rate of convergence between two numerical
results with different numbers of elements K along the corrugated surface S
and the half-space surface S
0
can be measured by computing the error
∑
∑
+
=
+
=
−
−
=
1
1
2
3
1
1
2
1 2 3 3
2
) , (
) 2 , ( ) , (
) , (
K
i
i
K
i
i i
HS
K u
K u K u
K Err
x
x x
β
η ;
0 1 2
, S S
i i
∪ ∈
−
x x (4.4)
where ) , (
3
K u
i
x and ) 2 , (
1 2 3
K u
i−
x denote the displacements at equivalent
nodes for the models with K and 2K elements, respectively. For a vertical
incidence, the results for
HS
Err
2
are shown by Table 4.4. Apparently,
increase in the number of elements K minimizes the difference between two
results at equivalent nodes. The number of elements 928 = K is found to be

42
sufficient for all further computations in this study. As before, similar
calculations have been performed for a grazing incidence. The same
consistency can be observed from these results and thus they are omitted.

Table 4.4: The relative error ) , (
2
K Err
HS
β
η based on the numerical results
with different numbers of elements K at equivalent nodes along the
corrugated interface S and the half-space surface S
0
subjected to a vertical
plane harmonic SH wave as 2 . 0 = ε , 1 = = b a , 8 = m , 6 / 1
2
= μ and
2 / 1
2
=
β
v .
HS
Err
2
\
β
η ) 464 , (
2 β
η
HS
Err ) 928 , (
2 β
η
HS
Err ) 1856 , (
2 β
η
HS
Err
25 . 0 =
β
η
0.5658% 0.1512% 0.0402%
1 =
β
η
3.9940% 1.0741% 0.1595%
2 =
β
η
5.7237% 1.7576% 0.3381%
Based on the error analyses of
HS
Err
0
,
HS
Err
1
and
HS
Err
2
it is possible
to determine all the parameters required for evaluating the displacement field
for the half-space problem. This concludes the preliminary investigation of the
numerical solution. The results for circular type of the soft inclusions are
considered next.

43
-3 -2 -1 0 1 2 3
0
0.5
1
1.5
2
2.5
3
Mesh Data, ε=0.2, m=2

-3 -2 -1 0 1 2 3
0
0.5
1
1.5
2
2.5
3
Mesh Data, ε=0.2, m=8

Figure 4.1: The geometry of the half-space problem where the dash line
denotes the smooth surface limit S’ and the solid line denotes the corrugation
S and the half-space surface S
0
. Two period coefficients are considered:
2 = m (left) and 8 = m (right). In addition, 1 = = b a , 2 . 0 = ε and a h 2 = .

Figure 4.1 depicts two characteristic models for circular type of
corrugations. It should be noted that this figure shows the two extremes for
the scatterer corrugation due to the corrugation period. For longer
corrugation period, the corrugation essentially generates an elliptical
scatterer (Fig. 4.1, left). However, for the shorter corrugation period, the
scatterer becomes highly irregular (Fig. 4.1, right). These two non-symmetric
models are used to evaluate the numerical results for the half-space
problems. The embedment depth is chosen as a h 2 = , and the response
along the half-space surface S
0
due to a vertical incidence is investigated
first.

44
It should be pointed out that the corrugation response is a complex
valued function consisting of both amplitude and phase. However, the
principal aim of this study is to evaluate the maximum surface response due
to the incident wave and the corrugated surface S. Thus, only the
displacement amplitude results are shown in this study.
Since at low frequencies the presence of the corrugation generated
minor changes in the surface displacement field when compared to the
corresponding smooth scatterer results, these results are omitted. Increase in
the frequency to 2 =
β
η produced the surface displacement field depicted
by Fig. 4.2. Based on the surface displacements, the half-space surface
response can be grouped into the near field ( a x 3
1
≤ ) and the far field
( a x 3
1
> ). Clearly, the surface motion appears to be very sensitive upon the
presence of the corrugation in both fields.

45
-10 -8 -6 -4 -2 0 2 4 6 8 10
0
0.5
1
1.5
2
2.5
3
3.5
Displacement vs. x
1
, γ=0, η
β
=2, h=2, S
0
=40, m=8
x
1
Displacement |u
3
|
ε=0
ε=0.2

Figure 4.2: Half-space inclusion anti-plane strain model displacement
amplitude along the half-space surface S
0
subjected to a vertical plane
harmonic SH wave for different corrugation peak amplitude ε as a function
of location x
1
(Fig. 2.1) when 2 =
β
η , 8 = m , 1 = = b a , 1
1
= μ , 6 / 1
2
= μ ,
1
1
=
β
v , 2 / 1
2
=
β
v , a h 2 = , 128 = P and 928 = K .

The motion for a grazing incidence is considered as well. Figure 4.3
displays the half-space surface displacements for a shorter corrugation
period at an intermediate frequency. In the near and far fields, the rough and
the smooth inclusion motions have similar patterns although the amplitudes
of the two may be different. The far-field surface response can be divided into
the illuminated portion ( a x 3
1
− < ) and the shadow portion ( a x 3
1
> ). In the
illuminated region, the surface motions oscillate far into the half space.
However, there is no oscillation observed in the shadow part of the surface

46
motions, and the displacements attenuate to the free-field result. Thus for the
grazing incidence the inclusion produces a shielding effect in the shadow
region. In the region atop the inclusion, strong local site effect is found due to
the presence of the corrugation.
-10 -8 -6 -4 -2 0 2 4 6 8 10
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Displacement vs. x
1
, γ=90, η
β
=1, h=2, S
0
=40, m=8
x
1
Displacement |u
3
|
ε=0
ε=0.2

Figure 4.3: Half-space inclusion anti-plane strain model displacement
amplitude along the half-space surface S
0
subjected to a grazing plane
harmonic SH wave for different corrugation peak amplitude ε as a function
of location x
1
(Fig. 2.1) when 1 =
β
η , 8 = m , 1 = = b a , 1
1
= μ , 6 / 1
2
= μ ,
1
1
=
β
v , 2 / 1
2
=
β
v , a h 2 = , 128 = P and 928 = K .

For a stiff inclusion the half-space surface response for a vertical
incidence is considered next. As before, at low frequencies, the presence of
the corrugation produced minor impact in the resulting motion so these
results are omitted. At a higher frequency the motion is depicted by Fig. 4.4.

47
Apparently, in the region a x 3
1
− ≤ , the presence of the corrugation produced
small variation in the response when compared with the corresponding
smooth inclusion results. However, in the region a x 3
1
> , the resulting
surface response attenuates much faster to the free-field result than the
smooth interface motion.
-15 -10 -5 0 5 10 15
0
0.5
1
1.5
2
2.5
3
3.5
Displacement vs. x
1
, γ=0, η
β
=2, h=2, S
0
=40, m=2
x
1
Displacement |u
3
|
ε=0
ε=0.2

Figure 4.4: Half-space inclusion anti-plane strain model displacement
amplitude along the half-space surface S
0
subjected to a vertical plane
harmonic SH wave for different corrugation peak amplitude ε as a function
of location x
1
(Fig. 2.1) when 2 =
β
η , 2 = m , 1 = = b a , a h 2 = , 128 = P and
928 = K . Stiff inclusion case: 1
1
= μ , 8
2
= μ , 1
1
=
β
v and 2
2
=
β
v .

The resulting motion due to a stiff inclusion subjected to a grazing
incidence at an intermediate frequency and for a larger corrugation period
( 2 = m ) is shown by Fig. 4.5. In the illuminated region of the half-space

48
surface the corrugated and the smooth inclusion motions have similar
patterns but different amplitudes. Furthermore, the corrugated-inclusion
surface motion attenuates faster to the free-field result than the
corresponding smooth inclusion response. In the shadow portion of the
half-space surface, the responses are non-oscillatory and the amplitudes of
these two motions are found to be similar.
-10 -8 -6 -4 -2 0 2 4 6 8 10
0
0.5
1
1.5
2
2.5
3
3.5
4
Displacement vs. x
1
, γ=90, η
β
=1, h=2, S
0
=40, m=2
x
1
Displacement |u
3
|
ε=0
ε=0.2

Figure 4.5: Half-space inclusion anti-plane strain model displacement
amplitude along the half-space surface S
0
subjected to a grazing plane
harmonic SH wave for different corrugation peak amplitude ε as a function
of location x
1
(Fig. 2.1) when 1 =
β
η , 2 = m , 1 = = b a , a h 2 = , 128 = P and
928 = K . Stiff inclusion case: 1
1
= μ , 8
2
= μ , 1
1
=
β
v and 2
2
=
β
v .

It should be noticed that in general for soft inclusion cases, increase in
the frequency consistently generated pronounced change in the resulting

49
motion. However, for a stiff inclusion, the same increase may not always
produce such significant effect of the corrugation upon the half-space surface
response. Figures 4.2 to 4.5 display the extreme cases. The others,
producing less prominent effects, are omitted in order to reduce the number
of figures.
The same calculations have been performed for two elliptical types of
corrugated inclusions as well ( 1 = a , 75 . 0 = b and 1 = a , 75 . 1 = b ). These
results show similar phenomena observed for the circular-inclusion models
and thus they are omitted. Hence, the half-space results can be summarized
as follows.
The half-space surface response strongly depends upon the corrugation
amplitude especially at higher frequencies and shorter corrugation periods.
For larger corrugation amplitudes and at higher frequencies, the motion along
the far field of the illuminated portion of the half-space surface may attenuate
faster to the free-field response than the corresponding smooth inclusion
result. In general, the surface displacement directly atop the scatterer clearly
detects the presence of the corrugation in the response. Highly oscillatory

50
surface motion can be observed at considerable distances in the far field for
the vertical incidence and in the illuminated portion of the half space for the
grazing incidence. The response in the shadow region for the grazing
incidence is found to be non-oscillatory with the scatterer producing the
shielding effect.
This concludes the presentation of the numerical results for the
half-space inclusion problem. The half-space cavity problem results are
considered next.
4.2. Half-space cavity problem results
In addition to the surface S and frequency, η
β
, the total number of
elements, K, used for modeling of this problem depends upon the length of
the half-space surface, S
0
. As for the half-space cavity models, this length
can be determined by comparing the smooth surface limit response of a
half-space semi-circular canyon model with the exact solution (Trifunac,
1973). For a vertical incidence, these results are tabulated in Table 4.5 in
terms of an error
HS
Err
1
defined by equation (4.3) and
0
S S
i
∪ ∈ x .

51
Table 4.5: The relative error ) , (
0 1
S Err
HS
β
η as a function of frequency η
β
and
the length of the half-space surface S
0
for a semi-circular canyon subjected to
a vertical plane harmonic SH wave.
0
\ S
β
η 10
0
= S 20
0
= S 40
0
= S 80
0
= S
25 . 0 =
β
η
0.1681% 0.0347% 0.0101% 0.0059%
1 =
β
η
0.1145% 0.0713% 0.0632% 0.0518%
2 =
β
η
0.2007% 0.1265% 0.0938% 0.0696%
It is apparent from Table 4.5 that, when the length of the half-space
surface 20
0
≥ S , the relative error uniformly increases with increase of the
frequency. Based on the results of Table 4.5, the computational range of the
half-space surface is chosen as { } a x S 20 :
0
≤ . Further increase in the
surface length generates minor changes in the surface displacement field.
For a grazing incidence, the results display similar phenomena and thus
they are omitted.
After computations of the error
HS
Err
1
, the convergence error
HS
Err
2
,
defined by equation (4.4), is considered next. For a vertical incidence, the
results for
HS
Err
2
are shown in Table 4.6. Apparently, the difference between
the two results at the same nodes decreases with the increasing number of
elements K. Based on Table 4.6, the number of elements is chosen as

52
928 = K for all the calculations. For a grazing incidence, similar conclusions
can be obtained, and thus these results for
HS
Err
2
are omitted.

Table 4.6: The relative error ) , (
2
K Err
HS
β
η based on the numerical results
with different numbers of elements K at equivalent nodes along the
corrugated surface S and the half-space surface S
0
subjected to a vertical
plane harmonic SH wave as 2 . 0 = ε , 1 = = b a and 8 = m .
HS
Err
2
\
β
η ) 464 , (
2 β
η
HS
Err ) 928 , (
2 β
η
HS
Err ) 1856 , (
2 β
η
HS
Err
25 . 0 =
β
η
0.5233% 0.1398% 0.0550%
1 =
β
η
4.4226% 1.1075% 0.1930%
2 =
β
η
5.7160% 1.2848% 0.2390%
Based on the results of the errors
HS
Err
1
and
HS
Err
2
all the parameters
required for the calculations of the surface displacement field can be
determined. The numerical results for this model are presented next.
Figure 4.1 depicts two non-symmetric characteristic models for a circular
type of corrugations used to evaluate the numerical results. As before, the
displacements shown are the amplitude spectra of the total wave field. The
response along the half-space surface S
0
subjected to a vertical incidence is
considered first.

53
The presence of the corrugation at low frequencies generates only minor
changes in the resulting motion so these results are omitted. By increasing
the frequency of a vertical incident wave and by decreasing the corrugation
period produces the surface motion shown by Fig. 4.6. The surface motion
appears to be very sensitive upon the presence of the corrugation. In the far
field, the corrugated and smooth cavity motions exhibit similar highly
oscillatory patterns for all frequencies and corrugation shapes. However, the
amplitudes of the two motions may be very different especially for higher
frequencies and shorter corrugation periods. In that case, the corrugated
surface motion may attenuate faster (but not always) to the free-field result
than the corresponding smooth cavity response. In the near field, the
corrugated and smooth cavity results display different patterns and
amplitudes even at intermediate frequencies. Thus the near-field motion
clearly detects the presence of the cavity roughness in the surface
displacement field.

54
-10 -8 -6 -4 -2 0 2 4 6 8 10
0
0.5
1
1.5
2
2.5
3
Displacement vs. x
1
, γ=0, η
β
=2, h=2, S
0
=40, m=8
x
1
Displacement |u
3
|
ε=0
ε=0.2

Figure 4.6: Half-space cavity anti-plane strain model displacement amplitude
along the half-space surface S
0
subjected to a vertical plane harmonic SH
wave for different corrugation peak amplitude ε as a function of location x
1

(Fig. 2.1) when 2 =
β
η , 8 = m , 1 = = b a , 1
1
= μ , 1
1
=
β
v , a h 2 = , 128 = P
and 928 = K .

Figure 4.7 shows the half-space surface motion for a grazing incidence,
a shorter corrugation period and at an intermediate frequency. Apparently, the
response is strongly affected by the presence of the corrugation. In both the
illuminated and the shadow regions the corrugated and the smooth cavity
motions have similar patterns. In the illuminated part, strong oscillations in
the surface motions are observed indicating constructive-destructive
interaction of the scattered waves that can be observed far away from the
scatterer. However, the amplitudes of the rough cavity motion are significantly

55
different from the amplitudes of the corresponding smooth cavity response. In
the region atop the cavity, strong local site effect is found due to the
corrugation. In the shadow portion both smooth and corrugated cavity
surface motions attenuate significantly away from the scatterer.
-10 -8 -6 -4 -2 0 2 4 6 8 10
0
1
2
3
4
5
6
Displacement vs. x
1
, γ=90, η
β
=1, h=2, S
0
=40, m=8
x
1
Displacement |u
3
|
ε=0
ε=0.2

Figure 4.7: Half-space cavity anti-plane strain model displacement amplitude
along the half-space surface S
0
subjected to a grazing plane harmonic SH
wave for different corrugation peak amplitude ε as a function of location x
1

(Fig. 2.1) when 1 =
β
η , 8 = m , 1 = = b a , 1
1
= μ , 1
1
=
β
v , a h 2 = , 128 = P
and 928 = K .

As before, the same calculations have been performed for two elliptical
types of corrugated cavities. These results show similar phenomena
observed in the circular cavity models and thus they are omitted. Hence, the
half-space results can be summarized as follows. Increase in the corrugation

56
amplitude may result in significant changes in the half-space surface
response when compared with the smooth scatterer result. For a vertical
incidence, the far-field half-space surface motion may approach faster the
free-field result than the corresponding smooth scatterer motion. This is
displayed especially for larger corrugation amplitudes, shorter corrugation
periods and at higher frequencies. The half-space surface near field clearly
detects the presence of the cavity roughness in the surface response. For a
grazing incidence and a rough scatterer, the motion in the illuminated region
is highly oscillatory and the amplitudes may be very different from the
corresponding smooth cavity motion. This is especially significant for larger
corrugation amplitudes, shorter corrugation periods and at higher frequencies.
In the shadow portion of the half-space surface the presence of the
corrugation produces minor change in the surface response and the cavity
creates the shielding effect for that region.
This concludes the analysis of the numerical results for the half-space
cavity problem. The full-space inclusion problem results are considered next.

57
4.3. Full-space inclusion problem results
For this problem, the initial number of elements K is determined by using
the smooth interface response. Namely, as the corrugation amplitude
approaches zero, the interface response of the problem reduces to the one
for which the exact solution is available (e.g., Pao and Mow, 1973 for
1 = = b a ). Therefore, at each frequency, the number of elements has been
increased to match the exact solution. This is illustrated in Tables 4.7 and 4.8
which display the error as a function of frequency and the number of nodes
for soft and stiff inclusions where the error
FS
Err
1
is introduced as
∑
∑
=
=
−
=
K
i
i exact
K
i
i i exact
FS
u
u u
K Err
1
2
1
2
3
1
) (
) ( ) (
) , (
x
x x
β
η ; S
i
∈ x (4.5)
with ) (
i exact
u x and ) (
3 i
u x denoting the exact and the smooth surface limit
( 0 → ε ) displacements, respectively.

58
Table 4.7: The relative error ) , (
1
K Err
FS
β
η as a function of the frequency η
β

and the number of elements K for a circular soft inclusion ( 6 / 1
2
= μ , 2 / 1
2
=
β
v )
embedded in a full space subjected to a plane harmonic SH wave.
K \
β
η
64 = K 128 = K 256 = K 512 = K
25 . 0 =
β
η
0.0517% 0.0129% 0.0035% 0.0012%
1 =
β
η
0.8929% 0.2572% 0.0830% 0.0308%
2 =
β
η
2.1869% 0.9388% 0.7693% 0.7538%

Table 4.8: The relative error ) , (
1
K Err
FS
β
η as a function of the frequency η
β

and the number of elements K for a circular stiff inclusion ( 8
2
= μ , 2
2
=
β
v )
embedded in a full space subjected to a plane harmonic SH wave.
K \
β
η
64 = K 128 = K 256 = K 512 = K
25 . 0 =
β
η
0.0452% 0.0106% 0.0024% 0.0006%
1 =
β
η
0.2038% 0.0608% 0.0207% 0.0083%
2 =
β
η
0.6700% 0.2996% 0.2011% 0.1775%
From Tables 4.7 and 4.8, it is evident that the number of elements
128 = K is sufficient to provide accurate results for the range of frequencies
considered here. It is of interest to note that for the same number of elements,
the relative error
FS
Err
1
for the stiff inclusion is found to be smaller than the
corresponding error for the soft inclusion.

59
Table 4.9: The relative error ) , (
2
K Err
FS
β
η based on the numerical results
with different numbers of elements K at equivalent nodes along the
corrugated interface S as 2 . 0 = ε , 1 = = b a , 8 = m , 6 / 1
2
= μ and
2 / 1
2
=
β
v .
FS
Err
2
\
β
η ) 64 , (
2 β
η
FS
Err ) 128 , (
2 β
η
FS
Err ) 256 , (
2 β
η
FS
Err
25 . 0 =
β
η
0.6032% 0.1555% 0.1427%
1 =
β
η
6.0395% 1.5699% 0.2654%
2 =
β
η
10.6048% 3.3494% 0.6341%
For a nonzero corrugation amplitude ε, the convergence rate between
two numerical results, with different numbers of elements along the
corrugated interface S, can be measured by computing the error
∑
∑
=
=
−
−
=
K
i
i
K
i
i i
FS
K u
K u K u
K Err
1
2
3
1
2
1 2 3 3
2
) , (
) 2 , ( ) , (
) , (
x
x x
β
η ; S
i i
∈
−1 2
,x x (4.6)
where ) , (
3
K u
i
x and ) 2 , (
3
K u
i
x denote the displacements at node x
i
for the
model with K and 2K elements, respectively. Therefore, ) , (
3
K u
i
x and
) 2 , (
1 2 3
K u
i−
x are the displacements evaluated at the same nodes for models
with K and 2K elements, respectively. The results for
FS
Err
2
and a soft

60
inclusion are shown in Table 4.9. Similar results were obtained for the stiff
inclusion and are omitted.
Apparently, when the number of elements K increases, the difference
between two results at equivalent nodes becomes smaller. The number of
elements 128 = K is found to provide sufficient accuracy for the
displacement field in this study.
Therefore, all the parameters required for computation of the interface
response for the anti-plane full-space inclusion problem can be determined
from the analysis of the error functions ) , (
1
K Err
FS
β
η and ) , (
2
K Err
FS
β
η . The
results for circular based soft inclusions are considered next.
First, illustrative examples for the geometry of the problem are shown by
Fig. 4.8. The models depict the two extremes in the inclusion roughness due
to the corrugation period. For convenience, the results are presented as a
function of the angle θ (Fig. 4.8).

61

Figure 4.8: Full-space problem geometry with a smooth circular cavity S’
(dash line, the principal axes 1 = = b a ) and the corresponding corrugated
cavities (solid lines) of amplitude 2 . 0 = ε and period coefficients 2 = m (left)
and 8 = m (right), respectively.

-4 -3 -2 -1 0 1 2 3 4
0
0.5
1
1.5
2
2.5
3
Displacement vs. θ, η
β
=2, m=8
θ
Displacement |u
3
|
ε=0
ε=0.1
ε=0.2

Figure 4.9: Full-space inclusion anti-plane strain model displacement
amplitude along the corrugated interface S for different corrugation peak
amplitude ε as a function of angle θ (Fig. 4.8) when 2 =
β
η , 8 = m ,
1 = = b a , 1
1
= μ , 6 / 1
2
= μ , 1
1
=
β
v , 2 / 1
2
=
β
v and 128 = K .

At a low frequency the presence of the corrugation produced minor
changes in the inclusion response when compared to the smooth scatterer

62
motion, so these results are omitted. Increasing the frequency and by
changing the period generated the results depicted by Fig. 4.9. Apparently,
the presence of the corrugation produced significant change in the interface
displacement field. This phenomenon is consistently observed for larger
corrugation amplitudes and shorter corrugation periods.
For the stiff inclusion at an intermediate frequency the motion is shown
by Fig. 4.10. Evidently, the scatterer roughness caused great impact upon the
stiff inclusion motion. It should be noted that for the soft inclusion, the
increase in frequency consistently produced significant effect upon the
corrugation motion. However, for the stiff inclusion the same increase in
frequency may not always result in such dramatic change in the corrugation
motion.

63
-4 -3 -2 -1 0 1 2 3 4
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Displacement vs. θ, η
β
=2, m=8
θ
Displacement |u
3
|
ε=0
ε=0.05
ε=0.1
ε=0.2

Figure 4.10: Full-space inclusion anti-plane strain model displacement
amplitude along the corrugated interface S for different corrugation peak
amplitude ε as a function of angle θ (Fig. 4.8) when 2 =
β
η , 8 = m ,
1 = = b a and 128 = K . Stiff inclusion case: 1
1
= μ , 8
2
= μ , 1
1
=
β
v and
2
2
=
β
v .

Analogous calculations performed for elliptically based inclusions show
similar phenomena observed for the circular models and thus they are
omitted. Therefore, for the range of parameters considered in this case, the
numerical results can be summarized as follows. At low frequencies for both
soft and stiff inclusions the presence of the corrugation does not significantly
alter the resulting motion when compared with the smooth scatterer results.
At intermediate frequencies and for a soft inclusion, increase in the frequency
generates more pronounced changes in the corrugated interface

64
displacement field when compared with the corresponding stiff-inclusion
response. The resulting motion strongly depends upon the corrugation
amplitude and the corrugation period.
This concludes the discussion of the numerical results for the full-space
inclusion problem. The full-space cavity problem results are examined next.
4.4. Full-space cavity problem results
The initial number of elements K for this problem is determined by using
the smooth surface response for which the exact solution is available (e.g.,
Pao and Mow, 1973 for 1 = = b a ). At each frequency, the number of
elements has been increased to match the exact solution. For that purpose,
the error
FS
Err
1
, defined by equation (4.5), is computed for different numbers
of elements and frequencies (Table 4.10). It is evident from the results of
Table 4.10, that the number of elements 128 = K is sufficient to provide
accurate results for the range of frequencies considered here.

65
Table 4.10: The relative error ) , (
1
K Err
FS
β
η as a function of the frequency η
β

and the number of elements K for a circular cavity embedded in a full space
subjected to a plane harmonic SH wave.
K \
β
η
64 = K 128 = K 256 = K 512 = K
25 . 0 =
β
η
0.0604% 0.0148% 0.0037% 0.0009%
1 =
β
η
0.3531% 0.0872% 0.0217% 0.0054%
2 =
β
η
2.1498% 0.8463% 0.6984% 0.6887%

Table 4.11: The relative error ) , (
2
K Err
FS
β
η based on the numerical results
with different numbers of elements K at equivalent nodes along the
corrugated surface S as 2 . 0 = ε , 1 = = b a and 8 = m .
FS
Err
2
\
β
η ) 64 , (
2 β
η
FS
Err ) 128 , (
2 β
η
FS
Err ) 256 , (
2 β
η
FS
Err
25 . 0 =
β
η
1.1884% 0.3101% 0.2011%
1 =
β
η
7.2016% 1.8378% 0.3769%
2 =
β
η
11.2235% 2.7016% 0.5849%
Since this error function
FS
Err
1
depends only upon the smooth cavity
response, the role of the corrugation presence must be taken into account for
the testing as well. Thus, for a nonzero corrugation amplitude ε, the rate of
convergence between the two results with different numbers of elements K
can be measured by computing the error
FS
Err
2
defined by equation (4.6).
The results for
FS
Err
2
in Table 4.11 demonstrate the convergence of the

66
solution with the increase of the number of elements K. The number of
elements 128 = K is found to be sufficient for accurate evaluation of the
displacement field in this case.
Based on Tables 4.10 and 4.11 the values of all the parameters required
for computation of the response for the full-space cavity problem can be
determined.
The numerical results for a circular based cavity are considered next.
Illustrative examples for the geometry of the problem are depicted by Fig. 4.8.
At a low frequency, the presence of the corrugation produced minor
changes in the cavity response when compared to the smooth scatterer
motion, so these results are omitted. Increase in the frequency to 2 =
β
η
produced the results depicted by Fig. 4.11. Evidently, the presence of the
corrugation produced significant change in the surface displacement field
when compared with the corresponding smooth surface results. Therefore,
for shorter corrugation periods the resulting motion appears to be very
sensitive to the increase of the corrugation amplitude.

67
-4 -3 -2 -1 0 1 2 3 4
0
0.5
1
1.5
2
2.5
3
3.5
Displacements vs. θ, η
β
=2, m=8
θ
Displacement |u
3
|
ε=0
ε=0.05
ε=0.1
ε=0.2

Figure 4.11: Full-space cavity anti-plane strain model displacement amplitude
along the corrugated surface S for different corrugation peak amplitude ε as
a function of angle θ (Fig. 4.8) when 2 =
β
η , 8 = m , 1 = = b a , 1
1
= μ ,
1
1
=
β
v and 128 = K .

As before, the same calculations have been performed for elliptically
based corrugations. These results show similar features observed for the
circular based model and thus they are omitted. Therefore, the full-space
results show that increase in the corrugation amplitude may produce a
significant change in the corrugated surface displacement field when
compared with the smooth cavity response. The cavity motion strongly
depends upon the corrugation amplitude especially at higher frequencies and
for shorter corrugation periods.

68
This concludes the analysis of the numerical results for the anti-plane
strain models. The solutions for the plane strain models are considered next.

69
Chapter 5: Plane Strain Model Solutions
As before, the direct boundary integral equation method is used to solve
for the unknown scattered waves. The half-space inclusion problem is
considered first.
5.1. Half-space inclusion problem solution
For this problem the boundary integral equations for plane-strain model
in domain B
1
become (Niu and Dravinski, 2003b)
∫ ∫
+
− =
0
) ( ) , , ( ) , ( ) , ( ) ( ) (
) 1 ( ) 1 ( ) 1 ( ) 1 ( ) 1 ( ) 1 (
S S
x
sc
j ji
S
x
sc
j ji
sc
j
sc
ji
dS u t PV dS f g u c x n y x n x y x y y ;
0
S S∪ ∈ y (5.1)
while for domain B
2

∫ ∫
− = −
S
x j ji
S
x j ji j
sc
ji
dS u t PV dS f g u c ) ( ) , , ( ) , ( ) , ( ) ( ) (
) 2 ( ) 2 ( ) 2 ( ) 2 ( ) 2 ( ) 2 (
x n y x n x y x y y ; S ∈ y
(5.2)
Here, ) , (
) (
y x
h
ji
g are the full-space plane-strain displacement Green’s
functions which denote the x
j
-direction displacement component at an
observation point x due to a harmonic unit force applied at the x
i
direction at a
source point y, and ) , , (
) (
n y x
h
ji
t are the corresponding traction Green’s
functions for domain B
h
, h=1:2 (Kobayashi, 1987) at a surface with unit

70
normal n. In addition,
) (h
j
u and
) (h
j
f are the unknown displacement and
traction components along the corrugated interface S,
) (h sc
ji
c represent the
free terms (Niu and Dravinski, 2003a; Paris and Caňas, 1997), and PV
denotes the principal value integrals. Therefore the traction integrals in the
above equations are evaluated in the principal value sense (Paris and Caňas,
1997).
The presence of the half-space surface S
0
in the integral equations
occurs due to the fact that the full-space traction Green’s functions adopted in
the study do not satisfy the traction-free boundary conditions at the
half-space surface S
0
. The advantage of using the full-space Green’s
functions follows from the ease of their numerical evaluations.
Using the traction-free and continuity conditions the integral equations
become
∫ ∫
∫ ∫
− + =
− +
+
+
S
x
ff
j ji
S S
x
ff
j ji
ff
j
sc
ji
S
x j ji
S S
x j ji j
sc
ji
dS f g dS u t PV u c
dS f g dS u t PV u c
) , ( ) , ( ) ( ) , , ( ) ( ) (
) , ( ) , ( ) ( ) , , ( ) ( ) (
) 1 ( ) 1 ( ) 1 (
) 1 ( ) 1 ( ) 1 ( ) 1 ( ) 1 (
0
0
n x y x x n y x y y
n x y x x n y x y y
;
0
S S∪ ∈ y (5.3)

71
0 ) , ( ) , ( ) ( ) , , ( ) ( ) (
) 2 ( ) 2 ( ) 2 (
= − + −
∫ ∫
S
x j ji
S
x j ji j
sc
ji
dS f g dS u t PV u c n x y x x n y x y y ; S ∈ y
(5.4)
where
) , ( ) , ( ) , (
) ( ) ( ) (
) ( ) ( ) (
) 2 ( ) 1 (
) 2 ( ) 1 (
) 2 ( ) 1 (
n x n x n x
y y y
x x x
j j j
j j j
j j j
f f f
u u u
u u u
= =
= =
= =
; S ∈ y x, (5.5)
Here
ff
j
u and
ff
j
f are the known free-field displacements and tractions
along the corrugated interface S, respectively. The integral equations must be
solved for the unknown traction and displacement fields along the corrugated
interface S and the half-space surface S
0
. Then the displacement fields u
(1)

and u
(2)
can be evaluated throughout the half space by using the integral
representations
∫ ∫
+
− + =
0
) ( ) , , ( ) , ( ) , ( ) ( ) (
) 1 ( ) 1 ( ) 1 ( ) 1 ( ) 1 (
S S
x
sc
j ji
S
x
sc
j ji
ff
i i
dS u t PV dS f g u u x n y x n x y x y y ;
1
B ∈ y ;
0
S S∪ ∉ y (5.6)
∫ ∫
+ − =
S
x j ji
S
x j ji i
dS u t PV dS f g u ) ( ) , , ( ) , ( ) , ( ) (
) 2 ( ) 2 ( ) 2 (
x n y x n x y x y ;
2
B ∈ y ; S ∉ y
(5.7)

72
The surfaces S and S
0
are discretized using P and (K-P)
one-dimensional linear elements, respectively as shown in Fig. 3.1.
Consequently, the integral equations (5.3) and (5.4) become
[ ] [ ]
[ ] [ ] ) (
) 1 (
) (
) 1 (
) (
) 1 (
) (
) ( ) (
) 1 (
) (
) ( ) (
1
1 2
2 ) 1 ( 21
2
) 1 ( 21
1
1 1
1 ) 1 ( 11
2
) 1 ( 11
1
1
) 1 (
2
) 1 (
2 ) 1 ( 21
2
) 1 ( 21
1
) 1 (
2
) 1 (
21
1
) 1 (
1
) 1 (
1 ) 1 ( 11
2
) 1 ( 11
1
) 1 (
1
) 1 (
11
l F
k f
k f
B B
k f
k f
B B
k u
k u
A A l u l c
k u
k u
A A l u l c
P
k
lk lk
P
k
lk lk
K
k
lk lk
sc
K
k
lk lk
sc
=






+
−






+
−






+
+ +






+
+
∑ ∑
∑ ∑
= =
= =
0
S S
l
∂ ∪ ∂ ∈ y ; ) 1 ( : 1 + = K l (5.8)
[ ] [ ]
[ ] [ ] ) (
) 1 (
) (
) 1 (
) (
) 1 (
) (
) ( ) (
) 1 (
) (
) ( ) (
2
1 2
2 ) 1 ( 22
2
) 1 ( 22
1
1 1
1 ) 1 ( 12
2
) 1 ( 12
1
1
) 1 (
2
) 1 (
2 ) 1 ( 22
2
) 1 ( 22
1
) 1 (
2
) 1 (
22
1
) 1 (
1
) 1 (
1 ) 1 ( 12
2
) 1 ( 12
1
) 1 (
1
) 1 (
12
l F
k f
k f
B B
k f
k f
B B
k u
k u
A A l u l c
k u
k u
A A l u l c
P
k
lk lk
P
k
lk lk
K
k
lk lk
sc
K
k
lk lk
sc
=






+
−






+
−






+
+ +






+
+
∑ ∑
∑ ∑
= =
= =
0
S S
l
∂ ∪ ∂ ∈ y ; ) 1 ( : 1 + = K l (5.9)
[ ] [ ]
[ ] [ ] 0
) 1 (
) (
) 1 (
) (
) 1 (
) (
) ( ) (
) 1 (
) (
) ( ) (
1 2
2 ) 2 ( 21
2
) 2 ( 21
1
1 1
1 ) 2 ( 11
2
) 2 ( 11
1
1 2
2 ) 2 ( 21
2
) 2 ( 21
1 2
) 2 (
21
1 1
1 ) 2 ( 11
2
) 2 ( 11
1 1
) 2 (
11
=






+
−






+
−






+
+ −






+
+ −
∑ ∑
∑ ∑
= =
= =
P
k
lk lk
P
k
lk lk
P
k
lk lk
sc
P
k
lk lk
sc
k f
k f
B B
k f
k f
B B
k u
k u
A A l u l c
k u
k u
A A l u l c

S
l
∂ ∈ y ; P l : 1 = (5.10)
[ ] [ ]
[ ] [ ] 0
) 1 (
) (
) 1 (
) (
) 1 (
) (
) ( ) (
) 1 (
) (
) ( ) (
1 2
2 ) 2 ( 22
2
) 2 ( 22
1
1 1
1 ) 2 ( 12
2
) 2 ( 12
1
1 2
2 ) 2 ( 22
2
) 2 ( 22
1 2
) 2 (
22
1 1
1 ) 2 ( 12
2
) 2 ( 12
1 1
) 2 (
12
=






+
−






+
−






+
+ −






+
+ −
∑ ∑
∑ ∑
= =
= =
P
k
lk lk
P
k
lk lk
P
k
lk lk
sc
P
k
lk lk
sc
k f
k f
B B
k f
k f
B B
k u
k u
A A l u l c
k u
k u
A A l u l c

S
l
∂ ∈ y ; P l : 1 = (5.11)
where

73
[ ]
[ ]
[ ] [ ]
∑ ∑
∑
∑
= =
=
=






+
−






+
−






+
+ +






+
+ =
P
k
ff
ff
lk lk
P
k
ff
ff
lk lk
K
k
ff
ff
lk lk
ff sc
K
k
ff
ff
lk lk
ff sc
k f
k f
B B
k f
k f
B B
k u
k u
A A l u l c
k u
k u
A A l u l c l F
1
2
2 ) 1 ( 21
2
) 1 ( 21
1
1
1
1 ) 1 ( 11
2
) 1 ( 11
1
1
2
2 ) 1 ( 21
2
) 1 ( 21
1 2
) 1 (
21
1
1
1 ) 1 ( 11
2
) 1 ( 11
1 1
) 1 (
11 1
) 1 (
) (
) 1 (
) (
) 1 (
) (
) ( ) (
) 1 (
) (
) ( ) ( ) (
(5.12)
[ ]
[ ]
[ ] [ ]
∑ ∑
∑
∑
= =
=
=






+
−






+
−






+
+ +






+
+ =
P
k
ff
ff
lk lk
P
k
ff
ff
lk lk
K
k
ff
ff
lk lk
ff sc
K
k
ff
ff
lk lk
ff sc
k f
k f
B B
k f
k f
B B
k u
k u
A A l u l c
k u
k u
A A l u l c l F
1
2
2 ) 1 ( 22
2
) 1 ( 22
1
1
1
1 ) 1 ( 12
2
) 1 ( 12
1
1
2
2 ) 1 ( 22
2
) 1 ( 22
1 2
) 1 (
22
1
1
1 ) 1 ( 12
2
) 1 ( 12
1 1
) 1 (
12 2
) 1 (
) (
) 1 (
) (
) 1 (
) (
) ( ) (
) 1 (
) (
) ( ) ( ) (
(5.13)
Here the collocation point y
l
is located at node l , u
j
(k) and u
j
(k+1) are the
x
j
-direction displacements at nodes k and k+1, respectively, K is the total
number of elements. Nodes 1 through P belong to the corrugated interface S
while the nodes P+1 to K+1 are on the half-space surface S
0
. In addition, the
integration constants
) (h ji
mlk
A and
) (h ji
mlk
B (h,i,j,m=1:2) are specified by
∫
∫
−
−
=
=
1
1
) ( ) (
1
1
) ( ) (
) ( ) , (
) ( ) , , (
ξ ξ ξ
ξ ξ ξ
d J N l g B
d J N l t A
k
m
h
ji
h ji
mlk
k
m
k h
ji
h ji
mlk
n
; 2 : 1 , , , = m j i h (5.14)
where ξ, N
m
, and J
k
are the local coordinate, the shape functions, and the
Jacobian, respectively, while n
k
is the unit normal on the element
k
S ∂ . For

74
linear elements, the shape functions and Jacobians are given by equations
(3.12) and (3.13). Finally, the displacement and traction Green’s functions for
full-space plane-strain model are defined as (Kobayashi, 1987)
) , , ˆ ˆ (
4
) , (
2 1 i j ji ji
r r g g
i
g − = δ
μ
y x ; y x− = r (5.15)
}
ˆ
] , , , 2 [
ˆ
] , ) , , 2 , ( 2 ) , [(
ˆ
] , , {[
4
) , , (
2 2
1
dr
g d
r n
n
r
r r
r
g
r n
n
r
r r r n r n
n
r
dr
g d
r n r n
n
r i
t
i j i j i j j i i j j i ji
i j j i ji ji
μ
λ
μ
λ
δ
μ
λ
δ
+
∂
∂
− +
∂
∂
− + +
∂
∂
−
+ +
∂
∂
= n y x

(5.16)
where
) (
1
) ( ) (
1
) ( ˆ
) 1 (
1
2 ) 1 (
1
) 1 (
0 1
r k H
r k k
k
r k H
r k
r k H g
α
α β
α
β
β
β
+ − = (5.17)
) ( ) ( ) ( ˆ
) 1 (
2
2 ) 1 (
2 2
r k H
k
k
r k H g
α
β
α
β
+ − = (5.18)
Here H
0
(1)
, H
1
(1)
and H
2
(1)
are the Hankel functions of the first kind of order
zero, order one and order two, respectively, r is the distance between the
source y and observation point x, while n denotes the unit normal vector on a
surface where the traction is being evaluated.
After placing the collocation point y
l
at each node along the surfaces S
and S
0
results in 2(K+P+1) equations which can be combined as

75
HSI HSI HSI
F U A = ;
) 1 ( 2 ) 1 ( 2 + + × + +
∈
P K P K HSI
C A ;
1 ) 1 ( 2 × + +
∈
P K HSI
C U ;
1 ) 1 ( 2 × + +
∈
P K HSI
C F
(5.19)
where the matrix A
HSI
is expressed in terms of the integration constants and
the free terms while the vector F
HSI
contains the free-field results. The vector
U
HSI
contains the unknown displacement and traction fields. Based on the
linear system (5.19), the unknown vector U
HSI
can be computed using the
standard Gauss elimination method. Now the displacement fields throughout
the half space can be evaluated by using the integral representations (5.6)
and (5.7).
This completes the solution of the half-space inclusion problem. The
solution of the half-space cavity problem is considered next.
5.2. Half-space cavity problem solution
For this problem, the boundary integral equations for plane-strain model
are given by equation (5.1). Using the traction-free boundary conditions, the
integral equations become
∫ ∫
− = +
+ S
x
ff
j ji
S S
x
sc
j ji
sc
j
sc
ji
dS f g dS u t PV u c ) , ( ) , ( ) ( ) , , ( ) ( ) (
) 1 ( ) 1 ( ) 1 ( ) 1 ( ) 1 (
0
n x y x x n y x y y ;
0
S S∪ ∈ y (5.20)

76
where
ff
j
f is the known free-field tractions along the corrugated surface S.
These integral equations must be solved for the unknown scattered field
along the corrugated surface S and the half-space surface S
0
. Then the
scattered wave field u
(1)
can be evaluated throughout the half space by using
the integral representations (5.6) for the half-space inclusion problem.
When the boundaries S and S
0
are discretized using one-dimensional
linear elements (Fig. 3.1), the above integral equations become
[ ]
[ ] ) (
) 1 (
) (
) ( ) (
) 1 (
) (
) ( ) (
1
1
) 1 (
2
) 1 (
2 ) 1 ( 21
2
) 1 ( 21
1
) 1 (
2
) 1 (
21
1
) 1 (
1
) 1 (
1 ) 1 ( 11
2
) 1 ( 11
1
) 1 (
1
) 1 (
11
l F
k u
k u
A A l u l c
k u
k u
A A l u l c
K
k
sc
sc
lk lk
sc sc
K
k
sc
sc
lk lk
sc sc
=






+
+ +






+
+
∑
∑
=
=
;
0
S S
l
∂ ∪ ∂ ∈ y ; ) 1 ( : 1 + = K l (5.21)
[ ]
[ ] ) (
) 1 (
) (
) ( ) (
) 1 (
) (
) ( ) (
2
1
) 1 (
2
) 1 (
2 ) 1 ( 22
2
) 1 ( 22
1
) 1 (
2
) 1 (
22
1
) 1 (
1
) 1 (
1 ) 1 ( 12
2
) 1 ( 12
1
) 1 (
1
) 1 (
12
l F
k u
k u
A A l u l c
k u
k u
A A l u l c
K
k
sc
sc
lk lk
sc sc
K
k
sc
sc
lk lk
sc sc
=






+
+ +






+
+
∑
∑
=
=
;
0
S S
l
∂ ∪ ∂ ∈ y ; ) 1 ( : 1 + = K l (5.22)
where
[ ] [ ]
∑ ∑
= =






+
−






+
− =
P
k
ff
ff
lk lk
P
k
ff
ff
lk lk
k f
k f
B B
k f
k f
B B l F
1
2
2 ) 1 ( 21
2
) 1 ( 21
1
1
1
1 ) 1 ( 11
2
) 1 ( 11
1 1
) 1 (
) (
) 1 (
) (
) ( (5.23)

77
[ ] [ ]
∑ ∑
= =






+
−






+
− =
P
k
ff
ff
lk lk
P
k
ff
ff
lk lk
k f
k f
B B
k f
k f
B B l F
1
2
2 ) 1 ( 22
2
) 1 ( 22
1
1
1
1 ) 1 ( 12
2
) 1 ( 12
1 2
) 1 (
) (
) 1 (
) (
) ( (5.24)
Here the collocation point y
l
is located at node l , and the nodes 1 through P
belong to the corrugated surface S while the nodes P+1 to K+1 are at the
half-space surface S
0
. All the other terms in the integral equations have been
already defined.
After placing the collocation point y
l
at each node along the surfaces S
and S
0
, there will be 2(K+1) equations of the type (5.21) and (5.22) and they
can be combined as
HSC sc HSC
F U A = ;
) 1 ( 2 ) 1 ( 2 + × +
∈
K K HSC
C A ;
1 ) 1 ( 2 × +
∈
K sc
C U ;
1 ) 1 ( 2 × +
∈
K HSC
C F (5.25)
Here the matrix A
HSC
is expressed in terms of the integration constants and
the free terms while the vector F
HSC
contains the free-field results. Based on
the last result, the unknown displacement vector U
sc
along the corrugated
surface S and the half-space surface S
0
can be computed using the standard
Gauss elimination method. Now the total displacement field throughout the
half space can be evaluated by using the integral representations (5.6).

78
This completes the solution of the half-space problem. The solution of
the full-space inclusion problem is considered next.
5.3. Full-space inclusion problem solution
For a plane-strain model in domain B
1
the general boundary integral
equations (Niu and Dravinski, 2003a) become
∫ ∫
− + =
S
x j ji
S
x j ji
inc
i j
sc
ji
dS u t PV dS f g u u c ) ( ) , , ( ) , ( ) , ( ) ( ) ( ) (
) 1 ( ) 1 ( ) 1 ( ) 1 ( ) 1 ( ) 1 (
x n y x n x y x y y y ;
S ∈ y (5.26)
while for domain B
2
the integral equations are given by equation (5.2). Here S
represents the corrugated interface and all the other terms have been defined
following equations (5.1) and (5.2).
By substituting the continuity conditions (2.18) into the integral equations
results in the following
) ( ) , ( ) , ( ) ( ) , , ( ) ( ) (
) 1 ( ) 1 ( ) 1 (
y n x y x x n y x y y
inc
i
S
x j ji
S
x j ji j
sc
ji
u dS f g dS u t PV u c = − +
∫ ∫
; S ∈ y
(5.27)
together with equation (5.4). In addition,
) , ( ) , ( ) , (
) ( ) ( ) (
) ( ) ( ) (
) 2 ( ) 1 (
) 2 ( ) 1 (
) 2 ( ) 1 (
n x n x n x
y y y
x x x
j j j
j j j
j j j
f f f
u u u
u u u
= =
= =
= =
; S ∈ y x, (5.28)

79
These integral equations have to be solved first for the unknown
displacement field u(x) and traction field f(x,n) along the corrugation S.
Subsequently, the displacement fields u
(1)
(x) and u
(2)
(x) can be evaluated
throughout the full space using the integral representations (Niu and
Dravinski, 2003a)
∫ ∫
− + =
S
x j ji
S
x j ji
inc
i i
dS u t PV dS f g u u ) ( ) , , ( ) , ( ) , ( ) ( ) (
) 1 ( ) 1 ( ) 1 (
x n y x n x y x y y ;
1
B ∈ y ;
S ∉ y (5.29)
together with equation (5.7), respectively.
In order to solve the integral equations, the corrugated interface S is
discretized using linear elements as shown by Fig. 3.2. Consequently, the
integral equations (5.27) and (5.4) become
[ ] [ ]
[ ] [ ] ) (
) 1 (
) (
) 1 (
) (
) 1 (
) (
) ( ) (
) 1 (
) (
) ( ) (
1
1 2
2 ) 1 ( 21
2
) 1 ( 21
1
1 1
1 ) 1 ( 11
2
) 1 ( 11
1
1 2
2 ) 1 ( 21
2
) 1 ( 21
1 2
) 1 (
21
1 1
1 ) 1 ( 11
2
) 1 ( 11
1 1
) 1 (
11
l u
k f
k f
B B
k f
k f
B B
k u
k u
A A l u l c
k u
k u
A A l u l c
inc
K
k
lk lk
K
k
lk lk
K
k
lk lk
sc
K
k
lk lk
sc
=






+
−






+
−






+
+ +






+
+
∑ ∑
∑ ∑
= =
= =
;
S
l
∂ ∈ y ; K l : 1 = (5.30)
[ ] [ ]
[ ] [ ] ) (
) 1 (
) (
) 1 (
) (
) 1 (
) (
) ( ) (
) 1 (
) (
) ( ) (
2
1 2
2 ) 1 ( 22
2
) 1 ( 22
1
1 1
1 ) 1 ( 12
2
) 1 ( 12
1
1 2
2 ) 1 ( 22
2
) 1 ( 22
1 2
) 1 (
22
1 1
1 ) 1 ( 12
2
) 1 ( 12
1 1
) 1 (
12
l u
k f
k f
B B
k f
k f
B B
k u
k u
A A l u l c
k u
k u
A A l u l c
inc
K
k
lk lk
K
k
lk lk
K
k
lk lk
sc
K
k
lk lk
sc
=






+
−






+
−






+
+ +






+
+
∑ ∑
∑ ∑
= =
= =
;
S
l
∂ ∈ y ; K l : 1 = (5.31)

80
[ ] [ ]
[ ] [ ] 0
) 1 (
) (
) 1 (
) (
) 1 (
) (
) ( ) (
) 1 (
) (
) ( ) (
1 2
2 ) 2 ( 21
2
) 2 ( 21
1
1 1
1 ) 2 ( 11
2
) 2 ( 11
1
1 2
2 ) 2 ( 21
2
) 2 ( 21
1 2
) 2 (
21
1 1
1 ) 2 ( 11
2
) 2 ( 11
1 1
) 2 (
11
=






+
−






+
−






+
+ −






+
+ −
∑ ∑
∑ ∑
= =
= =
K
k
lk lk
K
k
lk lk
K
k
lk lk
sc
K
k
lk lk
sc
k f
k f
B B
k f
k f
B B
k u
k u
A A l u l c
k u
k u
A A l u l c
;
S
l
∂ ∈ y ; K l : 1 = (5.32)
[ ] [ ]
[ ] [ ] 0
) 1 (
) (
) 1 (
) (
) 1 (
) (
) ( ) (
) 1 (
) (
) ( ) (
1 2
2 ) 2 ( 22
2
) 2 ( 22
1
1 1
1 ) 2 ( 12
2
) 2 ( 12
1
1 2
2 ) 2 ( 22
2
) 2 ( 22
1 2
) 2 (
22
1 1
1 ) 2 ( 12
2
) 2 ( 12
1 1
) 2 (
12
=






+
−






+
−






+
+ −






+
+ −
∑ ∑
∑ ∑
= =
= =
K
k
lk lk
K
k
lk lk
K
k
lk lk
sc
K
k
lk lk
sc
k f
k f
B B
k f
k f
B B
k u
k u
A A l u l c
k u
k u
A A l u l c
;
S
l
∂ ∈ y ; K l : 1 = (5.33)
where l denotes the collocation point when y is at node l , and all the other
terms have been already defined. Therefore, the integral equations can be
combined and rewritten in a matrix form as
inc FSI FSI
U U A = ;
K K FSI
C
4 4 ×
∈ A ;
1 4 ×
∈
K FSI
C U ;
1 4 ×
∈
K inc
C U (5.34)
Here
K K
C
×
denotes a K K× dimensional complex vector space. The
matrix A
FSI
is expressed in terms of the integration constants and the free
terms. Based on the linear system, the vector U
FSI
containing the unknown
displacements and tractions along the corrugated interface S can be
computed by using standard Gauss elimination procedure. Once the
displacement and traction fields along the corrugation are known, the total

81
displacements throughout the full space can be evaluated by using the
integral representations (5.29) and (5.7).
This completes solution of the inclusion problem. The full-space cavity
problem solution is considered next.
5.4. Full-space cavity problem solution
For this problem the general boundary integral equations are given by
equation (5.26). By substituting the boundary conditions (2.22) into the
integral equations results in the following
) ( ) ( ) , , ( ) ( ) (
) 1 ( ) 1 ( ) 1 ( ) 1 (
y x n y x y y
inc
i
S
x j ji j
sc
ji
u dS u t PV u c = +
∫
; S ∈ y (5.35)
These integral equations have to be solved first for the unknown
displacement field u
(1)
(x) along the corrugation S. Subsequently, the
displacement field u
(1)
(x) can be evaluated throughout the full space using
the integral representations (Niu and Dravinski, 2003a)
∫
− =
S
x j ji
inc
i i
dS u t PV u u ) ( ) , , ( ) ( ) (
) 1 ( ) 1 ( ) 1 (
x n y x y y ;
1
B ∈ y ; S ∉ y (5.36)
In order to solve the integral equations, the corrugated boundary S is
discretized using linear elements (Fig. 3.2). Consequently, the integral
equations become

82
[ ]
[ ] ) (
) 1 (
) (
) ( ) (
) 1 (
) (
) ( ) (
1
1
) 1 (
2
) 1 (
2 ) 1 ( 21
2
) 1 ( 21
1
) 1 (
2
) 1 (
21
1
) 1 (
1
) 1 (
1 ) 1 ( 11
2
) 1 ( 11
1
) 1 (
1
) 1 (
11
l u
k u
k u
A A l u l c
k u
k u
A A l u l c
inc
K
k
lk lk
sc
K
k
lk lk
sc
=






+
+ +






+
+
∑
∑
=
=
;
S
l
∂ ∈ y ; K l : 1 = (5.37)
[ ]
[ ] ) (
) 1 (
) (
) ( ) (
) 1 (
) (
) ( ) (
2
1
) 1 (
2
) 1 (
2 ) 1 ( 22
2
) 1 ( 22
1
) 1 (
2
) 1 (
22
1
) 1 (
1
) 1 (
1 ) 1 ( 12
2
) 1 ( 12
1
) 1 (
1
) 1 (
12
l u
k u
k u
A A l u l c
k u
k u
A A l u l c
inc
K
k
lk lk
sc
K
k
lk lk
sc
=






+
+ +






+
+
∑
∑
=
=
;
S
l
∂ ∈ y ; K l : 1 = (5.38)
where l denotes the collocation point when y is at node l , and all the other
terms have been defined. After applying all collocation points K l : 1 = , the
above equations can be combined in a matrix form as
inc FSC FSC
u U A = ;
K K FSC
C
2 2 ×
∈ A ;
1 2 ×
∈
K FSC
C U ;
1 2 ×
∈
K inc
C u (5.39)
Here
K K
C
×
denotes a K K× dimensional complex vector space. The
matrix A
FSC
is expressed in terms of the integration constants and the free
terms. The unknown displacement vector U
FSC
along the corrugated surface
S can be computed by using standard Gauss elimination procedure. Once
the displacement field along the corrugation is known, the total displacements

83
throughout the full space can be evaluated by using the integral
representations (5.36).
This completes the solutions of the plane strain problems.
Corresponding results are considered next.

84
Chapter 6: Plane Strain Model Numerical Results
For the plane strain problems the corrugation parameters are tabulated
in Table 6.1. In particular, a soft elastic inclusion ( 6 / 1
2
= μ , 2 / 1
2
=
β
v , 3 =
β
I )
is considered for the inclusion problems.

Table 6.1: The corrugation parameters for the plane strain model.
Medium Scatterer Parameter
Half Space
Inclusion 2 . 0 ; 0 = ε , 8 ; 2 = m ,
1 = a ,
75 . 1 ; 1 ; 75 . 0 = b ,
2
1
= λ , 1
1
= μ ,
1
1
= ρ , 1 ; 25 . 0 =
β
η
1
2
= λ ,
6 / 1
2
= μ ,
3 / 2
2
= ρ
928 = K ,
128 = P ,
a h 2 = ,
ο
80 ; 0 = γ Cavity -
Full Space
Inclusion
1
2
= λ ,
6 / 1
2
= μ ,
3 / 2
2
= ρ
128 = K
Cavity -
6.1. Half-space inclusion problem results
First, a transparency test is considered. Namely, when the half space
and the inclusion have the same material properties the displacements along
the half-space surface S
0
should agree with the free-field results. Therefore,
the surface length S
0
has been increased to match the free-field response at
each frequency. In addition, a convergence test is considered as well. For a
nonzero corrugation amplitude, the rate of convergence between two
numerical results with different numbers of elements K along the corrugated

85
interface S and the half-space surface S
0
can be measured. It is found that
increase in the number of elements K minimizes the difference between two
results at equivalent nodes. Finally, a validation test is considered as well.
When the material properties for the inclusion are replaced by a testing
material with high impedance contrast ( 300 =
β
I ), the resulting motion should
match that for the corresponding half-space cavity problem.
Consequently, the computational range of the half-space surface is
determined as { } a x S 20 :
0
≤ . Any further increase in the range of S
0

produced a negligible effect upon the resulting motion. This length
{ } a x S 20 :
0
≤ satisfies the criterion that S
0
should be twice the wavelength of
the incident longitudinal wave as suggested by Kobayashi and Nishimura
(1982b). Therefore, the number of total elements 928 = K is used for the
subsequent calculations in this study.
Based on the above analyses it is possible to determine all the
parameters required for evaluation of the total wave field for the half-space
problem. The numerical results for this model are considered next.

86
Figure 4.1 depicts two characteristic models for a circular based
inclusion ( 1 = = b a ) with peak amplitude 2 . 0 = ε and two different period
coefficients. These two non-symmetric models are used for the analysis of
the numerical results. The response along the half-space surface S
0
due to a
vertical P incidence is investigated first.
-20 -15 -10 -5 0 5 10 15 20
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
Displacement vs. x
1
, γ=0, η
β
=1, h=2, S
0
=40, m=2
x
1
Displacement |u
P
1
|
ε=0
ε=0.2
-20 -15 -10 -5 0 5 10 15 20
0.5
1
1.5
2
2.5
3
Displacement vs. x
1
, γ=0, η
β
=1, h=2, S
0
=40, m=2
x
1
Displacement |u
P
2
|
ε=0
ε=0.2

Figure 6.1: Half-space inclusion plane strain model displacement amplitudes
along the half-space surface S
0
due to a vertical P incidence for different
corrugation peak amplitude ε as a function of location x
1
(Fig. 2.1) when
1 =
β
η , 2 = m , 1 = = b a , 2
1
= λ , 1
1
= μ , 1
1
= ρ , 1
2
= λ , 6 / 1
2
= μ , 3 / 2
2
= ρ ,
0 = γ , a h 2 = , 128 = P and 928 = K .

As before, only the surface displacement amplitudes for the total wave
field are shown. Since at the low frequency the presence of the corrugation
produced minimal change in the half-space surface response, these results
are omitted. By increasing the frequency to a higher value (Fig. 6.1) it is

87
evident that both components of the surface motion appear to be very
sensitive upon the presence of the corrugation. The half-space surface
motion can be grouped into the far field ( a x 5
1
> ) and the near field ( a x 5
1
≤ ).
In the far field, the corrugated and smooth inclusion motions have similar
patterns however the corresponding amplitudes are very different. In the near
field, the u
1
displacement component atop the scatterer clearly detects the
presence of the inclusion roughness in the surface displacement field.
-20 -15 -10 -5 0 5 10 15 20
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
Displacement vs. x
1
, γ=80, η
β
=1, h=2, S
0
=40, m=2
x
1
Displacement |u
P
1
|
ε=0
ε=0.2
-20 -15 -10 -5 0 5 10 15 20
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Displacement vs. x
1
, γ=80, η
β
=1, h=2, S
0
=40, m=2
x
1
Displacement |u
P
2
|
ε=0
ε=0.2

Figure 6.2: Half-space inclusion plane strain model displacement amplitudes
along the half-space surface S
0
due to an oblique P incidence for different
corrugation peak amplitude ε as a function of location x
1
(Fig. 2.1) when
1 =
β
η , 2 = m , 1 = = b a , 2
1
= λ , 1
1
= μ , 1
1
= ρ , 1
2
= λ , 6 / 1
2
= μ , 3 / 2
2
= ρ ,
ο
80 = γ , a h 2 = , 128 = P and 928 = K .

For an oblique P incidence, a longer corrugation period and at a higher
frequency the results are depicted by Fig. 6.2. Apparently, the near-field

88
surface response is strongly affected by the presence of the corrugation. A
shielding effect can be observed as well. In the far field, both the smooth and
the corrugated inclusion surface responses have similar patterns and
amplitudes.
-20 -15 -10 -5 0 5 10 15 20
0
0.5
1
1.5
2
2.5
Displacement vs. x
1
, γ=0, η
β
=1, h=2, S
0
=40, m=2
x
1
Displacement |u
SV
1
|
ε=0
ε=0.2
-20 -15 -10 -5 0 5 10 15 20
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
Displacement vs. x
1
, γ=0, η
β
=1, h=2, S
0
=40, m=2
x
1
Displacement |u
SV
2
|
ε=0
ε=0.2

Figure 6.3: Half-space inclusion plane strain model displacement amplitudes
along the half-space surface S
0
due to a vertical SV incidence for different
corrugation peak amplitude ε as a function of location x
1
(Fig. 2.1) when
1 =
β
η , 2 = m , 1 = = b a , 2
1
= λ , 1
1
= μ , 1
1
= ρ , 1
2
= λ , 6 / 1
2
= μ , 3 / 2
2
= ρ ,
0 = γ , a h 2 = , 128 = P and 928 = K .

The response due to an SV incidence is investigated next. At the low
frequency the presence of the corrugation produced minor changes in the
surface response thus these results are omitted. At a higher frequency and
for a longer corrugation period the surface response is shown by Fig. 6.3. As

89
before, the half-space surface displacement amplitudes can be divided into
the far field and the near field.
In the far field, both the corrugated and smooth inclusion motions have
similar patterns and amplitudes. However, in the near field, the corrugation is
clearly detected in the surface displacement field.
-20 -15 -10 -5 0 5 10 15 20
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Displacement vs. x
1
, γ=80, η
β
=1, h=2, S
0
=40, m=8
x
1
Displacement |u
SV
1
|
ε=0
ε=0.2

-20 -15 -10 -5 0 5 10 15 20
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
Displacement vs. x
1
, γ=80, η
β
=1, h=2, S
0
=40, m=8
x
1
Displacement |u
SV
2
|
ε=0
ε=0.2

Figure 6.4: Half-space inclusion plane strain model displacement amplitudes
along the half-space surface S
0
due to an oblique SV incidence for different
corrugation peak amplitude ε as a function of location x
1
(Fig. 2.1) when
1 =
β
η , 8 = m , 1 = = b a , 2
1
= λ , 1
1
= μ , 1
1
= ρ , 1
2
= λ , 6 / 1
2
= μ , 3 / 2
2
= ρ ,
ο
80 = γ , a h 2 = , 128 = P and 928 = K .

For an oblique SV incidence at a higher frequency and for a shorter
corrugation period the half-space surface response is depicted by Fig. 6.4. In
the far field, the smooth and the corrugated inclusion responses have similar
patterns but very different amplitudes. As before, the far-field surface

90
response can be divided into the illuminated portion ( a x 5
1
− < ) and the
shadow portion ( a x 5
1
> ). In the illuminated region, the amplitudes of the
rough inclusion motion are significantly reduced by the presence of the
corrugation when compared with the corresponding smooth scatterer results.
However, in the shadow region, the maximum amplitudes in both
displacement components of the surface response are increased due to the
presence of the inclusion roughness. In the near field, a strong site effect is
observed and the surface motion clearly detects the existence of the
corrugation.
Finally, a Rayleigh incidence is considered as well. At a higher frequency
and for a shorter corrugation period the half-space surface motion is shown
by Fig. 6.5. As before, the surface response can be separated into the
illuminated, the shadow and the near-field parts.
In both the illuminated and the shadow portions, the smooth and the
corrugated inclusion response have similar patterns but significantly different
amplitudes. In the illuminated region, the smooth inclusion surface response
attenuates faster to the free-field displacements when compared with the

91
corresponding corrugated inclusion response. In the shadow portion, a more
pronounced shielding effect is observed for the rough scatterer motion than
for the smooth one. In the near field, a strong site effect is observed, and the
low amplitudes in both displacement components are further decreased by
the presence of the corrugation.
-20 -15 -10 -5 0 5 10 15 20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Displacement vs. x
1
, η
β
=1, h=2, S
0
=40, m=8
x
1
Displacement |u
R
1
|
ε=0
ε=0.2
-20 -15 -10 -5 0 5 10 15 20
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Displacement vs. x
1
, η
β
=1, h=2, S
0
=40, m=8
x
1
Displacement |u
R
2
|
ε=0
ε=0.2

Figure 6.5: Half-space inclusion plane strain model displacement amplitudes
along the half-space surface S
0
due to a Rayleigh incidence for different
corrugation peak amplitude ε as a function of location x
1
(Fig. 2.1) when
1 =
β
η , 8 = m , 1 = = b a , 2
1
= λ , 1
1
= μ , 1
1
= ρ , 1
2
= λ , 6 / 1
2
= μ , 3 / 2
2
= ρ ,
a h 2 = , 128 = P and 928 = K .

The analogous calculations have been performed for two elliptically
based inclusions as well ( 1 = a , 75 . 0 = b and 1 = a , 75 . 1 = b ). These results
show similar phenomena observed for the circular inclusion models and thus

92
they are omitted. Hence, the half-space results can be summarized as
follows.
The presence of the corrugation may produce significant changes in the
half-space surface responses when compared with the corresponding
smooth scatterer results. This is particularly displayed at higher frequencies.
For a vertical incidence, the smooth and the corrugated inclusion surface
motions may have similar patterns, but their amplitudes may be very different
especially at the higher frequency and in the near field.
For oblique P and SV or a Rayleigh incidence, the shorter corrugation
period and at the higher frequency the rough inclusion motion amplitudes in
the illuminated portion of the half-space surface may be very different from
the amplitudes of the corresponding smooth scatterer motion. In the shadow
region of the half-space surface, the presence of the corrugation may
generate a pronounced shielding effect in the surface response. In the near
field and at the higher frequency, the half-space surface motion clearly
detects the presence of the corrugation.

93
This concludes the discussion of the numerical results for the half-space
inclusion problem. The results for the half-space cavity model are considered
next.
6.2. Half-space cavity problem results
The total number of elements K used for modeling of the half-space
problem depends upon the length of the half-space surface S
0
. This length
can be determined based on the smooth cavity response for which the
surface displacements are available (e.g., Luco and Barros, 1994). Figures
6.6 to 6.8 show the displacement components at the half-space surface S
0
for
three different incident waves when compared with the corresponding Luco
and Barros’ results. It should be noticed that for a Rayleigh incidence (Fig. 6.8)
Luco and Barros’ displacement amplitudes have to be scaled by
2
2
2
R
k
k
β
. Based
on these comparisons the computational range of the half-space surface is
determined as { } a x S 20 :
0
≤ . Any further increase in the range of S
0

produced a negligible effect upon the resulting motion. The length
{ } a x S 20 :
0
≤ also satisfies the criterion that S
0
should be twice the

94
wavelength of the incident longitudinal wave as suggested by Kobayashi and
Nishimura (1982b).
-5 -4 -3 -2 -1 0 1 2 3 4 5
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Displacement vs. x
1
, γ=0, η
β
=0.5, h=1.5, S
0
=40
x
1
Displacement |u
P
1
|
Luco & Barros
Yu & Dravinski
-5 -4 -3 -2 -1 0 1 2 3 4 5
0.5
1
1.5
2
2.5
Displacement vs. x
1
, γ=0, η
β
=0.5, h=1.5, S
0
=40
x
1
Displacement |u
P
2
|
Luco & Barros
Yu & Dravinski

Figure 6.6: Half-space cavity plane strain model displacement amplitude
along the half-space surface S
0
due to a vertical P incidence as a function of
location x
1
(Fig. 2.1) when 5 . 0 =
β
η , 1 = = b a , 2
1
= λ , 1
1 1
= =ρ μ , 0 = γ ,
a h 5 . 1 = , 128 = P and 928 = K . Solid lines denote the results of this study
while the open circles denote those of Luco and Barros (1994).

-5 -4 -3 -2 -1 0 1 2 3 4 5
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
Displacement vs. x
1
, γ=0, η
β
=0.5, h=1.5, S
0
=40
x
1
Displacement |u
SV
1
|
Luco & Barros
Yu & Dravinski

-5 -4 -3 -2 -1 0 1 2 3 4 5
0
0.5
1
1.5
2
2.5
3
Displacement vs. x
1
, γ=0, η
β
=0.5, h=1.5, S
0
=40
x
1
Displacement |u
SV
2
|
Luco & Barros
Yu & Dravinski

Figure 6.7: Half-space cavity plane strain model displacement amplitude
along the half-space surface S
0
due to a vertical SV incidence as a function of
location x
1
(Fig. 2.1) when 5 . 0 =
β
η , 1 = = b a , 2
1
= λ , 1
1 1
= =ρ μ , 0 = γ ,
a h 5 . 1 = , 128 = P and 928 = K . Solid lines denote the results of this study
while the open circles denote those of Luco and Barros (1994).

95
-5 -4 -3 -2 -1 0 1 2 3 4 5
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
Displacement vs. x
1
, η
β
=0.5, h=1.5, S
0
=40
x
1
Displacement |u
R
1
|
Luco & Barros
Yu & Dravinski

-5 -4 -3 -2 -1 0 1 2 3 4 5
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Displacement vs. x
1
, η
β
=0.5, h=1.5, S
0
=40
x
1
Displacement |u
R
2
|
Luco & Barros
Yu & Dravinski

Figure 6.8: Half-space cavity plane strain model displacement amplitude
along the half-space surface S
0
due to a Rayleigh incidence as a function of
location x
1
(Fig. 2.1) when 5 . 0 =
β
η , 1 = = b a , 2
1
= λ , 1
1 1
= =ρ μ , 0 = γ ,
a h 5 . 1 = , 128 = P and 928 = K . Solid lines denote the results of this study
while the open circles denote those of Luco and Barros (1994).

For a nonzero corrugation amplitude, the convergence rate between two
numerical results with different numbers of elements along the corrugated
cavity surface S and half-space surface S
0
is investigated as well. It can be
shown that increase in the number of elements K minimizes the difference
between the two results at the same nodes. Thus the number of elements
928 = K is used for all the subsequent calculations.
Based on the above analyses it is possible to determine all the
parameters required for evaluation of the total wave field for the problem. The
numerical results for this model are considered next.

96
Figure 4.1 depicts two characteristic models for a circular type of
corrugations. These two non-symmetric extremes are used for the analysis of
the numerical results. The response along the half-space surface S
0
due to a
vertical P incidence is investigated first.
Since at the low frequency ( 25 . 0 =
β
η ) the presence of the corrugation
resulted in minimal change in the half-space surface smooth cavity response,
these results are omitted. By increasing the frequency to a higher value the
surface motion, shown by Fig. 6.9, appears to be very sensitive upon the
presence of the corrugation. The half-space surface motion can be grouped
into the far field ( a x 5
1
> ) and the near field ( a x 5
1
≤ ).
In the far field, the corrugated and smooth cavity motions have similar
patterns but different amplitudes. The corrugated surface motion attenuates
faster to the free-field results than the corresponding smooth cavity response.
In the near field, the motion atop the scatterer clearly detects the presence of
the cavity roughness in the surface displacement field.

97
-20 -15 -10 -5 0 5 10 15 20
0
0.5
1
1.5
2
2.5
Displacement vs. x
1
, γ=0, η
β
=1, h=2, S
0
=40, m=8
x
1
Displacement |u
P
1
|
ε=0
ε=0.2

-20 -15 -10 -5 0 5 10 15 20
0
0.5
1
1.5
2
2.5
3
3.5
Displacement vs. x
1
, γ=0, η
β
=1, h=2, S
0
=40, m=8
x
1
Displacement |u
P
2
|
ε=0
ε=0.2

Figure 6.9: Half-space cavity plane strain model displacement amplitude
along the half-space surface S
0
due to a vertical P incidence for different
corrugation peak amplitude ε as a function of location x
1
(Fig. 2.1) when
1 =
β
η , 8 = m , 1 = = b a , 2
1
= λ , 1
1 1
= =ρ μ , 0 = γ , a h 2 = , 128 = P and
928 = K .

Figure 6.10 shows the half-space surface displacements for an oblique P
incidence at a higher frequency and for a short corrugation period. Apparently,
this surface response is strongly affected by the presence of the corrugation.
As before, the far-field surface response can be divided into the illuminated
portion ( a x 5
1
− < ) and the shadow portion ( a x 5
1
> ). In the illuminated region
of the far field the rough cavity motion is highly oscillatory and of different
amplitude than the smooth cavity motion. In the shadow portion the
amplitudes and the patterns of these two motions are similar. In the near field
strong site effect can be observed due to the scatterer’s roughness.

98
-20 -15 -10 -5 0 5 10 15 20
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Displacement vs. x
1
, γ=80, η
β
=1, h=2, S
0
=40, m=8
x
1
Displacement |u
P
1
|
ε=0
ε=0.2

-20 -15 -10 -5 0 5 10 15 20
0
0.5
1
1.5
2
2.5
Displacement vs. x
1
, γ=80, η
β
=1, h=2, S
0
=40, m=8
x
1
Displacement |u
P
2
|
ε=0
ε=0.2

Figure 6.10: Half-space cavity plane strain model displacement amplitude
along the half-space surface S
0
due to an oblique P incidence for different
corrugation peak amplitude ε as a function of location x
1
(Fig. 2.1) when
1 =
β
η , 8 = m , 1 = = b a , 2
1
= λ , 1
1 1
= =ρ μ ,
ο
80 = γ , a h 2 = , 128 = P and
928 = K .

-20 -15 -10 -5 0 5 10 15 20
0.5
1
1.5
2
2.5
3
Displacement vs. x
1
, γ=0, η
β
=1, h=2, S
0
=40, m=8
x
1
Displacement |u
SV
1
|
ε=0
ε=0.2

-20 -15 -10 -5 0 5 10 15 20
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Displacement vs. x
1
, γ=0, η
β
=1, h=2, S
0
=40, m=8
x
1
Displacement |u
SV
2
|
ε=0
ε=0.2

Figure 6.11: Half-space cavity plane strain model displacement amplitude
along the half-space surface S
0
due to a vertical SV incidence for different
corrugation peak amplitude ε as a function of location x
1
(Fig. 2.1) when
1 =
β
η , 8 = m , 1 = = b a , 2
1
= λ , 1
1 1
= =ρ μ , 0 = γ , a h 2 = , 128 = P and
928 = K .

Just like for the P incidence an SV incidence at the low frequency
( 25 . 0 =
β
η ) shows that the presence of the corrugation produced minor

99
changes in the surface motion. Consequently, these results are omitted.
However, at a higher frequency the surface response (Fig. 6.11) is very
sensitive to the presence of the cavity roughness. As before, the half-space
surface response can be grouped into the far field ( a x 5
1
> ) and the near
field ( a x 5
1
≤ ). In the far field, the corrugated and the smooth cavity motions
have similar patterns but different amplitudes. Both the rough and smooth
scatterer surface responses are highly oscillatory with similar amplitudes for
the horizontal components but different amplitudes for the vertical
components. In the near field, a strong site effect is clearly detected in the
surface displacement field demonstrating the importance of the cavity
roughness upon the surface motion.
Figure 6.12 displays the half-space surface response due to an oblique
SV incidence for a higher frequency and a shorter corrugation period. As
before, the surface response can be separated into the illuminated portion
( a x 5
1
− < ), the shadow portion ( a x 5
1
> ) and the near field ( a x 5
1
≤ ).
In the illuminated region, the surface motion of the rough cavity is highly
oscillatory and it may be significantly magnified by the presence of the

100
corrugation when compared with the corresponding smooth cavity results.
However, in the shadow region both the smooth and rough cavity motions
attenuate with distance and with fewer oscillations. The rough scatterer
produces a strong shielding effect in the remote shadow area. In the near
field, the surface motion clearly detects the presence of the cavity roughness
especially in the horizontal displacement component u
1
.
-20 -15 -10 -5 0 5 10 15 20
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Displacement vs. x
1
, γ=80, η
β
=1, h=2, S
0
=40, m=8
x
1
Displacement |u
SV
1
|
ε=0
ε=0.2
-20 -15 -10 -5 0 5 10 15 20
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Displacement vs. x
1
, γ=80, η
β
=1, h=2, S
0
=40, m=8
x
1
Displacement |u
SV
2
|
ε=0
ε=0.2

Figure 6.12: Half-space cavity plane strain model displacement amplitude
along the half-space surface S
0
due to an oblique SV incidence for different
corrugation peak amplitude ε as a function of location x
1
(Fig. 2.1) when
1 =
β
η , 8 = m , 1 = = b a , 2
1
= λ , 1
1 1
= =ρ μ ,
ο
80 = γ , a h 2 = , 128 = P and
928 = K .

Finally, for a Rayleigh incidence at a higher frequency and for a shorter
corrugation period the half-space surface motion is shown by Fig. 6.13.

101
Again, this surface response can be separated into the illuminated part, the
shadow part and the area atop the scatterer.
In the illuminated region, highly oscillatory motion of different amplitudes
is generated due to the presence of the corrugation when compared with the
smooth scatterer response. In the shadow part, the attenuation of the rough
cavity surface motion is significant. The presence of the cavity roughness
produces a pronounced shielding effect for this area. In the area atop the
rough cavity, a strong site effect is observed demonstrating its sensitivity
upon the presence of the cavity roughness.
-20 -15 -10 -5 0 5 10 15 20
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Displacement vs. x
1
, η
β
=1, h=2, S
0
=40, m=8
x
1
Displacement |u
R
1
|
ε=0
ε=0.2

-20 -15 -10 -5 0 5 10 15 20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
Displacement vs. x
1
, η
β
=1, h=2, S
0
=40, m=8
x
1
Displacement |u
R
2
|
ε=0
ε=0.2

Figure 6.13: Half-space cavity plane strain model displacement amplitude
along the half-space surface S
0
due to a Rayleigh incidence for different
corrugation peak amplitude ε as a function of location x
1
(Fig. 2.1) when
1 =
β
η , 8 = m , 1 = = b a , 2
1
= λ , 1
1 1
= =ρ μ , a h 2 = , 128 = P and
928 = K .

102
The analogous calculations have been performed for two elliptical types
of corrugated cavities as well ( 1 = a , 75 . 0 = b and 1 = a , 75 . 1 = b ). These
results display similar phenomena observed in the circular cavity models and
thus they are omitted. Hence, the half-space results can be summarized as
follows.
For a rough cavity and shorter corrugation periods increase in the
frequency may produce dramatic effect upon the half-space surface
response when compared with the corresponding smooth cavity results.
However, it should be noted that for a rough inclusion the same increase in
the frequency may not generate such significant change in the resulting
motion when compared with the corresponding rough cavity response.
For vertical incidences and shorter corrugation periods and at higher
frequencies the far-field half-space surface motion may approach faster the
free-field results than the corresponding smooth scatterer motion. The
half-space surface near field clearly detects the presence of the cavity
roughness in the surface response.

103
For oblique P and SV or a Rayleigh incidence, the rough cavity motion
amplitudes in the illuminated portion of the half-space surface may be very
different from the amplitudes of the corresponding smooth cavity motion. This
is especially significant for shorter corrugation periods and at higher
frequencies. In the shadow portion of the half-space surface the presence of
the corrugation may generate a pronounced shielding effect in the surface
response.
This concludes the analysis of the numerical results for the half-space
cavity problem. The full-space inclusion problem results are considered next.
6.3. Full-space inclusion problem results
Initially, a transparency test is performed. When the domains B
1
and B
2

have the same material properties the displacements along the interface S
should agree with the incident-field results. In addition, the role of the
corrugation presence must be taken into account, i.e., for a nonzero
corrugation amplitude (ε) the rate of convergence between the two results
with different numbers of elements (K) has been investigated. It is shown that
when the number of elements (K) increases, the difference between two

104
results at equivalent nodes becomes smaller. Finally, a validation test is
applied as well. Namely, when the material for the inclusion is replaced by an
extremely soft material ( 300 =
β
I ) the resulting motion should match the
available displacement response for the corresponding cavity model.
Based on these tests, the number of elements 128 = K is found to be
sufficient for accurate evaluation of the displacement field for the range of
frequencies considered in this study.
The full-space problem for a circular based cavity ( 1 = = b a ) with a
horizontal P incidence is considered first. The geometry of the problems is
depicted by Fig. 4.8. These two models are used for the analysis of the
numerical results.

105
0.5
1
1.5
30
210
60
240
90
270
120
300
150
330
180 0
Displacement |u
P
R
| vs. θ, η
β
=1, m=8
ε=0.2
ε=0

0.5
1
1.5
2
30
210
60
240
90
270
120
300
150
330
180 0
Displacement |u
P
θ
| vs. θ, η
β
=1, m=8
ε=0.2
ε=0

Figure 6.14: Radial and tangential displacement amplitudes along the
corrugated interface S for full-space plane strain inclusion models with
different corrugation amplitudes ε and a P incidence as a function of angle
θ (Fig. 4.8) when 1 =
β
η , 8 = m , 1 = = b a , 2
1
= λ , 1
1
= μ , 1
1
= ρ , 1
2
= λ ,
6 / 1
2
= μ , 3 / 2
2
= ρ and 128 = K .

Since at the low frequency ( 25 . 0 =
β
η ) the presence of the corrugation
only slightly altered the smooth inclusion surface response, these results are
omitted. At a higher frequency, Fig. 6.14 depicts a response along the
corrugated interface for a short corrugation period. For convenience, the
amplitude is presented for the radial and tangential displacement
components as a function of the angle θ (Fig. 4.8). Apparently, the presence
of the corrugation produced significant changes in the displacement field
along the interface when compared with the corresponding smooth scatterer

106
results. For the range of parameters considered in this study the resulting
motion appears to be more sensitive to the short corrugation period than the
long one.
0.5
1
1.5
2
30
210
60
240
90
270
120
300
150
330
180 0
Displacement |u
SV
R
| vs. θ, η
β
=1, m=8
ε=0.2
ε=0
1
2
3
30
210
60
240
90
270
120
300
150
330
180 0
Displacement |u
SV
θ
| vs. θ, η
β
=1, m=8
ε=0.2
ε=0

Figure 6.15: Radial and tangential displacement amplitudes along the
corrugated interface S for full-space plane strain inclusion models with
different corrugation amplitudes ε and an SV incidence as a function of
angle θ (Fig. 4.8) when 1 =
β
η , 8 = m , 1 = = b a , 2
1
= λ , 1
1
= μ , 1
1
= ρ ,
1
2
= λ , 6 / 1
2
= μ , 3 / 2
2
= ρ and 128 = K .

For an SV incidence at a low frequency the presence of the corrugation
generated minor changes in the resulting motion so these results are omitted.
At a higher frequency, Fig. 6.15 displays the interface displacement field for a
short corrugation period. Apparently, the presence of the corrugation

107
produced dramatic changes in the displacement field when compared with
the corresponding smooth scatterer results.
The same calculations have been performed for elliptically based
inclusions ( 75 . 0 , 1 = = b a and 75 . 1 , 1 = = b a ) as well. These results show
similar phenomena observed in the circular based model. Hence, they are
omitted in order to reduce the number of figures. For the range of parameters
considered in this study the full-space results can be summarized as follows.
For both P and SV incidences the presence of the corrugation may produce a
significant change in the corrugated interface displacement field when
compared with the smooth scatterer response. This is especially displayed at
higher frequencies and for shorter corrugation periods.
This concludes the analysis of the numerical results for the full-space
inclusion problem. The cavity problem results are considered next.
6.4. Full-space cavity problem results
The number of elements (K) is determined by using the smooth surface
response. Namely, as the corrugation amplitude approaches zero, the
surface response of the problem reduces to the one for which analytical

108
solution is available (e.g., Mow and Mente, 1963 for 1 = = b a ). Therefore, at
each frequency, the number of elements has been increased to match the
analytical solution. Figures 6.16 and 6.17 display the displacement results for
a smooth scatterer due to an incident SV wave at a low and a higher
frequencies, respectively. The total number of 128 elements produced
excellent agreement between the analytical results and the results of this
study.
In addition, the role of the corrugation presence must be taken into
account as well. Namely, when the corrugation amplitude ε is not zero, the
rate of convergence between the two results with different numbers of
elements K has to be checked. It is shown that when the number of elements
K increases, the difference between two results at equivalent nodes becomes
smaller. Based on these tests the number of elements 128 = K is found to
be sufficient for accurate evaluation of the displacement field for the range of
frequencies considered in this study.

109
0.5
1
1.5
30
210
60
240
90
270
120
300
150
330
180 0
Displacement |u
SV
R
| vs. θ, η
β
=0.25
Yu & Dravinski
Mow & Mente
0.5
1
1.5
2
30
210
60
240
90
270
120
300
150
330
180 0
Displacement |u
SV
θ
| vs. θ, η
β
=0.25
Yu & Dravinski
Mow & Mente

Figure 6.16: Full-space plane strain cavity model displacement amplitude
along the smooth surface S’ as a function of angle θ (Fig. 4.8) when
25 . 0 =
β
η , 1 = = b a , 2
1
= λ , 1
1 1
= =ρ μ and 128 = K . Solid lines denote the
results of this study while solid circles are those of Mow and Mente (1963).

0.5
1
1.5
2
30
210
60
240
90
270
120
300
150
330
180 0
Displacement |u
SV
R
| vs. θ, η
β
=1
Yu & Dravinski
Mow & Mente
0.5
1
1.5
2
30
210
60
240
90
270
120
300
150
330
180 0
Displacement |u
SV
θ
| vs. θ, η
β
=1
Yu & Dravinski
Mow & Mente

Figure 6.17: Full-space plane strain cavity model displacement amplitude
along the smooth surface S’ as a function of angle θ (Fig. 4.8) when 1 =
β
η ,
1 = = b a , 2
1
= λ , 1
1 1
= =ρ μ and 128 = K . Solid lines denote the results of
this study while solid circles are those of Mow and Mente (1963).

110
The full-space problems for a circular based cavity ( 1 = = b a ) with a
horizontal P incidence are considered next. The geometry of the problems is
depicted by Fig. 4.8.
Since at the low frequency ( 25 . 0 =
β
η ) the presence of the corrugation
only slightly altered the cavity surface response these results are omitted.
Figure 6.18 depicts the response along the corrugated surface S for a longer
corrugation period and for different corrugation amplitudes at an intermediate
frequency. Apparently, the presence of the corrugation produced minor
changes in the radial displacement component of the cavity surface response
when compared with the smooth scatterer motion. However, the tangential
displacement component exhibits greater sensitivity to the presence of the
cavity roughness.

111
0.5
1
1.5
2
2.5
30
210
60
240
90
270
120
300
150
330
180 0
Displacement |u
P
R
| vs. θ, η
β
=1, m=2
ε=0.2
ε=0

0.5
1
1.5
2
30
210
60
240
90
270
120
300
150
330
180 0
Displacement |u
P
θ
| vs. θ, η
β
=1, m=2
ε=0.2
ε=0

Figure 6.18: Radial and tangential displacement amplitude along the
corrugated surface S for full-space plane strain cavity models with different
corrugation amplitudes ε and a P incidence as a function of angle θ (Fig.
4.8) when 1 =
β
η , 2 = m , 1 = = b a , 2
1
= λ , 1
1 1
= =ρ μ and 128 = K .

For the same frequency and a shorter corrugation period the surface
displacement field is shown by Fig. 6.19. Clearly, the presence of the
corrugation generated a pronounced change in both displacement
components of the surface displacement field when compared with the
corresponding smooth cavity results. The surface motion seems to be very
sensitive to both the amplitude and the period of the corrugation. However,
for the range of parameters considered in this study the surface response
appears to be more sensitive to the presence of the corrugation with short
corrugation periods than with the long ones.

112
1
2
3
4
30
210
60
240
90
270
120
300
150
330
180 0
Displacement |u
P
R
| vs. θ, η
β
=1, m=8
ε=0.2
ε=0

1
2
3
4
5
30
210
60
240
90
270
120
300
150
330
180 0
Displacement |u
P
θ
| vs. θ, η
β
=1, m=8
ε=0.2
ε=0

Figure 6.19: Radial and tangential displacement amplitude along the
corrugated surface S for full-space plane strain cavity models with different
corrugation amplitudes ε and a P incidence as a function of angle θ (Fig.
4.8) when 1 =
β
η , 8 = m , 1 = = b a , 2
1
= λ , 1
1 1
= =ρ μ and 128 = K .

For an SV incidence at a low frequency the presence of the corrugation
produced minor changes in the resulting motion so these results are omitted.
At a higher frequency, Fig. 6.20 displays the surface displacement field for a
shorter corrugation period. Apparently, the presence of the corrugation
produced a significant change in the surface displacement field when
compared with the corresponding smooth surface results. Clearly, the
resulting motion appears to be very sensitive to the presence of the
corrugation.

113
0.5
1
1.5
2
2.5
30
210
60
240
90
270
120
300
150
330
180 0
Displacement |u
SV
R
| vs. θ, η
β
=1, m=8
ε=0.2
ε=0

1
2
3
4
5
30
210
60
240
90
270
120
300
150
330
180 0
Displacement |u
SV
θ
| vs. θ, η
β
=1, m=8
ε=0.2
ε=0

Figure 6.20: Radial and tangential displacement amplitude along the
corrugated surface S for full-space plane strain cavity models with different
corrugation amplitudes ε and an SV incidence as a function of angle θ
(Fig. 4.8) when 1 =
β
η , 8 = m , 1 = = b a , 2
1
= λ , 1
1 1
= =ρ μ and 128 = K .

The same calculations have been performed for elliptical types of
corrugated cavities ( 75 . 0 , 1 = = b a and 75 . 1 , 1 = = b a ) as well. These
results show similar phenomena observed for the circular based models and
thus they are omitted in order to reduce the number of figures. Thus for the
range of parameters considered in this study the full-space cavity results can
be summarized as follows. Presence of the corrugation may produce a
significant change in the corrugated surface displacement field when
compared with the corresponding smooth cavity response. This is especially
detected at higher frequencies and for shorter corrugation periods. However,

114
it should be noticed that for a rough inclusion the presence of the corrugation
may not generate such dramatic change in the response when compared
with the corresponding rough cavity results.
This concludes the analysis of the numerical results for all the problems.
A summary and conclusions are considered next.

115
Chapter 7: Summary and Conclusions
Anti-plane strain and plane strain models for scattering of plane
harmonic SH, P, SV and Rayleigh waves by a corrugated scatterer
embedded in a full space or in a half space was investigated by using a direct
boundary integral equation method. The scatterer was assumed to be of an
elliptical shape with a superimposed periodic corrugated surface of arbitrary
amplitude and period.
An extensive analysis of the accuracy of the method has been presented.
The response along the corrugated surface for the full-space models and
along the half-space surface for the half-space models was evaluated for a
wide range of parameters present in the problems.
Based on the presented results the following conclusions were obtained.
The key features of the resulting surface response are found to be similar for
both full-space and half-space models. At higher frequencies, larger
corrugation amplitudes and for shorter corrugation periods the corrugation
produced significant changes in the response when compared with the
corresponding smooth scatterer result.

116
For a rough scatterer embedded in a half space and subjected to a
vertical incidence the smooth and the corrugated scatterer surface responses
may have similar patterns but very different amplitudes. This is especially
displayed at higher frequencies and for shorter corrugation periods. The
far-field surface response for a rough scatterer may attenuate faster to the
free-field results than the corresponding smooth scatterer motion. In the near
field, it was shown that the half-space surface response may be significantly
different from the equivalent smooth scatterer results. The difference
depends on the frequency and the corrugation shape.
For the half-space problems with a grazing SH, oblique P and SV or a
Rayleigh incidence, the illuminated half-space surface displacements for the
rough scatterer may greatly differ from the corresponding smooth scatterer
results. Strong oscillations in the surface motion reach far into the illuminated
region especially for shorter corrugation periods and at higher frequencies. In
the shadow portion of the half-space surface, the presence of the corrugation
may generate a significant shielding effect in the resulting motion.
Furthermore, atop the rough scatterer the surface response could be

117
significantly affected by the presence of the corrugation. This local site effect
depends upon the frequency, corrugation shape and/or the impedance
contrast between the materials.
For the plane strain rough cavity model increase in the frequency may
produce dramatic effect upon the cavity and the half-space surface response
when compared with the corresponding smooth cavity results. However, for a
rough inclusion the same increase in the frequency may not generate such
significant change in the resulting motion when compared with the
corresponding rough cavity response. The difference between the cavity and
the inclusion problem results for the plane strain model was not found in the
anti-plane strain model results.
Therefore, presented results clearly demonstrate the importance of the
corrugation of the scatterer upon the resulting motion. It was demonstrated
that by neglecting the scatterer’s roughness may result in a significant error in
the estimate of the resulting motion.
The presented solution places no restrictions on the incident wave and
the corrugation shape. However, more complex scatterers and higher

118
frequencies will require larger number and/or higher order of the elements
compared to those used in this study.
The new contributions developed in this study are summarized as
follows. The formulation of the problems for scattering of plane harmonic SH,
P, SV and Rayleigh waves by a fully embedded periodically corrugated
scatterer was completed by using a direct boundary integral equation method.
The solution places no restriction on the corrugation amplitude and period.
The free terms
) 2 ( sc
c and
) 2 ( sc
ji
c for the inclusion problems were formulated
and evaluated. In addition, an efficient algorithm for evaluation of the
numerical results in a MATLAB environment was developed. Extensive
validation of the numerical results was performed. It was showed that by
neglecting the corrugated nature of the scatterer may lead to significant
errors in the motion estimate thus establishing the importance of the
scatterer’s roughness on the resulting motion.

119
References
Abubakar I. (1962a). Reflection and refraction of plane SH waves at irregular
interfaces. I. Journal of Physics of the Earth; 10(1):1-14.

Abubakar I. (1962b). Reflection and refraction of plane SH waves at irregular
interfaces. II. Journal of Physics of the Earth; 10(1):15-20.

Aki K. (1988). Local site effects and strong ground motion. Proceedings of the
Special Conference on Earthquake Engineering and Soil Dynamics 2, Am.
Soc. Civil Eng., Park City, Utah.

Aki K., Larner K. L. (1970). Surface motion of a layered medium having an
irregular interface due to incident plane SH waves. Journal of Geophysical
Research; 75(5):933-954.

Alterman Z., Karal F.C. Jr. (1968). Propagation of elastic waves in layered
media by finite difference methods. Bulletin of the Seismological Society of
America; 58(1):367-398.

Asano S. (1960). Reflection and refraction of elastic waves at a corrugated
boundary surface. Part I. The case of incidence of SH wave. Bulletin of the
Earthquake Research Institute; 38:177-197.

Asano S. (1961). Reflection and refraction of elastic waves at a corrugated
boundary surface. Part II. Bulletin of the Earthquake Research Institute;
39:367-466.

Asano S. (1966). Reflection and refraction of elastic waves at a corrugated
interface. Bulletin of the Seismological Society of America; 56(1):201-221.

Bard P.Y., Bouchon M. (1980). The seismic response of sediment-filled
valleys. Part 1. The case of incident SH waves. Bulletin of the Seismological
Society of America; 70(4):1263-1286.

120
Bard P.Y., Bouchon M. (1985). The two-dimensional resonance of
sediment-filled valleys. Bulletin of the Seismological Society of America;
75(2):519-541.

Beskos D.E. (1987). Boundary elements methods in dynamic analysis.
Applied Mechanics Reviews; 40(1):1-23.

Beskos D.E. (1997). Boundary elements methods in dynamic analysis: Part II
(1986-1996). Applied Mechanics Reviews; 50(3):149-197.

Boore D.M. (1972). A note on the effect of simple topography on seismic SH
waves. Bulletin of the Seismological Society of America; 62(1):275-284.

Boore D.M. (1973). The effect of simple topography on seismic waves:
implications for the accelerations recorded at Pacoima Dam, San Fernando
Valley, California. Bulletin of the Seismological Society of America;
63(5):1603-1609.

Bouchon M., Schultz C.A., Toksöz M.N. (1996). Effect of three-dimensional
topography on seismic motion. Journal of Geophysical Research;
101(B3):5835-5846.

Burton A.J., Miller G.F. (1971). The application of integral equation methods
to the numerical solution of some exterior boundary-value problems.
Proceedings of the Royal Society of London Series A; 323:201-210.

Chen Z., Dravinski M. (2007a). Numerical evaluation of harmonic Green’s
functions for triclinic half-space with embedded sources-Part I: A 2D model.
International Journal for Numerical Methods in Engineering; 69:347-366.

Chen Z., Dravinski M. (2007b). Numerical evaluation of harmonic Green’s
functions for triclinic half-space with embedded sources-Part II: A 3D model.
International Journal for Numerical Methods in Engineering; 69:367-389.

121
Clayton R., Engquist B. (1977). Absorbing boundary conditions for acoustic
and elastic wave equations. Bulletin of the Seismological Society of America;
67(6):1529-1540.

Datta S.K., El-Akily N. (1978). Diffraction of elastic waves in half space I:
integral representations and matched asymptotic expansions. Modern
Problems in Elastic Wave Propagation. John Wiley & Sons: New York;
197-218.

Dravinski M. (1982). Influence of interface depth upon strong ground motion.
Bulletin of the Seismological Society of America; 72(2):597-614.

Dravinski M. (1983). Scattering of plane harmonic SH wave by dipping layers
or arbitrary shape. Bulletin of the Seismological Society of America;
73(5):1303-1319.

Dravinski M. (2007). Scattering of waves by a sedimentary basin with a
corrugated interface. Bulletin of the Seismological Society of America;
97(1B):256-264.

Dravinski M., Mossessian T.K. (1987). Scattering of plane harmonic P, SV
and Rayleigh waves by dipping layers or arbitrary shape. Bulletin of the
Seismological Society of America; 77(1):212-235.

Dravinski M., Mossessian T.K. (1988). On evaluation of the Green functions
for harmonic line loads in a viscoelastic half space. International Journal for
Numerical Methods in Engineering; 26:823-841.

Dravinski M., Niu Y. (2002). Three-dimensional time-harmonic Green’s
functions for a triclinic full-space using a symbolic computation system.
International Journal for Numerical Methods in Engineering; 53:455-472.

122
Dravinski M., Zheng T. (2000). Numerical evaluation of three-dimensional
time-harmonic Green’s functions for a nonisotropic full-space. Wave Motion;
32:141-151.

Elfouhaily T.M., Guérin C.A. (2004). A critical survey of approximate
scattering wave theories from random rough surfaces. Wave Random Media;
14:R1-R40.

Eshraghi H., Dravinski M. (1989). Scattering of elastic waves by
nonaxisymmetric three-dimensional dipping layer. Numerical Methods for
Partial Differential Equations; 5:327-345.

Eshraghi H., Dravinski M. (1991). Transient scattering of elastic waves by
three-dimensional non-axisymmetric dipping layers. International Journal for
Numerical Methods in Engineering; 31:1009-1026.

Fu L.Y., Wu R.S., Campillo M. (2002). Energy partition and attenuation of
regional phases by random free surface. Bulletin of the Seismological Society
of America; 92(5):1992-2007.

Fu L.Y. (2005). Rough surface scattering: comparison of various
approximation theories for 2D SH waves. Bulletin of the Seismological
Society of America; 95(2):646-663.

Hong T.L., Helmberger D.V. (1977). Generalized ray theory for dipping
structure. Bulletin of the Seismological Society of America; 67(4):995-1008.

Kobayashi S. (1987). Chapter 4: Elastodynamics. Boundary Element
Methods in Mechanics. Beskos D.E. (editor). Elsevier Science Publishers B.
V.: Amsterdam.

Kobayashi S., Nishimura N. (1982a). Chapter 7: Transient stress analysis of
tunnels and caverns of arbitrary shape due to traveling waves. Developments
in Boundary Element Methods-II. Applied Science Publishers: London.

123
Kobayashi S., Nishimura N. (1982b). Dynamic analysis of underground
structures by the integral equation method. Numerical Methods in
Geomechanics, Edmonton 1982: Proceedings of the Fourth International
Conference on Numerical Methods in Geomechanics, Edmonton, Canada/
May 31-June 4, 1982. Eisenstein Z. (editor). A.A. Balkema: Rotterdam.

Kuo J.T., Nafe J.E. (1962). Period equation for Rayleigh waves in a layer
overlying a half space with a sinusoidal interface. Bulletin of the
Seismological Society of America; 52(4):807-822.

Kupradze V.D. (1963). Progress in Solid Mechanics. North-Holland
Publishing Company: Amsterdam.

Lee J.J., Langston C.A. (1983). Wave propagation in a three-dimensional
circular basin. Bulletin of the Seismological Society of America;
73(6):1637-1653.

Lee V.W. (1977). On deformations near circular underground cavity subjected
to incident plane SH-waves. Proceedings of the Symposium on Applications
of Computer Methods in Engineering, University of Southern California;
951-962.

Luco J.E., De Barros F.C.P. (1994). Dynamic displacements and stresses in
the vicinity of a cylindrical cavity embedded in a half-space. Earthquake
Engineering and Structural Dynamics; 23:321-340.

Mal A.K., Singh S.J. (1991). Deformation of Elastic Solids. Prentice-Hall Inc.:
New Jersey.

Miklowitz J. (1983). Wavefront fields in the scattering of elastic waves by
surface-breaking and sub-surface cracks. Review of Progress in Quantitative
Nondestructive Evaluation 2A. Plenum Press: New York; 413-440.

124
Mossessian T.K., Dravinski M. (1987). Application of a hybrid method for
scattering of P, SV, and Rayleigh waves by near-surface irregularities.
Bulletin of the Seismological Society of America; 77(5):1784-1803.

Mow C.C., Mente L.J. (1963). Dynamic stresses and displacements around
cylindrical discontinuities due to plane harmonic shear waves. Journal of
Applied Mechanics; 30:598-604.

Niu Y., Dravinski M. (2003a). Three-dimensional BEM for scattering of elastic
waves in general anisotropic media. International Journal for Numerical
Methods in Engineering; 58:979-998.

Niu Y., Dravinski M. (2003b). Direct 3D BEM for scattering of elastic waves in
a homogeneous anisotropic half-space. Wave Motion; 38:165-175.

Pao Y.H., Mow C.C. (1973). Diffraction of elastic waves and dynamic stress
concentrations. Crane, Russak & Company Inc.: New York.

Paris F., Caňas J. (1997). Boundary element method. Oxford University
Press Inc.: New York.

Rizzo F.J., Shippy D.J., Rezayat M. (1985). A boundary integral equation
method for radiation and scattering of elastic waves in three dimensions.
International Journal for Numerical Methods in Engineering; 21:115-129.

Sanchez-Sesma F.J. (1983). Diffraction of elastic waves by
three-dimensional surface irregularities. Bulletin of the Seismological Society
of America; 73(6):1621-1636.

Shaw R.P. (1979). Chapter 6: Boundary integral equation methods applied to
wave problems. Developments in Boundary Element Methods-I. Applied
Science Publishers: London.

125
Smith W.D., Bolt B.A. (1976). Rayleigh’s principle in finite element
calculations of seismic wave response. Geophys. J. R. astr. Soc.;
45:647-655.

Trifunac M.D. (1971). Surface motion of a semi-cylindrical alluvial valley for
incident plane SH waves. Bulletin of the Seismological Society of America;
61(6):1755-1770.

Trifunac M.D. (1973). Scattering of plane SH waves by a semi-cylindrical
canyon. Earthquake Engineering and Structural Dynamics; 1:267-281.

Warnick K.F., Chew W.C. (2001). Numerical simulation methods for rough
surface scattering. Wave Random Media; 11:R1-R30.

Wong H.L., Jennings P.C. (1975). Effect of canyon topography on strong
ground motion. Bulletin of the Seismological Society of America;
65(4):1239-1257.

Wong H.L. (1982). Effect of surface topography on the diffraction of P, SV,
and Rayleigh waves. Bulletin of the Seismological Society of America;
72(4):1167-1183.

Zheng T., Dravinski M. (1999). Amplification of waves by an orthotropic basin:
sagittal plane motion. Earthquake Engineering and Structural Dynamics;
28:565-584.

126
Appendix: Evaluation of Computer Programs
All the computer programs for the wave scattering problems at hand
were written using MATLAB. To evaluate the efficiency of these programs,
the average CPU times for running the codes are tabulated in Table A.1.
Apparently, the CPU times for the plane strain model are much longer than
those for the anti-plane strain model. In general, the CPU time depends upon
the size of the linear system and the complexity of the Green’s functions.

Table A.1: The average CPU times for executing MATLAB programs with an
Intel Core 2 CPU Q6600 at 2.4 GHz and 3 GB RAM.
Model (Incident Wave) \ Scatterer Cavity Inclusion
Anti-Plane Strain
Model (SH)
Full-Space
Problem
3 seconds 10 seconds
Half-Space
Problem
250 seconds 255 seconds
Plane Strain Model
(P, SV and Rayleigh)
Full-Space
Problem
46 seconds 115 seconds
Half-Space
Problem
2750 seconds 2850 seconds

Asset Metadata

Creator Yu, Chih-Wei (author)

Core Title Scattering of a plane harmonic wave by a completely embedded corrugated scatterer

Contributor Electronically uploaded by the author (provenance)

School Andrew and Erna Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Mechanical Engineering

Publication Date 09/16/2008

Defense Date 09/03/2008

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag corrugated scatterer,OAI-PMH Harvest,plane harmonic wave,wave scattering

Language English

Advisor Dravinski, Marijan (committee chair), Flashner, Henryk (committee member), Proskurowski, Wlodek (committee member), Sadhal, Satwindar S. (committee member), Vashishta, Priya (committee member)

Creator Email chihweiy@usc.edu,mikeyucw@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-m1603

Unique identifier UC1168263

Identifier etd-Yu-2416 (filename),usctheses-m40 (legacy collection record id),usctheses-c127-93863 (legacy record id),usctheses-m1603 (legacy record id)

Legacy Identifier etd-Yu-2416.pdf

Dmrecord 93863

Document Type Dissertation

Rights Yu, Chih-Wei

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Repository Name Libraries, University of Southern California

Repository Location Los Angeles, California

Repository Email uscdl@usc.edu

Abstract (if available)

Abstract Anti-plane strain and plane strain models for steady-state scattering of elastic waves by a rough scatterer embedded either in a full space or in a half space is considered by using a direct boundary integral equation method. Both cavity and inclusion problems are investigated. The roughness of the scatterer is assumed to be periodic with arbitrary amplitude and period.