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Diffraction of elastic waves around layered strata with irregular topography
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Diffraction of elastic waves around layered strata with irregular topography
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Diffraction of Elastic Waves around Layered Strata with Irregular Topography
by
Wen-Young Liu
Ph.D. Advisor: Vincent W. Lee
___________________________
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
(CIVIL ENGINEERING)
May 2017
Copyright 2017 Wen-Young Liu
i
This Page Intentionally Left Blank
ii
Abstract
The purpose of the thesis aims to analyze the wave propagation responses in the irregular
layered elastic media due to seismic waves. As a result of excitation by incoming seismic waves,
the underground structures generate additional waves from diffraction and scattering. Such
generated waves always result in amplification or de-amplification of the input waves, which
may cause the deformation, distribution and concentration of stresses near the ground surface
and the structures above. Thus, it is important to understand the theoretical aspects and the
effects of such diffraction behavior.
Lee & Liu (2013) published a simplified mathematical formulation consisting of Fourier
half-range expansions, which does not require either Lee & Cao’s arc approximation or
complicated integration by Lin et al. (2010), to re-examine the cylindrical canyon model. This
new approach automatically satisfies the boundary conditions at the flat ground surface and
reduces the number of equations in half. In other words, the computational efficiency for
calculation increases 100 percent, compared to the previous method. The site amplification
responses computed from the new exact solutions are more precise and can be applied to higher
frequencies, compared to the results by Lee & Cao (1991). Two papers for demonstrating the
feasibility of the Fourier half-range expression approach to irregular elastic layers in Love and
SH Waves (2014), Rayleigh and Body P and SV Waves (2014) have been published and
incorporated into Chapter 3, 4 and 5.
iii
Acknowledgement
I am sincerely and deeply appreciated to my advisor, Professor Vincent W. Lee, for the
unlimited support and guidance he showed me throughout my thesis writing. Without his
generous help, I am sure that it would have not been possible for completion. Besides I would
like to thank to my parents, wife and children whom had been tolerated my neglectfulness during
my work. In addition, my heartily special thanks to Dr. Chi-Hsin Lin for his generous review
and valuable suggestions on the manuscript.
iv
Contents
Abstract ii
Acknowledgement iii
List of Figures vi
List of Tables ix
1 Introduction 1
1.1 General Introduction …..……..……………………………………..………….... 1
1.2 Ground Response Near a Site with Irregular Topography – Homogeneous Half-Space
……………………………………………………………………………………... 2
1.3 Ground Response On and Below a Site with Irregular Topography – Elastic Layered
Media …...…………..………………………………..…………………….……. 4
1.4 Existing Work on Irregular Multi-Layered Media .........…..…………..………... 11
1.5 Organization of Thesis ………..…..……………………….…………...……….. 18
2 Two-Dimensional Diffraction of P and SV Waves Around a Semi-Circular
Canyon in an Elastic Half-Space: Analytic Solution via Stress-Free Wave
Function 19
2.1 Introduction ………………………………………………………………..…… 19
2.2 Previous Work on Two-Dimensional Diffraction ……………………...………. 19
2.3 Analytic Solution of the Wave Equations with Zero-Stress on the Half-Space Surface
………………………………………………………………………….………….. 26
2.4 The Canyon Zero-Stress Boundary Conditions ………………………………... 30
2.5 Numerical Implementation: Incidence Angle:
60 ………...……………... 34
2.6 Results for Other Incidence Angles: 5 , 30 and 90 ……….………….... 41
2.7 Summary ……………………………………………………………………….. 46
3 Diffraction Around an Irregular Layered Elastic Media, I: Love and Body
SH Waves 49
3.1 Introduction …………………………………………………………………….. 49
3.2 Love Surface and Body SH Waves On and Below the Surface of an Elastic Layered
Media ………………………………………………………………………..…. 51
3.3 Love and Body SH Waves Incident On an Irregular Layered Elastic Media ..… 53
v
3.4 The Method of Weighted Residues (Moment) Method ……………………...… 57
3.5 Numerical Implementation …………………………………………………….. 60
3.6 The Diffracted Mode Shapes of Love and SH Body Waves …………………... 67
3.6.1 The Input Free-Field Waves ……………………...………………….... 67
3.6.2 The Diffracted Mode Shapes …………………...………………….….. 74
3.7 Generalization to More Complex Layered Interfaces …………………….……. 84
3.7.1 More General Irregular Layered Media …………...…..………………. 84
3.7.2 Defining Two Origins on the Lth Layer …………………...………..… 85
3.7.3 The Case of a Two-Layer Canyon-Hill Interface ……………......……. 89
4 Diffraction Around an Irregular Layered Elastic Media, II: Rayleigh and
Body P, SV Waves 104
4.1 Introduction ………………………………………………….………………. 104
4.2 Rayleigh and Body P, SV Waves Incident On and Below the Surface of Elastic
Layered Media ……………..…………………………...……………………. 106
4.3 Rayleigh and Body P, SV Waves Incident On an Irregular Elastic Layered Media
………………………………………………………………………………...… 109
4.4 Numerical Implementation ……………………………………………...…… 116
4.5 The Diffracted Mode Shapes of Rayleigh and P, SV Body Waves …………. 123
4.5.1 The Input Free-Field Waves ……………………………..………….. 123
4.5.2 The Diffracted Mode Shapes ……………………………...………… 130
4.6 Generalization to More Complex Layered Interfaces ………..……………… 148
4.6.1 More General Irregular Layered Media ……………………………... 148
4.6.2 The Case of a Two-Layer Canyon-Hill Interface ………………….... 150
5 Synthetic Scattered and Diffracted Time Histories of Love and Rayleigh
Waves 165
5.1 Introduction ……………………………………………………………….…. 165
5.2 Introduction of Love Wave ………………………………………..………… 165
5.3 Synthetic Time Histories of Love Wave ………………………………..…… 166
5.4 The Fourier and Response Spectral Amplitudes of Love Wave …….………. 168
5.5 Summary for Love Wave ……………………………………………………. 171
5.6 Introduction of Rayleigh Wave …………………………………………….... 172
5.7 Synthetic Time Histories of Rayleigh Wave ………………………………… 173
5.8 The Fourier and Response Spectral Amplitudes of Rayleigh Wave ……….... 175
5.9 Summary for Rayleigh Wave ………………………………………………... 181
5.10 Conclusion …………………………………………………………………… 182
Bibliography 183
vi
List of Figures
1.1 N-Layered Half-Space with Incident Love Waves ……………………….…. 5
1.2 N-Layered Half-Space with Incident Body SH Waves ……………………... 6
1.3 N-Layered Half-Space with Incident Rayleigh Waves …………………….... 7
1.4 N-Layered Half-Space with Incident Body P, SV Waves ………………...… 8
1.5 Irregular-Shaped N-Layered Half-Space Incident with Love Waves …...…. 14
1.6 Irregular-Shaped N-Layered Half-Space Incident with Rayleigh Waves ….. 15
1.7 Multiple Coordinate System Generalization Layers in Love Waves ….…… 16
1.8 Multiple Coordinate System Generalization Layers in Rayleigh Waves ….. 17
2.1 The model of two-dimensional semi-circular canyon ……………...……… 20
2.2 Approximating the half-space boundary by a large convex circular surface ... 23
2.3 Approximating the half-space by convex surfaces ……………………...…. 24
2.4 Approximating the half-space by concave surfaces ………………………... 24
2.5 Sine versus Cosine Expansion …………………………………...………… 27
2.6 Displacement amplitudes on or around canyon (for 60 , 10 ) ……… 39
2.7 Displacement amplitudes on or around canyon (for 60 , 20 ) …….... 40
2.8 Displacement amplitudes on canyon (for 5 ,
20 ) ………………….. 43
2.9 Displacement amplitudes on canyon (for 30 ,
20 ) ……………….… 44
2.10 Displacement amplitudes on or around the semi-circular canyon
(for 90 ,
20 ,
N = 136-144) …………………………………….…… 45
3.1 N-layered half-space with Love Waves …………………...……………..… 51
3.2 Irregular-Shaped N-layered half-space with Love Waves ………...……….. 53
3.3 Angle between radial vector and normal at a point …………………….. 55
3.4 Mode#1 Mode Shapes at f = 0.20, 0.67, 2.00 and 10.00 Hz ……………….. 69
3.5 Mode#2 Mode Shapes at f = 0.50, 1.00, 3.33 and 12.50 Hz ……………….. 70
3.6 Mode#3 Mode Shapes at f = 1.00, 1.25, 2.50 and 12.50 Hz ………..……… 71
3.7 SH Body Waves Mode Shapes at f=0.25, 1.00, 2.50 and 10.00 Hz ……..… 73
3.8 The Irregular Two-Layered Media with (i) Love Waves and (ii) SH Body Waves
…………………………………………………………………………….…… 74
3.9 Mode#1 Diffracted Mode Shapes at f = 13.33, 18.18, 20.00 & 25.00 Hz …. 76
3.10 Mode#2 (2-D) Diffracted Mode Shapes at f = 13.33, 18.18, 20.00 & 25.00 Hz 77
3.11 Mode#2 (3-D) Diffracted Mode Shapes at f = 13.33, 18.18, 20.00 & 25.00 Hz 78
3.12 Mode#3 Diffracted Mode Shapes at f = 13.33, 18.18, 20.00 & 25.00 Hz …. 79
3.13 Mode#4 Diffracted Mode Shapes at f = 13.33, 18.18, 20.00 & 25.00 Hz …. 80
3.14 Mode#5 Diffracted Mode Shapes at f = 13.33, 18.18, 20.00 & 25.00 Hz …. 81
3.15 SH Body Waves Diffracted Mode Shapes at f = 13.33, 18.18, 20.00 & 25.00 Hz
…………………………………………………………………………………. 83
3.16 Irregular Layered Media with Multiple Coordinate Systems …………….... 84
vii
3.17 Two Origin for Lth Layer ………………………………………………….. 86
3.18 Transformation using Graf’s Addition Theorem …………………………... 87
3.19 The Irregular Two-Layered Media with Love Waves ……………………... 89
3.20 Mode#1 Diffracted Mode Shapes at f = 1.00, 2.00, 3.33 & 5.00 Hz …….… 92
3.21 Mode#1 Diffracted Mode Shapes at f = 6.67, 10.00, 12.50 & 13.33 Hz …... 93
3.22 Mode#2 (2-D) Diffracted Mode Shapes at f = 1.00, 2.00, 2.50 & 3.33 Hz …. 94
3.23 Mode#2 (3-D) Diffracted Mode Shapes at f = 1.00, 2.00, 2.50 & 3.33 Hz ….. 95
3.24 Mode#2 (2-D) Diffracted Mode Shapes at f = 5.00, 6.67, 10.00 & 12.50 Hz … 96
3.25 Mode#2 (3-D) Diffracted Mode Shapes at f = 5.00, 6.67, 10.00 & 12.50 Hz …. 97
3.26 Mode#3 Diffracted Mode Shapes at f = 1.43, 2.50, 3.12 & 3.57 Hz ………. 98
3.27 Mode#3 Diffracted Mode Shapes at f = 5.00, 6.67, 10.00 & 11.11 Hz …..... 99
3.28 Mode#4 Diffracted Mode Shapes at f = 2.00, 2.50, 3.33 & 5.00 Hz ……... 100
3.29 Mode#4 Diffracted Mode Shapes at f = 5.88, 6.67, 9.09 & 10.00 Hz ……. 101
3.30 Mode#5 Diffracted Mode Shapes at f = 2.00, 2.50, 3.12 & 5.00 Hz ……... 102
3.31 Mode#5 Diffracted Mode Shapes at f = 5.88, 7.14, 9.09 & 10.00 Hz ……. 103
4.1 N-layered half-space with Rayleigh Waves ……………………………..... 106
4.2 Irregular-Shaped N-layered half-space with Rayleigh Waves …………..... 109
4.3 Angle between radial vector and normal at a point …………………… 112
4.4 Mode#1 Rayleigh Mode Shapes at f = 0.20, 0.67, 2.00 and 6.67 Hz ……... 125
4.5 Mode#2 Rayleigh Mode Shapes at f = 0.33, 0.67, 3.33 and 6.67 Hz ……... 126
4.6 Mode#3 Rayleigh Mode Shapes at f = 0.42, 0.83, 4.17 and 6.67 Hz …….. 127
4.7 Mode#11, 12 - Incident P, SV Body Waves Mode Shapes at f = 1.00, 3.33 Hz
……………………………………………………………………………….. 129
4.8 The Irregular Two-Layered Media with Incident SH Body Waves ……..... 130
4.9 Mode#1 x-comp. Diffracted Mode Shapes at f = 0.25, 1.25, 2.00 & 3.13Hz .. 132
4.10 Mode#1 z-comp. Diffracted Mode Shapes at f = 0.25, 1.25, 2.00 & 3.13Hz .. 133
4.11 Mode#2 (2-D) x-comp. Diffracted Mode Shapes at f = 1.25, 2.38, 3.57 & 7.14Hz
………………………………………………………………………………... 135
4.12 Mode#2 (3-D) x-comp. Diffracted Mode Shapes at f = 1.25, 2.38, 3.57 & 7.14Hz
………………………………………………………………………………... 136
4.13 Mode#2 (2-D) z-comp. Diffracted Mode Shapes at f = 1.25, 2.38, 3.57 & 7.14Hz
………………………………………………………………………………... 137
4.14 Mode#2 (3-D) z-comp. Diffracted Mode Shapes at f = 1.25, 2.38, 3.57 & 7.14Hz
………………………………………………………………………………... 138
4.15 Mode#3 x-comp. Diffracted Mode Shapes at f = 1.00, 2.08, 3.57 & 7.14Hz .. 139
4.16 Mode#3 z-comp. Diffracted Mode Shapes at f = 1.00, 2.08, 3.57 & 7.14Hz .. 140
4.17 Mode#4 x-comp. Diffracted Mode Shapes at f = 1.25, 2.08, 4.17 & 7.14Hz .. 141
4.18 Mode#4 z-comp. Diffracted Mode Shapes at f = 1.25, 2.08, 4.17 & 7.14Hz ... 142
4.19 Mode#5 x-comp. Diffracted Mode Shapes at f = 1.33, 2.00, 3.33 & 4.17Hz … 143
4.20 Mode#5 z-comp. Diffracted Mode Shapes at f = 1.33, 2.00, 3.33 & 4.17Hz ... 144
4.21 Mode#11 P-Body Waves Diffracted Mode Shapes at f = 1.25 & 5.26 Hz .. 145
4.22 Mode#12 SV-Body Waves Diffracted Mode Shapes at f = 1.25 & 5.56 Hz .. 146
4.23 Two Origin for Lth Layer ……………………………………………….... 148
4.24 Two Origin for (L-1)th Layer …………………………………………….. 149
4.25 The Irregular Two-Layered Media with Rayleigh Body Waves …………. 150
4.26 Mode#1 Diffracted Mode Shapes (x-) at f = 2.00, 3.33, 6.67 & 13.33 Hz .. 153
4.27 Mode#1 Diffracted Mode Shapes (z-) at f = 2.00, 3.33, 6.67 & 13.33 Hz .. 154
viii
4.28 Mode#2 (2-D) Diffracted Mode Shapes (x-) at f = 3.13, 5.26, 6.67 & 9.09 Hz .. 155
4.29 Mode#2 (3-D) Diffracted Mode Shapes (x-) at f = 3.13, 5.26, 6.67 & 9.09 Hz .. 156
4.30 Mode#2 (2-D) Diffracted Mode Shapes (z-) at f = 3.13, 5.26, 6.67 & 9.09 Hz .. 157
4.31 Mode#2 (3-D) Diffracted Mode Shapes (z-) at f = 3.13, 5.26, 6.67 & 9.09 Hz .. 158
4.32 Mode#3 Diffracted Mode Shapes (x-) at f = 2.00, 2.78, 3.13 & 5.26 Hz .... 159
4.33 Mode#3 Diffracted Mode Shapes (z-) at f = 2.00, 2.78, 3.13 & 5.26 Hz … 160
4.34 Mode#4 Diffracted Mode Shapes (x-) at f = 2.00, 2.50, 4.55 & 9.09 Hz … 161
4.35 Mode#4 Diffracted Mode Shapes (z-) at f = 2.00, 2.50, 4.55 & 9.09 Hz … 162
4.36 Mode#5 Diffracted Mode Shapes (x-) at f = 2.00, 2.50, 3.33 & 4.17 Hz … 163
4.37 Mode#5 Diffracted Mode Shapes (z-) at f = 2.00, 2.50, 3.33 & 4.17 Hz … 164
5.1 Horizontal, out-of-plane, synthetic displacement at x=-2, 0 and 2km ……. 167
5.2 Response- and Fourier-amplitude spectra of surface motions at x = -2, -1, 1, and
2km ……………………………………………………………………….. 169
5.3 Contours of Fourier amplitude spectra (FS) in the area surrounding the
inhomogeneous layer ( 2x 2 km and 0depth 2 km) at 12 periods: T =
0.15, 0.20, 0.30, 0.40, 0.50, 0.75, 1.00, 1.50, 2.00, 4.00, 5.00, and 7.50 s … 170
5.4 Contours of pseudo-relative velocity spectra (PSV) in the area surrounding the
inhomogeneous layer ( 2x 2 km and 0depth 2 km), at 12 periods: T =
0.15, 0.20, 0.30, 0.40, 0.50, 0.75. 1.00, 1.50, 2.00, 4.00, 5.00, and 7.50 s - and for
5% of critical damping ………………………………………………….... 171
5.5 Horizontal, in-plane, synthetic displacement at x=-2.0, 0.0 and 2.0km ….. 174
5.6 Vertical, in-plane, synthetic displacement at x=-2.0, 0.0 and 2.0km …….. 174
5.7 Response- and Fourier-amplitude spectra – horizontal components …….... 176
5.8 Response- and Fourier-amplitude spectra – vertical components: T= 0.15, 0.20,
0.30, 0.40, 0.50, 0.75, 1.00, 1.50, 2.00, 4.00, 5.00, and 7.50 s, at the corresponding
frequencies: f = 6.67, 5.00, 3.33, 2.50, 2.00, 1.33, 1.00, 0.67, 0.50, 0.25, 0.20, and
0.133 Hz ………………………………………………………………….. 177
5.9 Fourier-spectral amplitudes – horizontal components ………………….… 178
5.10 Fourier-spectral amplitudes – vertical component ……………………….. 179
5.11 PSV-spectral amplitudes – horizontal component, for 5% damping ……... 180
5.12 PSV-spectral amplitudes – vertical component, for 5% damping ………... 181
ix
List of Tables
2.1 Free-field Amplitudes vs Incidence Angles …………………………...…… 41
3.1 Two-Layer Velocity Model ………………………………………………... 67
3.2 Amplification Factors at Various Frequencies …………………………….. 82
4.1 Two-Layer Velocity Model ………………………………………………. 123
5.1 Time-History Parameters …………………………………………………. 166
5.2 Time-History Parameters …………………………………………………. 173
1
Chapter 1
Introduction
1.1 General Introduction
Study of elastic wave propagation is a fundamental topic of understanding earthquake site
response. From a practical civil engineering point of view, a site for constructing building
structures is required to consider the earthquake ground motions in the United States based on the
building codes. Design of infrastructural structures, such as bridges, roadways, tunnels, dams,
power plants, pipelines and transmission towers is required different standards on seismic design
and regulated by various public agencies, such as Federal Highway Administration (FHWA), state
transportation departments, energy commissions and local governments. Since those
infrastructural structures are often located at difficult, irregular terrains in mountains, canyons, soft
marsh near rivers or oceans, understanding the seismic site response plays a very critical factor for
site selection and structural design during early stage of those billion-dollar projects. In general,
infrastructural projects require different degrees of site-specific ground motion analysis depending
on the budget and importance of the structures. Response spectrum generated based on the U.S.
Geological Survey (USGS) earthquake database is commonly used for seismic design of structures
located at a site without potential seismic hazard. However, a project site located at seismic hazard
zones involving soft soil, liquefaction and landslide, will require an in-depth site-specific response
analysis. In such case, surface topography and soil stratum become the key points for study
earthquake ground motions because they can amplify or de-amplify the propagated seismic waves
under the site.
The site-specific ground motion computer program, SHAKE (Schnabel, Lysmer & Seed,
1972), which is developed to evaluate response spectrum at a site with layered soil strata is
commonly used by engineers. In fact, SHAKE is based on a simplified one-dimensional wave
propagation theory assuming a plane incident shear wave travels vertically from a horizontally flat
bedrock situated underneath the site. However, earthquake epicenters are not usually located near
the site, and seismic waves travel further in horizontal distance than in vertical distance.
2
Considering travel paths, surface waves generated by wave traveling along the near surface stratum
contribute a good share of site response, however, are seldom discussed in a site response study.
For two-dimensional and three-dimensional site responses on an irregular layered stratum
are commonly evaluated by finding approximated results using Finite Element Method (FEM).
FEM requires well-defined mesh and boundaries to generate reasonable results, and requires large
amount of computational resources to implement the numerical data. Commercial FEM software,
such as FLAC (Fast Lagrangian Analysis of Continua) and PLAXIS (brand name of a geotechnical
company), is commonly used in the field of geotechnical earthquake engineering. Engineers are
always required to carefully review and justify the results computed from the software since errors
from the numerical inputs and iterations are not avoidable. Therefore, spectral response calculated
from the analytical methods provides a valuable baseline reference to validate the results from
numerical FEM approach.
We next turn to the study of elastic layered media in a half-space for both out-of-plane and
in-plane surface and body waves. We will study the translational and stress components of motions
at points on or below a half-space surface. We will first described the (out-of-plane) Love Surface
Waves and Body SH-Waves, followed by the (in-plane) Rayleigh Surface Waves and Body P-,
SV-Waves. We will go through the theory of the generation and propagation of these waves,
followed by a historical review of the numerical development for the computation of these waves.
1.2 Ground Response Near a Site with Irregular Topography – Homogeneous
Half-Space
Study of ground response using analytical method based on elastic wave propagation
theory requires deriving sophisticated formulations and understanding of geometry
mathematically. Researchers always started with a simplified geometry approximation to
overcome the mathematical obstacles. For example, Lee & Cao (1990) used two circular arcs to
model a two-dimensional cylindrical canyon and obtained the site response spectral amplifications
around a canyon. Lee & Cao’s study provided valuable amplification characteristics for analysis
on structures such as bridges and dams near or across a canyon in real world. Sanchez-Sesma
(1991) applied the boundary integral formulation for diffraction of P-, SV- and Rayleigh Waves
3
by topographic features. Kojic & Trifunac (1991) applied Lee & Cao’s model to simulate the
response of an arch dam site. Lee & Wu (1994) extended Lee & Cao’s model to a canyon with an
irregular topography to make the model more versatile. Huang & Chiu (1999) applied Lee & Wu’s
method to evaluate the topography effects with comparisons to the real ground motion records at
Feisui Dam in Taiwan. Via et al (1999) used the indirect boundary element method (IBEM) to
simulate the elastic wave propagation in two-dimensional irregularly layered medium. Davis et al
(2001) used an approximate method in their studies of the failure of underground pipes involving
transverse SV-Wave incidence in 1994 Northridge Earthquake. Liang et al (2000, 2001, 2002,
2003 and 2004) continued applying the Lee & Cao’s method various topographies, including
shallow canyons and alluvial valleys. Lin et al (2010) investigated amplification effects on a flat
site with an underground tunnel by using a new method involving integral forms of waves to obtain
the exact solutions without using Lee & Cao’s arc approximation for the flat ground surface.
In 2013, me and my advisor published a paper “Two-Dimensioned Scattering and
Diffraction of P- and SV-Waves Around a Semi-Circular Canyon in an Elastic Half-Space: An
Analytic Solution via a Stree-Free Wave Function” in a simplified mathematical formulation
consisting of Fourier half-range expansions, which does not require either Lee & Cao’s arc
approximation or complicated integration by Lin et al. (2010), to re-examine the cylindrical canyon
model. This new formulation automatically and analytically satisfies the boundary conditions at
the flat ground surface and reduces the number of equations in half. This is a challenging step
which, over the years, many researchers have tried to accomplish. In other words, the
computational efficiency for calculation is greatly improved, compared to the previous method.
The site amplification responses computed from the new analytic solutions are more precise and
can be applied to higher frequencies, compared to the results by Lee & Cao (1991). The main
content of this paper will be presented in Chapter 2 as the preparatory work to demonstrate the
feasibility of the Fourier half-range expression approach to further complicated irregular multi-
layered medium.
4
1.3 Ground Response On and Below a Site with Irregular Topography –
Elastic Layered Media
The above section deals with homogeneous elastic half-space. In the real world, the half-
space is hardly homogeneous anywhere. Seismologists and Strong-Motion Engineers often
assume the half-space to be a layered elastic media instead, where each layer is gradually harder
than the one above, with the topmost layer the softest. In this section, we will thus study waves
present in an elastic layered media.
This dissertation is a continuation of our sequence of reports on waves in elastic layered
and media, which we are finished. In particular, I participated in the preparation and production
of the following three reports, from among a sequence (M.D. Trifunac, W.Y. Liu and V.W. Lee;
reports):
Report I Synthetic Translational Motions of Surface Waves On or Below a Layered Media
Report II Diffraction Around an Irregular Layered Elastic Media, I: Love and Body SH
Waves - 1
Report III Diffraction Around an Irregular Layered Elastic Media, II: Rayleigh and Body P,
SV Waves - 2
which has now been summarized and appeared on two journal papers:
1) Lee, V.W., Liu, W.Y., Trifunac, M.D., Orbovi ć, N. Scattering and diffraction of
earthquake motions in irregular elastic layers, I: Love and SH waves, “Soil Dynamics
and Earthquake Engineering”, Volume 66, November 2014, Pages 125–134.
2) Lee, V.W., Liu, W.Y., Trifunac, M.D., Orbovi ć, N. Scattering and diffraction of
earthquake motions in irregular, elastic layers, II: Rayleigh and body P and SV
waves, “Soil Dynamics and Earthquake Engineering”, Volume 66, November 2014,
Pages 220–230.
Subsequent chapters of this thesis will include extracts from the contents of these reports
and corresponding papers that we are writing and finishing. We will first describe the existence
and generation of surface Love and Body SH-waves, followed by surface Rayleigh and Body P-,
5
SV-Waves. We will then do a literature survey on the existing numerical algorithms for the
computation and generation of these waves. Chapter 3 and 4 of the present thesis will then deal
with the more complicated cases when irregularities exist among the parallel layered elastic media.
Starting from Report II, we considered a typical N-layered half-space with Love Waves or
Body SH-Waves incident from the left (Figure 1.1 here, from Report II, Figure 2.1). Here, each
layer has two components of harmonic or surface wave propagating up and down the layer and
travelling to the right. The waves here all have motions in the y-direction, the so-called out-of-
plane motions. At the bottom layer, only one wave present, namely the surface waves. The waves
on the topmost layer will satisfy the zero-stress boundary condition at the half-space surface (z=0).
The waves (harmonic or surface) between each layer satisfy the continuity equations of stress and
displacements at the common interface. The stress to be used at each layer will be the shear stress.
Finally the surface wave at the bottom semi-infinite layer will be a wave that decays exponentially
in amplitude towards increasing depths. The theory governing the existence and computation of
these waves has long been studied and will be reviewed further below.
Figure 1.1 N-Layered Half-Space with Incident Love Waves
6
For the case of Body SH-Wave incident from the semi-infinite medium at the bottom, we
will consider the model in Figure 1.2 (from Report II: Figure 2.2). It is the same N-layered media
model as in Figure 1.1, for the case of Love Surface Waves; but now the waves in all layers are
body waves. Figure 1.2 is essentially the same as Figure 1.1, with the surface wave above in the
bottom semi-infinite layer now replaced by incident and reflected Body SH-Waves here. The
difference between Figure 1.1 and 1.2 is at the bottommost layer, the Body SH-Wave contents of
two components of surface wave which are propagating up and down instead the decaying one to
infinity shown in the Love Wave case from Figure 1.1.
Figure 1.2 N-Layered Half-Space with Incident Body SH-Waves
Similarly for the case of Rayleigh Surface Waves, which have both components of in-plane
horizontal and vertical motions; from Report III, we considered a typical N-layered half-space with
Rayleigh Waves or Body P-, SV-Waves incident from the left (Figure 1.3 here, from Report III,
Figure 3.1). Here, each layer has four components of harmonic or surface wave propagating up
and down the layer and travelling to the right. At the bottommost layer, there are two waves
present, both are the surface wave. The waves on the topmost layer will satisfy the zero-stress
7
boundary condition at the half-space surface (z=0). The waves (harmonic or surface) between
each layer satisfy the continuity equations of stress and displacements at the common interface.
Figure 1.3 N-Layered Half-Space with Incident Rayleigh Waves
For the case of Body P-, SV-Wave incident from the semi-infinite medium at the bottom,
we will consider the model in Figure 1.4 (from Report III: Figure 3.2). It is the same N-layered
media model as in Figure 1.3, for the case of Rayleigh Surface Waves, but now the waves in all
layers are body waves. Figure 1.4 is essentially the same as Figure 1.3, with the surface wave
above in the bottom semi-infinite layer now replaced by incident and reflected Body P-, SV-Waves
here. The semi-infinite layer will now be replaced by the incident and reflected Body P- and SV-
Waves respectively.
8
Figure 1.4 N-Layered Half-Space with Incident Body P, SV Waves
For the computation of surface waves, we need to compute the phase velocities at each
frequency. For body waves, we will need to know the incident and reflected wave amplitudes at
each layer. The rest of the sections will be a historical review of previous and exiting algorithm
for the models studied in Chapter 3 and 4 to follow. Here, I would like to introduce the history
and development about our new applied method.
The numerical implementation of the boundary-valued problem such as Love surface
waves, Body SH-Waves, Rayleigh surface waves and Body P-, SV-Waves to study the propagation
of elastic waves in a layered media was first presented and formulated in the pioneering work by
Thomson (1950) and Haskell (1953). Thomson (1950) first set up the theoretical groundwork to
be extended later by Haskell (1953).
The Thomson-Haskell propagator matrices are based on the use of matrix multiplication
in the frequency-wave number domain. In theory and in principle, it provides the tool for the
computation of harmonic elastic waves in a layered, elastic media. It has been used to calculate
9
the roots of the zero-determinant of the propagation matrices to determine the phase velocities
of the Rayleigh and Love surface waves. It can also be used to find the mode shapes of the
displacement responses of the layered media in the vertical direction. Since its development in
the 50’s, the study of wave propagation in a layered elastic media has generated much attention
and widespread applications.
In the 60’s and 70’s, high-speed computations using mainframe computers become
available. Computer programs are developed to perform the calculations of these matrices, as
reported by Dorman et al (1960) and Press et al (1961). For the case of Love and SH-Waves, the
matrices in each layer are of order two. For the cases of Rayleigh, P- and SV-Waves, these
matrices in each layer are of order four. For an N-layer media, these matrices are thus of order
2N for Love, SH (scalar) elastic waves and of order 4N for Rayleigh, P and SV (vector) waves.
When the number of layers N becomes moderately large, it is thus seen that these calculations
become cumbersome. Furthermore, it is found from experience then that when the wave
frequencies are large, the matrix components are very large and the exponential wave functions
approach both overflow and underflow ranges numerically, making it impossible to obtain the
roots and mode shapes accurately. Such numerical instabilities exist even at moderate wave
frequencies for larger layer thickness. This problem is more noticeable in layers where the phase
velocity is less than the wave speed, so that both waves that are exponentially increasing and
decreasing in the vertical z-direction exist in the layer. In addition, the problem is more significant
for vector (Rayleigh, P- and SV-) waves than scalar (Love and SH-) waves.
Knopoff (1964) reported such numerical problem, first calling attention to the long-
standing difficulty in the solution experienced by many workers and researchers in this area.
Knopoff proposed an alternate matrix formulation to try to avoid the aforementioned numerical
difficulties. Duncan (1965), at the same time, after reading a preprint of Knopoff’s paper,
proposed another matrix formulation similar to that of Knopoff (1964).
Here is a brief summary of various proposed modifications of the solution methods for
the direct Reflection-Transmission (R-T) matrix method and the Thomson-Haskell propagator
matrix method, to avoid the inherent numerical instability. This covers the period from the
10
1960s to 2010. The published methods are developed mostly to the troublesome Rayleigh, P and
SV vector waves, but the procedures can also be applied to the Love and SH scalar waves.
Knopoff (1964), Duncan (1965), and Thrower (1965) replaced the original Thomson-
Haskell propagator matrix with its second-order minor matrices. Both Thrower (1965) and
Randall (1967) reported success using the alternate formulation. Further modification of the
alternate method was proposed by Watson (1970). Later Kennet (1974) revisited the R-T matrix
method and derived a new recursive algorithm for generating the generalized reflection-
transmission coefficients of the wave potentials to avoid the numerical instabilities. Those were
subsequently updated in Kennet and Kerry (1979), Luco and Apsel (1983), Kennet (1983, 2001)
and Chapman (2003). Those are now referred to as the generalized R-T coefficient method.
Chapman and Phinney (1972) adopted the same Gram-Schmidt orthogonization inner-
product method for radio waves developed by Pitteway (1965) to elastic waves, which claimed
to improve the numerical stability of the propagator algorithm, under some imposed restrictions
in the propagation step. Wang (1999) also proposed such othogonization and normalization
procedures in the propagation step.
Recently, an efficient and elegant method which combines the second-order minor
method of Knopoff (1965) and others with the Langer block-diagonal decomposition has been
presented by Chapman (2003) in the paper titled, “Yet another elastic Plane-wave Layer-matrix
algorithm”. Ma et al (2012) just recently proposed another such method in formulating the
Thomson-Haskell propagator matrix, also using the orthonormalization and the Langer block-
diagonal decomposition methods.
With so many “efficient” and “elegant” methods for layered-media matrix calculations
presented, it would be tedious and time consuming to studied them, implement them to evaluate
which is the preferred method to be used. As pointed out by Ma et al (2012), the three methods
above, namely,
1) recursive reflection-transmission matrix method,
2) orthonormalization method,
3) minor matrix method,
11
and are, as stated in their paper: “… all somewhat related and solve the numerical instability in
the original Thomson-Haskell propagator matrix method equally well …” This makes a lot of
sense since they are all numerical method implemented to solve the same boundary-valued
problem with identical physics.
The numerical scheme used in this work is an improved modification of the Thomson-
Haskell transfer matrix method using the above methodology proposed very recently by Liu
(2010). The same method has successfully been generalized from layered elastic media to that of
layered azimuthally anisotropic media (Liu et al, 2012). Instead of directly multiplying the
Thomson-Haskell propagator matrix, this method defines for each layer an intermediate stiffness
matrix and state vector and performs the propagation through an intermediate step at each layer.
This modified scheme keeps the simplicity of the original propagator method with the intermediate
step able to efficiently avoid and exclude the exponential growth terms. Here’s a brief summary
of the scheme for Rayleigh waves as in Liu (2010). It is then modified to cover the case of
incident P, SV Body waves.
In this dissertation, we have used the method of Liu (2010) to generate the surface and
body waves of parallel elastic layers (Report I). Examples of such waves are in Chapter 3, Figures
3.1 to 3.3 for Mode#1 through Mode#3 of Love Waves, Figure 3.4 for incident SH Body Waves,
Chapter 4.1 to 4.3 for Mode#1 through Mode#3 of Rayleigh Waves, Figure 4.4 for incident Body
P-, SV-Waves. Description of these figures can be found at those chapters.
1.4 Existing Work on Irregular Multi-Layered Media
The use of elastic parallel layers in an elastic waves represents the classic theory and model
for the studies of seismic surface Love and Rayleigh Waves. For Strong-Motion Earthquake
Engineers, we use the waves to models as input the local surface and sub-surface topographies at
a specified site. These of course include surface canyons, valley and foundations, and sub-surface
cavities, tunnels, pipes etc. In the last few decades, it was realized that these laterally parallel layer
homogeneous model is still not realistic enough. In the real earth, these parallel layers are often
irregular in shapes. It does go back to the case of solving wave diffraction problems in irregular
topographies. As for the case of homogeneous half-space, analytic solution exists for only a very
12
limited number of cases. Lee and Liu (2013) did provide such need analytic solution in Chapter 2
of this thesis.
The researches of irregular shaped in multi-layered media due to various wave propagating
sources were developed constantly in the recent two decades. Chen (1990) applied the method of
seismogram synthesis for multi-layered media with irregular topography over a half-space incident
by an arbitrary source. Later, a creative method named “Global Generalized
Reflection/Transmission Matrices Method (GGRTM) was developed for two-dimensional, SH-
Wave (1990, 1995), two-dimensional P- and SV-Waves (1996) and in-depth analyzed to get an
alternative version for classic theory of Love Waves in laterally homogeneous multilayered media
(1999).
In the meantime, a boundary integral formulation was developed to analyze the two-
dimensional irregularly layered, elastic media problems by Sanchez-Sesma et al (1991); further
application for alluvial valleys incident by P-, SV- and Rayleigh Waves from Sanchez-Sesma
(1993) and a useful developed methodology so-called “Indirect Boundary Element Method”
(IBEM) to simulate the elastic wave propagation in two-dimensional irregularly layered media by
Vai et al (1999).
Chen (1990) stated that these non-analytic methods fall into three types, as in the case of
homogeneous in half-space:
1) Numerical methods, including finite difference, boundary integral methods,
2) High-frequency asymptotic methods which included asymptotic ray theory, Maslov and
Kirchhoff Method,
3) Plane-wave decomposition methods, which is the similar approach we used here, namely
to decompose the solution as expansion into a set of plane waves.
Figure 1.5 is a model where the parallel layered with “bump” at some region, where is flat
at most of other areas. We use the “Weighted Residual (Moment) Method” to analyze the
diffracted mode shapes of Love and Body SH-Waves. The results are indicated in Chapter 3.
Figure 1.6 is a model which is similar to Figure 1.5; except the incident wave as Rayleigh and
13
Body P-, SV-Waves instead of Love and Body SH-Waves. The same application in Moment
Method to analyze and have the results shown in Chapter 4.
Later, we applied the multiple coordinate system to generalize more complex layered
interfaces. The waves in all layers can be used to apply the boundary conditions at all interfaces.
Figure 1.7 is indicated the Love and Body SH-Waves, and Figure 1.8 is indicated the Rayleigh and
Body P-, SV-Waves. The results of interface in Canyon-Canyon case and Canyon-Hill case are
presented in Chapter 3 and 4.
14
Figure 1.5 Irregular-Shaped N-Layered Half-Space Incident with Love Waves
15
Figure 1.6 Irregular-Shaped N-Layered Half-Space Incident with Rayleigh Waves
16
Figure 1.7 Multiple Coordinate System Generalization Layers in Love Waves
17
Figure 1.8 Multiple Coordinate System Generalization Layers in Rayleigh Waves
18
1.5 Organization of Thesis
My thesis is to apply the Fourier half-range expression of elastic waves to re-verify the
classical wave propagation theory on reflection, diffraction and scattering due to the topographical
irregularities, and apply this method to solve complicated irregular multi-layered stratum, which
was not able to achieve analytically by previous formulation. This dissertation is organized into
the following topics:
Chapter 1 – Introduction: A general engineering importance of site spectral response and literature
review are presented.
Chapter 2 – Two-Dimensional Diffraction of P- and SV-Waves Around a Semi-Circular Canyon
in an Elastic Half-Space: This is a reproduction of the paper by Lee and Liu which has been
publication in 2014. The semi-circular canyon site response is to be re-investigated using Fourier
half-range expression to solve the exact solution and improve the precision of computation.
Chapter 3 – Diffraction Around an Irregular Layered Elastic Media – Love and Body SH Waves:
Site response at an irregular multi-layered stratum due to SH-Waves and surface waves (Love
Wave) is to be investigated by weighted-residual method along with Fourier half-range wave
expressions. Diffracted mode shapes and spectral amplification characteristics of Love Waves and
SH-Waves at different frequencies were examined and discussed in a published paper by Lee, Liu
et al in 2014.
Chapter 4 – Diffraction Around an Irregular Layered Elastic Media – Rayleigh and Body P-, SV-
Waves: Site response at an irregular multi-layered stratum due to Rayleigh and P-, SV-Waves is
to be investigated by weighted-residual method along with Fourier half-range wave expressions.
Diffracted mode shapes and spectral amplification characteristics of Rayleigh waves and P-, SV-
Waves at different frequencies were examined and discussed in a published paper by Lee, Liu et
al in 2014.
Chapter 5 – The synthetic scattered and diffracted time histories of Love and Rayleigh Waves are
summarized in this chapter.
19
Chapter 2
Two-Dimensional Diffraction of P and SV Waves
Around a Semi-Circular Canyon in an Elastic Half-
Space: Analytic Solution via Stress-Free Wave
Function
2.1 Introduction
A well-defined boundary-valued problem of wave diffraction in elastic half-space should
have closed-form analytic solutions. The two-dimensional (2-D) diffraction around a semi-
circular canyon in elastic half-space subjected to seismic plane and cylindrical waves has long
been a challenging boundary-value problem. The diffracted waves will, in all cases, consist of
both longitudinal (P-) and shear (S-) rotational waves. Together at the half-space surface, these
in-plane longitudinal P- and shear SV-Waves are not orthogonal over the infinite half-space flat
plane boundary. Thus, to simultaneously satisfy both the zero normal and shear stresses there,
some approximation of the geometry and/or wave functions often has to be made, or in some cases,
relaxed (disregarded). This chapter is to re-examines this two-dimensional (2-D) boundary-value
problem from the applied mathematics points of view, and to redefine the proper form of the
orthogonal cylindrical wave functions for both the longitudinal P- and shear SV-Waves, so that
they can together simultaneously satisfy the zero-stress boundary conditions at the half-space
surface. With the zero-stress boundary conditions satisfied at the half-space surface, the most
difficult part of the problem will be solved, and the remaining boundary conditions at the finite
canyon surface are then comparatively less complicated to solve.
2.2 Previous Work on Two-Dimensional Diffraction
Here we present a brief summary of the method used in our recent work Lin et al (2010).
Figure 2.1 shows a model of a semi-circular canyon in elastic half-space. The incident wave is
20
taken to be a longitudinal plane P-Wave represented by the potential function
i it
e
with
harmonic frequency . It has an angle of incidence
with respect to the x-axis and the
propagating vector in the xy plane (Figure 2.1).
The incident potential wave takes the form:
exp cos sin
i
ik x y
(2.1)
The presence of the half-space surface 0 y results in reflected plane P-Wave potential
r
with reflection angle
and reflected plane SV-Wave potential
r
with reflection angle
(both with respect to x axis):
1
2
exp cos sin
exp cos sin
r
r
Kikx y
Kikx y
(2.2a)
Figure 2.1 The model of two-dimensional semi-circular canyon
Here k
and k
are the P- and SV-Wave numbers of the waves with speeds
and .
1
and
2
are respectively the coefficients of the reflected P- and SV- plane waves,
where
22
12
22 22
sin 2 sin 2 cos 2
2sin 2 cos2
,
sin 2 sin 2 cos 2 sin 2 sin 2 cos 2
(2.2b)
21
Together all the incident and reflected plane waves form the input free-field P- and SV-
potentials:
1
exp cos sin exp cos sin
ff i r
ik x y K ik x y
(2.3a)
2
exp cos sin
ff r
Kikx y
(2.3b)
which can both be expanded in terms of cylindrical wave (Bessel) functions:
0
()(sin cos)
ff
nn n
n
J kr a n b n
(2.4a)
0
()(sin cos)
ff
nn n
n
J kr c n d n
(2.4b)
The free-field P- and SV- potentials,
ff
and
ff
, together will satisfy the zero-stress
boundary conditions at the half-space surface. The boundary condition at the flat half-space surface:
For all , ra
0
0
yyx
y
, or
0,
0
r
(2.5)
Following the form of free-field potentials in (2.4), we can try to take the scattered waves
to be of the form
(1)
0
(1)
0
() sin cos
() sin cos
s
nn n
n
s
nn n
n
H kr A n B n
H kr C n D n
(2.6)
If we are to calculate the displacement and stress components, we can first use the
following set of formulas, starting from, for m, n = 0, 1, 2:
(1) (1)
sin sin
cos cos
mn mn ss
mn m mn m
mn mn
AnCn
Hkr H kr
BnDn
(2.7)
22
one can get, for displacements:
(3) (3)
11 12
(3) (3)
21 22
sin cos
(, , ) + (, , )
cos sin
cos sin
(, , ) + (, , )
sin cos
mn mn
r
mn mn
mn mn
mn mn
AnCn
u m nk r m nk r
BnDn
AnCn
u m nk r m nk r
BnDn
DD
DD
(2.8)
and stresses:
(3) (3)
11 12
sin cos
(, , ) + (, , )
cos sin
mn mn
r
mn mn
AnCn
mnk r mnk r
BnDn
EE
(3) (3)
21 22
sin cos
(, , ) + (, , )
cos sin
mn mn
mn mn
AnCn
mnk r mnk r
BnDn
EE (2.9)
(3) (3)
41 42
cos sin
(, , ) + (, , )
sin cos
mn mn
r
mn mn
AnCn
mn k r mn k r
BnDn
EE
where
(3)
ij
D and
(3)
ij
E are the displacement and stress functions defined in much the same way as
in Pao and Mow (1973). Note that we allow the Hankel and trigonometric functions above to have
different orders of m and n . Note also that the sine and cosine functions are switched in the u
and
r
terms from the original wave potentials.
The boundary condition at the flat half-space surface:
For all , ra
0
0
yyx
y
, or
0,
0
r
(2.5) above would give:
(3) (3)
21 22
0,
(, , ) (, , ) sin
nn
nnk r A nnk r D n
EE
0
(3) (3)
21 22
0, 0
( , , ) + ( , , ) cos 0
n
nn
n
nnk r B nnk r C n
EE
(2.10a)
(3) (3)
41 42
0, 0
(, , ) + (, , ) sin
rn n
n
nnk r B n nk r C n
EE
(3) (3)
41 42
0, 0
( , , ) + ( , , ) cos 0
nn
n
nnk r A nnk r D n
EE
(2.10b)
23
Using the (2.10a) and (2.10b), it will be found that it is not possible, to apply the zero-
stress boundary conditions at the half-space surface of (2.5), and to solve for, and get the
coefficients of the cylindrical scattered waves in (2.6).
For the last thirty years, the following approximate procedure has been employed to
account for this difficult step of satisfying the zero-stress boundary conditions at the half-space
surface. The plane, flat half-space boundary was replaced by a concave surface with large radius,
Ra , the canyon radius, as in Figure 2.2. This then became the boundary-value problem with
zero-stress at two circular surfaces, the canyon and ”half-space” circular surfaces (Lee and Cao
1989; Cao and Lee, 1989, 1990; Todorovska and Lee, 1990, 1991a,b; Lee and Wu, 1994a,b; Liang
et al, 2000, 2001a,b, 2002, 2003a, 2006c)
Figure 2.2 Approximating the half-space boundary by a large convex circular surface
The same approximation of the half-space has been used for diffraction problems by sub-
surface inhomogeneities, as shown in Figures 2.3 and 2.4 for convex (Manoogian and Lee, 1996,
1999; Liang et al, 2003b,c, 2004a,b,c, 2008) and concave (Davis et al, 2001) surfaces
24
Figure 2.3 Approximating the half-space by convex surfaces
Figure 2.4 Approximating the half-space by concave surfaces
25
Such approximations of the flat surface were studied by Lee and Cao (1989), and Cao and
Lee (1989, 1990). They presented numerical solutions to the diffraction problems of surface
circular canyons of various depths for incident plane P- and SV-Waves. Since then, such
approximations of the flat half space by a large circular surface were studied by us and by our
collegues. Todorovska and Lee (1990, 1991a,b) used the same method for anti-plane SH Waves
and for incident Rayleigh Waves on circular canyons. Lee and Karl (1993a,b) extended the method
to diffraction of P- and SV-Waves by circular underground cavities. Lee and Wu (1994a,b) used
the same method for arbitrary-shaped two-dimensional canyons. Davis et al (2001) used this
method in their studies of the failure of underground pipes involving transverse SV-Wave
incidence. Liang et al (2001a,b, 2002, 2003a,b,c,d, 2004a,b, 2005, 2006a,b,c,d, 2007a,b, 2008)
continued the analyses by the same method, for problems involving circular-arc canyons and
valleys, and underground pipes in elastic and poro-elastic half-space.
Recently, this approximate method has met with criticism. It was noted that the large circular
approximation does not converge quickly to that of the flat half-space when the radius, R, of the
circular surface approaches infinity, because as R , the Bessel Functions used in the
transformation approach zero. It was also suggested that without an accurate method to satisfy the
free-stress boundary conditions at the half-space surface, it might just be better to relax the free-
stress conditions than to use the large circle approximations (Todorovska and Al Rjoub, 2006).
Many other computational and numerical methods have also been developed for elastic wave
diffraction problems in the half-space.
1) Numerical methods such as finite difference, finite element and boundary element
methods, and
2) Analytical methods such as the method of wave function expansion, perturbation
method, the method of integral equations and integral transform.
In dealing with most practical problems, which involve arbitrary-shaped obstacles and
material nonlinearities, numerical methods offer more general approach to the solutions of such
problems. In contrast, analytical methods provide closed-form solutions with better accuracy and
relatively simpler numerical implementation. However the limitation of analytical schemes is that
those can only treat linearly-elastic or visco-elastic media with simple geometries. Therefore only
26
a limited set of problems have been solved analytically. In spite of this, analytic solutions still
offer the best benchmarks to test and to verify other approximate solutions. We will use an analytic
solution scheme in this study.
2.3 Analytic Solution of the Wave Equations with Zero-Stress on the Half-
Space Surface
Consider first the scattered P-Wave potential (, )
it
r e
in 2-D cylindrical (polar)
coordinates. As before, we write down the general solution of the waves of the form:
(1)
0
(1)
0
(1)
0
either ( )sin( ),
or ( )cos( ),
or (both) ( ) sin( ) cos( )
nn
n
nn
n
nn n
n
A H k r n
B H k r n
H k r A n B n
(2.11)
where kk
for P-Waves, and kk
for SV-Waves. Here it is either a sine or cosine series or
one of both. In full-space problems, where 02 , one would normally include sine and
cosine terms in the solutions because they together form an orthogonal complete set of functions.
But in the half-space, where 0 , only the sin n or the cos n functions are by
themselves orthogonal. In other words, on the half-space, the sine and cosine functions are not
independent. In fact, as pointed out by Lee et al (2006), one can express a given sine function in
terms of the cosine functions in a half-space range. For m=1,2,3…
0
2 2
odd
0
odd
0
2 2
odd
22
0
cos
sin
cos
sin
2
0
cos
2
0
2
odd
with cos
0
n
n
nm
n
n
mn
nm
n
n
nm
mn
n
m
n
m
m
mn
sn
m
mn
m
nm
snd mn
0
1
2 for n > 0
even
n
nm
(2.12)
27
Figure 2.5 Sine versus Cosine Expansion
Figure 2.5 (from Lee et al, 2006) is one such typical plot of the expansion of sin( ) m for
m=10 in terms of cosine functions in the range , . The solid line is the actual sine function.
The calculated values using the cosine series summation in (2.22) are plotted as ‘*’ in the figure.
They agree well, even among the critical end points 0, , the half-space surface where the
stresses are to be calculated. The RMS of the difference of the actual and calculated values is
0.00205.
RMS
This is indeed knowledge from a common subject in 1st year graduate level
Engineering Mathematics on Fourier series, where a function, () f , defined in a half-range [0, ] ,
can both be expressed as a sine series and a cosine series, obtained respectively by the odd and
even extensions of the function to the full range [, ] . In our case, it is the “odd” sine function
being given an “even” extension and expressed as a cosine series.
Note that the corresponding expansion of the “even” cosine function in terms of the “odd”
sine series is not as numerically desirable as the “odd” extension of the cosine function would
28
result in the function being discontinuous at 0 , since with an odd extension, the right and left
limits are different: cos 0 1 and cos 0 1
(the case of odd extension).
A complete independent solution in the half-space should thus not include both sines and
cosines, but must select only one of the two. We will proceed now by taking one of the two forms
for each of the P- and SV-potentials. Start with the scattered potentials both as a sine series:
(1)
1
(1)
1
()sin
()sin
s
nn
n
s
nn
n
AHkr n
CH k r n
(2.13)
Note that we change the summation to be from 1 to n , since
00
0 AC .
The next step is to check how these wave functions are to satisfy the zero-stresses at the
half-space surface. Using (2.12), the wave potentials in (2.13) can be expressed in terms of the
cosine functions:
(1) (1)
0
odd
11
(1) (1)
0
odd
11
() sin ( ) cos
() sin ( ) cos
s n
n
mm mm mn
nm
mm
s n
n
mm mm mn
nm
mm
HkrA m H krA s n
HkrC m H krC s n
(2.14)
or
(1) (1)
0
1 1, odd
(1) (1)
0
11, odd
() sin ( ) cos
() sin ( ) cos
s n
mm m mnm
n
mmmn
s n
mm m mnm
n
mmmn
HkrA m H kr sA n
HkrC m H kr sC n
(2.15)
Note that the cosine summation here starts from 0 to n . The (2.15) is essentially the
same as taking the P and SV potentials each as a cosine series. In other words, (2.13) and (2.15)
thus show that the scattered wave potentials in the half-space can be expressed either as a sine
series or as a cosine series.
29
The P-and SV-Wave potential functions as represented by (2.13) and (2.15) can now be y
shown to be the wave functions that implicitly satisfy the zero-stress boundary conditions on the
half-space surface, namely, (2.5):
1) Zero Normal Stress on the Half-Space Surface:
0,
0
0
y
y
Using (2.13) and (2.15), write the P-Wave potential as a sine series, and the SV-Wave
potential as a cosine series:
(1)
1
() sin
s
nn
n
H kr A n
from (2.13)
(1)
1
0
odd
() cos
s n
m
mmnm
n
mn
Hkr sB n
from (2.15)
then the normal stress will take the form, from (2.7):
(3) (3)
1
21 22
odd
1
(, , ) ( , , ) sin
n
m
nmnm
mn
n
nn k r A m nk r s C n
EE (2.16)
and thus for all , ra
0,
0
0
y
y
is satisfied.
2) Zero Shear Stress on the Half-Space Surface:
0,
0
0
yx r
y
Write the P-Wave potential as a cosine series, and the SV-Wave potential as a sine series:
(1)
1
0
odd
() cos
s n
m
mmnm
n
mn
Hkr s A n
from (2.15)
(1)
0
() sin
s
nn
n
H kr C n
from (2.13)
then the normal stress will take the form, from (2.7):
(3) (3)
1
41 42
odd
1
(, , ) (, , ) sin
m
rmmn n
mn
n
mnk r A s nnk r C n
EE (2.17)
and thus for all
, ra
0,
0
0
yx r
y
is again satisfied.
30
The scattered waves
s
and
s
of (2.13) are thus a complete set of cylindrical wave
functions that each by themselves and together both satisfy the zero-stress boundary conditions of
(2.7) at the half-space surface.
2.4 The Canyon Zero-Stress Boundary Conditions
With the zero-stress boundary conditions at the half-space surface taken care of, we now
turn to the boundary conditions at the surface of the canyon:
For 0 , at ra
0
r
ra
r
(2.18)
The P- and SV- Wave potentials will now include both the free-field waves and scattered
waves. Together they will satisfy the zero-stress boundary conditions at the surface of the canyon:
ffs
ffs
(2.19a)
so that
0
0
ff s
rr r
ra
ra
ff s
rr r
ra
ra
(2.19b)
1) The Zero Normal Stress Boundary Condition on the Canyon Surface:
0
r
ra
The free-field P- and SV- Wave potentials together will have normal stress given by
(1)
11
0
(1)
12
(, , ) sin cos
( , , ) sin cos
ff
rnn
n
ra
nn
nnk a a n b n
nnk a c n d n
E
E
(2.20)
starting with the P- and SV- scattered wave potentials both as sine series from (2.13), the normal
stress from the scattered waves takes the form
(3) (3)
11 12
0
(, , ) sin + (, , ) cos
s
rn n
n
ra
nnk a A n nnk a C n
EE
(2.21)
31
the stress-free boundary conditions at the canyon surface , ra , gives, for 0 :
(3) (3)
11 12
0
(1)
11
0
(1)
12
or
(, , ) sin + (, , ) cos
( , , ) sin cos
( , , ) sin cos
sff
rr
ra ra
mm
m
mm
m
mm
m
mmk a A m m mk a C m
mmk a a m b m
mmk a c m d m
EE
E
E
0
(2.22)
Note that the summation on the RHS is from 1 to n , since
00
0 ac .
For 1,2,3 n multiply both sides of (2.22) by sin n
and integrate both sides along
from 0 to . The orthogonality of the sin n
functions in the half space [0, ]
gives,
for 1, 2, 3 n
(3) (3)
11 12
1
( , , ) + ( , , ) ( )
2
ff
nnmm
m
nnk a A mmk a s C n
EE s
(2.23)
where ()
ff
n s is the
th
n term of the normal stress from the free-field waves, given by
(1) (1)
11 12
(1) (1)
0
11 12
odd
( ) (, , ) (, , )
2
( , , ) ( , , )
ff
nn
m
mmnm
mn
nnnkaannkac
mmk a b mmk a d s
sE E
EE
(2.24)
in exactly the same way,
2) The Zero Shear Stress Boundary Condition on the Canyon Surface:
0
r
ra
:
The free-field P- and SV- wave potentials together will have shear stress given by
(1)
41
0
(1)
42
(, , ) cos sin
( , , ) cos sin
ff
rnn
n
ra
nn
nnk a a n b n
nnk a c n d n
E
E
(2.25)
again from (2.13), the shear stress from the scattered waves takes the form
(3) (3)
41 42
0
(, , ) cos + (, , ) sin
s
rn n
n
ra
nnk a A n nnk aC n
EE (2.26)
32
the stress-free boundary conditions at the canyon surface, ra , gives, for 0 :
(3) (3)
41 42
1
(1)
41
1
(1)
42
or
(, , ) cos + (, , ) sin
( , , ) cos sin
( , , ) cos sin
sff
rr
ra ra
mm
m
mm
m
mm
mmk a A m mmk a C m
mmk a a m b m
mmk a c m d m
EE
E
E
1 m
(2.27)
Note that the summation on both sides of the equation is from 1 to m . As in the case
of (2.22) for normal stress, for 1,2,3 n multiply both sides of (2.17) by sin n
and integrate
both sides along from 0 to . Again the orthogonality of the sin n
functions in
the half space [0, ]
gives, for 1, 2, 3 n
(3) (3)
41 42
1
( , , ) + ( , , ) ( )
2
ff
nm m n
m
mmk a s A nnk a C n
EE t (2.28)
where ()
ff
n t is the
th
n term of the shear stress from the free-field waves, given by
(1) (1)
41 42
(1) (1)
41 42
1
( ) ( , , ) ( , , )
2
( , , ) ( , , )
ff
nn
mmnm
m
nnnkabnnkad
mmk a a mmk ac s
tE +E
EE
(2.29)
00
(0) ad . (2.23) for normal stress and (2.28) for shear stress constitute two sets of system of
equations of infinite order for the two sets of unknowns
n
A and
n
C .
Alternately, (2.23) and (2.28) can be rearranged to result in two sets of equations, one just
for
n
A and another one for
n
C , separately as follows.
From (2.28), one can write each
n
C for 1,2,3 n in terms of
m
A :
(3) (3)
1
42 41
odd
( ) ( ) ( )
2
ff
m
nnmm
mn
nC n m s A
Et E
(2.30)
33
with
(3)
41
(, , ) mmk a
E and
(3)
42
(, , ) nn k a
E respectively abbreviated as
(3)
41
() m E and
(3)
42
() n E in (2.30)
above. Rewrite the above equation for 1,2,3 m as:
(3) 1 (3)
1
42 41
odd
2
( ) ( ) ( )
ff
l
m ml l
lm
Cmm lsA
Et E
(2.31)
and substitute the expression into (2.28) to become for 1,2,3 n
(3)
11
(3) (3) 1 (3)
10
12 42 41
odd odd
( , , ) +
2
2
( ) ( ) ( ) ( ) ( )
n
ff ff
ml
nm ml l
mn l m
nn k a A
ms m m l s A n
E
EE t E s
(2.32)
with
(3)
12
(, , ) mmk a
E abbreviated as
(3)
12
() m E here. Rewrite (2.32) for 1,2,3 n as:
(3) (3)
1
11 41
even
(3) (3) 1
1
12 42
odd
2
( ) + ( )
2
2
( ) ( ) ( ) ( )
l
nnll
ln
ff ff
m
nm
mn
nA l A
nsmmm
EE
sEEt
(2.33a)
or
2
(3) (3)
1
11 41
even
(3) (3) 1
1
12 42
odd
( ) + ( )
2
( ) ( ) ( ) ( )
2
l
nnll
ln
ff ff
m
nm
mn
nA l A
nsmmm
EE
sEEt
(2.33b)
where
(3) (3) 1
1
12 42
odd
( ) ( ) even
0 odd
m
nm ml
ml
nl
ss E mE m n l
nl
(2.34)
for , 1,2,3 nl . (2.33a) and (2.33b) constitutes two sets of infinite equations, one for
13 5
,, , AA A of odd indices and one for
24 6
,, , AAA of even indices.
34
In exactly the same way, one can eliminate
n
A from (2.23) and (2.28) to get a system of
equations for
n
C . The result is. for 1,2,3 n :
(3) (3)
1
42 12
even
(3) (3) 1
1
41 11
odd
2
( ) ( )
2
2
( ) ( ) ( ) ( )
l
nnll
ln
ff ff
m
nm
mn
nC l C
nmmsm
EE
tEEs
(2.35a)
or
2
(3) (3)
1
42 12
even
(3) (3) 1
1
41 11
odd
( ) ( )
2
( ) ( ) ( ) ( )
2
l
nnll
ln
ff ff
m
nm
mn
nC l C
nmmsm
EE
tEEs
(2.35b)
where
(3) (3) 1
1
41 11
odd
( ) ( ) even
0 odd
m
nm ml
ml
nl
ss E mE m n l
nl
(2.36)
for , 1,2,3 nl . (2.35a) and (2.35b) constitutes two sets of infinite equations, one for
135
,,, CC C of odd indices and one for
24 6
,, , CCC of even indices.
2.5 Numerical Implementation: Incidence Angle: 60
In this section we will discuss the numerical implementation to solve for the coefficients
of the P- and SV-Wave potentials given in (2.21), using (2.41a) and (2.41b) for the { }
n
A
coefficients of the P- potentials and (2.46a) and (2.46b) for the { }
n
C coefficients of the SV-
potentials. Studying the (2.41a), (2.41b) and (2.46a), (2.46b) shows that we now have a much
simpler set of equations to solve compared to all previous work in trying to solve the same
boundary-valued problem.
Recall from all our previous work and from the present work that there are a total of four
sets of boundary conditions to be satisfied, namely, the zero-stress boundary conditions at the half-
space surface, zero normal and zero shear stresses: 0at 0
yyx
y , and the zero-stress
boundary conditions at the canyon surface, again zero normal and zero shear stresses in cylindrical
coordinates: 0at .
rr
ra
What was attempted in all previous work was to have two sets
of P- potential waves and two sets of SV- potential waves, each with coefficients for both sines
35
and cosines, with one set diffracted from the canyon and one set diffracted from the half-space
surfaces. This resulted in four sets of coefficients for the sine functions and another four sets of
coefficients for the cosine functions, or a total of eight sets of coefficients. Further, up till now, no
analytic equations were derived or found for the zero-stress boundary conditions at the half-space
surface that are solvable, since the P- and SV-Wave potential functions together are not orthogonal
to each other. Thus up to now, as already noted many previous works replaced the flat half-space
surface by a convex or concave surface of large radius to approximate the half-space. In the end,
a set of matrix equations involving the four sets of coefficients of the sine functions had to be
solved, together with another four sets for the cosine functions.
With the present approach, we now have an analytic set of wave functions written in a form
that do satisfy exactly the zero-stress boundary conditions at the half-space surface (Section 2.3)
and we need to have only one set of wave function coefficients for the P-Waves and one set for
the SV-Waves, instead of eight sets as before. Further, the solution to these wave coefficients are
found by applying the boundary conditions only at the canyon surface: (2.33a), (2.33b) and (2.35a),
(2.35b). One more simplifying step is that the equations have the P-Wave and SV- Wave
coefficients, namely the
n
A and
n
C respectively, decoupled from each other, so that they can
be solved independently.
One fact still remains, as (2.33a), (2.33b) and (2.35a), (2.35b) show, they are still, just as
in all previous work, a system of complex equations of infinite order, at each frequency. The
technique for solving a system of complex equations numerically remains the same. Since our
computers, mainframes before and PCs now, all have finite memory, we can only solve for systems
of complex (or real) equations of finite orders. So the only way to solve a theoretical system of
complex equations of infinite order by PC now is to truncate it to a finite order. Depending on the
frequency of the waves, we will have to take the finite truncated order N to be large enough so that
the results for N, N+1, N+2,… terms will remain the same numerically.
Take the case of plane P-Waves incident on the semi-circular canyon with an incident angle
of 60
with respect to the horizontal. At each frequency, we solve (2.33a), (2.33b) and
(2.35a), (2.35b) of finite order N, starting from some small value, in increment of 2 until the
36
solution converges. With the coefficients
n
A and
n
C of the diffracted wave potentials
s
and
s
computed, the resultant, total P- and SV-Wave potentials are found.
Following all previous work, knowing the P- and SV- potentials, one can use them to
compute all types of displacement, rotation, strain and stress of all components of motions. The
displacement components, computed here in cylindrical coordinates, are related to the wave
potentials by the following familiar equations, as given in all our previous work (Lee and Cao
1989; Cao and Lee, 1989, 1990; Todorovska and Lee, 1990, 1991a,b; Lee and Wu, 1994a,b; Liang
et al, 2000, 2001a,b, 2002, 2003a, 2006c):
1
Radial :
1
Angular :
r
U
rr
U
rr
(2.37)
(2.37) can also be transformed to rectangular components ,
x y
U U by
cos sin
sin cos
x r
y
U U
U U
(2.38)
We next, as before, computed the displacement amplitudes as:
22
22
Real( ) Imag( )
Real() Imag()
xx x
yy y
UU U
UU U
(2.39)
with Real(.) and Imag(.) respectively the real and imaginary part of the complex argument, as was
done in all above references. We next, as in previous work, using the notation of Trifunac (1973),
define the following dimensionless frequency:
22
ka
afa a
(2.40)
where the parameters ,, a are defined as before, 2 f is the cyclic frequency in Hz and
is the wave length of the shear waves at frequency , shear wave speed and wave number
k
.
In what follows, Figure 2.6 and Figure 2.7 are two-dimensional (2-D) plots of the
horizontal x- and vertical y- components of displacement amplitudes, plotted versus the
37
(dimensionless) horizontal distance x/a, from 3.0 to 3.0 at the half-space surface, where all the
significant diffraction occurs, all for the same angle of incidence of P-wave at 60 with respect
to the horizontal direction. In the range of x/a, from 1.0 to 1.0 , the displacement amplitudes
are plotted along the canyon surface instead. The two figures are respectively for displacement
amplitudes at two dimensionless frequencies of 10 and 20. (2.40) shows that, for example,
given a canyon of radius 1km a , a half-space with shear wave speed of 2 km s , say, this
would correspond to elastic waves at given cyclic frequencies respectively also of
10 and 20 Hz. f In terms of wavelengths, they would correspond to shear wavelengths
respectively 110 and 1 20 of the canyon’s diameter.
The first observation from the figures is that the plots of displacement amplitudes here are
all in a frequency range much higher than those calculated in all previous papers. This can be
attributed to the fact that the new, simple form of wave functions used in this paper automatically
satisfy the zero-stress boundary conditions on the half-space surface, resulting in a formulation
that is much simpler than before. Before, the wave functions used, because of them being non-
orthogonal, as stated in the earlier paragraph, required the introduction of additional waves, and
the use of large circular surfaces to approximate the half-space, and the resulting matrix equations
are much larger and more tedious to solve numerically. In one of our earlier paper on “Scattering
and Diffraction of Plane P Waves by Circular Cylindrical Canyons with Variable Depth-to-Width
Ratio” by Cao and Lee (1990) the highest dimensionless frequency presented for the amplitudes
was only up to 2 , much less than, in fact just 10
th
that the highest dimensionless frequency
presented here at 20 . In our more recent paper on “Diffraction of SV Waves by a Shallow
Circular-arc Canyon in a Saturated Poro-elastic Half-Space,” by Liang et al (2006c), the highest
dimensionless frequency presented was 5 , but still less than what can be done here.
Since the original matrix equations (2.41b), (2.46b) are infinite in order, we have to
truncate them to a finite order large enough to obtain convergent solutions. Take Figure 2.6, say,
the case of 10 . Here the displacement amplitudes are plotted for matrix equations of order
form 104 112 N in steps of 2. Figure 2.7 for the case of 20 are similarly for orders of
136 144 N . The resulting graphs show that the solutions are convergent at those orders of N.
38
In the figures, with a the canyon radius, the displacement amplitudes are plotted along the
horizontal distance from 5 xa to 5 xa . The range 1 xa on both sides of the canyon
are the distances on the half-space flat surfaces measured from the center of the semi-circular
canyon. The distances from 1 xa to 5 xa are to the left in the front side of the canyon
where the waves first arrived. 1 xa is thus the left rim of the canyon. The distances from
1 xa to 5 xa are to the right and in the back side of the canyon, and 1 xa is the right
rim of the canyon. The amplitudes plotted in the range 11 xa , are amplitudes on the
circular surface of the canyon.
Each graph in the two figures can be described in three parts; the front side, the back side
of half-space and the circular surface of the canyon. The x-component will have displacement
amplitudes that oscillate about the free-field amplitudes on both the front and back sides of the
canyon. At most frequencies, it is observed that there is a “dip and spike” in amplitude at both
rims: 1 xa . This is the corner point phenomena observed previously also for SH Waves
around circular canyon (Trifunac, 1973). The y-components will have amplitudes that oscillate
on the front side, have a spike at 1 xa , and exhibit a more shadowy behavior on the back side.
Both components are highly oscillatory on the canyon surface.
39
Figure 2.6 Displacement amplitudes on or around canyon (for 60 ,
10 )
40
Figure 2.7 Displacement amplitudes on or around canyon (for 60 ,
20 )
41
2.6 Results for Other Incidence Angles: 5 , 30 and 90
The calculations and plots in the previous section are repeated here for three other angles
of incidence: 5 , 30 and 90 , all with respect to the horizontal. 5 is the case of almost
horizontal incidence. As in all previous work, we cannot have the case of total P-Wave horizontal
incidence for 0 , as this will result in zero-amplitude free-field waves, and there will be no
scattered waves either. The case of P-Wave incidence at 30 is the case of oblique incidence
as that in the previous section. Finally the case of P-Wave incidence at 90 is the case of
vertical incidence.
Figure 2.8 shows plane P-Wave incidence at 5 for dimensionless frequency 20 .
Figure 2.9 is for of P-Waves at 30 and for 20 . Finally Figure 2.10 shows P-Waves with
vertical incidence 90
f and for 20 . Table 2.1 gives the free-field displacement
amplitudes of the horizontal and vertical components of motions for the four angles of incidence:
Figure
P-Wave Incident
Angles
Free-field Amplitudes
x-component y-component
8
5
0.97 0.35
9
30
1.73 1.00
6 and 7
60
1.12 1.69
10
90
0.00 2.00
Table 2.1 Free-field Amplitudes vs Incidence Angles
Note that the third row in Table 2.1 corresponds to the two figures in the previous section.
For the case of almost horizontal incidence, 5 , Table 2.1 and Figures 2.8 show that the free-
field amplitudes of both the x- and y- components are fairly small (at 0.97 and 0.35 respectively),
and the total displacement amplitudes in the front side (at 1 xa ) of the canyon are both
oscillatory about these free-field amplitudes. Again, as is the case in the previous section, the x-
component amplitudes are less oscillatory and the y-component amplitudes exhibit shadow
42
behavior at the back side of the half-space behind the canyon. What is noticeable is the highly
oscillatory nature of both component amplitudes at the canyon surface, with high amplification
factors. The same “dips and spikes” at both rims ( 1 xa ) are again observed as both figures.
These “spike” displacement amplitudes were not obvious in all previous work on P-Wave
incidences. This is because in all previous work, we were not able to get results at such high
dimensionless frequencies (to 20 here). Before, calculations were at lower frequencies, which
tend to smooth out these corner points. Corner points in wave propagation theory are often known
to create secondary wave sources. Further, it is observed that at the back rim of the canyon, the y-
component amplitude will exhibit this spike motions, and immediately dip to almost zero
amplitudes. Physically this is the phenomenon of the wave motion having a change from positive
(+ve) phase through zero phase to negative (–ve) phase (or vice-versa) in motions at those points
and the associated torsional motions.
The same observations and conclusions of Figure 2.8 can be made also for Figure 2.9, for
the case of P-Wave incidence at 30 . We observe the same ‘dip and spike’ and oscillatory
behavior. The only difference being at such oblique incidence, the free-field amplitudes of the x-
and y- components are now higher (1.73 and 1.00 respectively). It is best to compare Figure 2.9
to Figure 2.7 of the previous section at the same dimensionless frequencies ( 20 ) for P-Waves
incident at 60 . It is observed that Figure 2.9 for 30
exhibits higher amplification than
the corresponding Figure 2.7 for 60 at the same frequency. At the end of this section, after
studying all the figures, we will be able to conclude that as the incident angle increases from almost
horizontal at 5 to almost vertical at 90 , it is observed that the amplification amplitudes
factor reduces, especially at the rim of the canyon, where the dips and spikes of displacement can
occur. The case of vertical incidence ( 90 ) below is different, as it has zero horizontal free-
field motions (only vertical free-field motions) at the half-space surface.
Next for the case of vertical incidence ( 90 ) in Figure 2.10, the x- and y- component
free-field amplitudes are respectively 0 and 2.0. The case of 20 (Figure 2.10) is interesting. It
shows that all the motions just died down, exhibiting a much smaller oscillatory and amplification
behavior. It may be speculated that at that particular frequency and vertical incidence angle, there
is less diffraction, as the canyon looks more “flat” to the incident waves.
43
Figure 2.8 Displacement amplitudes on canyon (for 5 ,
20 )
44
Figure 2.9 Displacement amplitudes on canyon (for 30 ,
20 )
45
Figure 2.10 Displacement amplitudes on or around the semi-circular canyon
(for 90 ,
20 ,
N = 136-144)
46
2.7 Summary
Based on the plots and observations from Figure 2.6 to 2.10, our findings can be
summarized in terms of the following points. Some Points are not much different from previous
work. But some are new observations, as we can now do calculations at much higher frequencies.
1.
For the first time, being different from all previous work, using Fourier half-range
expansions, this paper presents the cylindrical wave functions that automatically
satisfy the zero-stress boundary conditions at the half-space surface, the so-called
stress-free wave functions. As a result the matrix equations are much simpler and
the results can be obtained at much higher frequencies.
2.
Previously in Cao and Lee, 1990, it was stated that the amplification of surface
displacement amplitudes on or near the canyon and half-space surface can be high,
but it was typically observed to be less than two times the free-field motions. In
the present work, as seen from Figures 2.6 to 2.10, the displacement amplitudes at
the canyon surface can be three to four times those of the free-field motions.
3.
Cao and Lee (1990) observed that “…Unlike for SH Waves, where a shadow zone
is always observed behind the canyon, the displacement amplitudes behind the
canyon for incident P Waves can be higher than those of the free-filed or even
those in front of the canyon….”. Based on the figures in this paper, this is partially
still true, but we can now elaborate further in the following points 5), 6) 7), 8) and
9) to follow.
4.
As stated above, on the side of the half-space in front of the canyon, the total
displacement amplitudes of both the x- and y- components at frequencies of 5
and beyond, are oscillatory around the free-field amplitudes.
5.
On the back side of the half-space, behind the canyon, the x-component
amplitudes, unlike for SH Waves, are still oscillatory, but the y-component
amplitudes do exhibit shadowy behavior, like for the SH Waves.
47
6.
Both the x- and y- components of displacement amplitudes are highly oscillatory
at the canyon surface, and the amplification factor can be as high as four above
that of the free-field motions.
7.
Spikes in displacement amplitudes are observed at both rims ( 1 xa ) in most
figures. These “spike” displacement amplitudes were not observed in our previous
work because we were not able then to get results at such high dimensionless
frequencies ( 5 ). Before, calculations were at lower frequencies, which tend to
overlook these corner points. Corner points in wave propagation theory are often
known to create secondary wave sources.
8.
As observed in the previous work, at the back rim of the canyon, the y-component
amplitude in most cases will exhibit the “dip and spike” motions, and immediately
decay to almost zero amplitudes. Physically this is the interference which leads to
standing waves having a change from +ve phase to –ve phase (or vice-versa) in
motions at the point, and corresponds to almost pure torsional or rocking motions
at these points.
9.
In Cao and Lee, 1990, it was stated that the incident angles of P-Waves affect the
pattern of surface displacements. This is still true here. As the incident angle
increases from almost horizontal at 5 to almost vertical at 90 , the
amplification amplitudes and factors reduce, especially at the rim of the canyon,
where spikes of displacement can occur. The case of vertical incidence ( 90 )
above has to be singled out, as it has zero horizontal free-field motions (only
vertical free-field motions) at the half-space surface.
10.
We observe that at some frequencies, there could be little or negligible diffraction,
like the case of normal incidence at 20 . The amplification factors from
diffraction are definitely not a linear function of frequency, and do not
automatically increase with increasing frequencies. The amplification factors,
which describe the degree of diffraction, can have peaks and valleys as functions
of frequencies.
48
11.
Finally, Cao and Lee (1990) stated that: “…The dimensionless frequency plays
an important role in determining the displacement patterns. Larger values of
will result in higher complexity of displacements and in higher amplifications….”.
We were able to extend these conclusions from 2 to now 20 and confirm
that the same conclusion hold now, as well.
12.
The concept of half-range expansion introduced in this paper resulted in the
scattered waves satisfying the zero-stress boundary conditions. This approach can
next be extended to other types of elastic plane P-, and SV- Waves, and also for
surface Rayleigh Waves in an elastic homogeneous half-space. The same concept
can also be used for waves incident upon poro-elastic half-space.
13.
The same concept of half-range expansion can also be extended to three-
dimensional elastic wave propagation involving a hemispherical canyon in an
elastic half-space (Zhu and Lee, 2013).
49
Chapter 3
Diffraction Around an Irregular Layered Elastic
Media, I: Love and Body SH Waves
3.1 Introduction
This chapter is a continuation of our sequence of reports on waves in elastic layered
media:
Report I Synthetic Translational Motions of Surface Waves On or Below a Layered Media
Report II Diffraction Around an Irregular Layered Elastic Media, I: Love and Body SH
Waves - 1
Report III Diffraction Around an Irregular Layered Elastic Media, II: Rayleigh and Body P,
SV Waves - 2
which has now been summarized and appeared on two journal papers:
1) Lee, V.W., Liu, W.Y., Trifunac, M.D., Orbovi ć, N. Scattering and diffraction of
earthquake motions in irregular elastic layers, I: Love and SH waves, “Soil Dynamics
and Earthquake Engineering”, Volume 66, November 2014, Pages 125–134.
2) Lee, V.W., Liu, W.Y., Trifunac, M.D., Orbovi ć, N. Scattering and diffraction of
earthquake motions in irregular, elastic layers, II: Rayleigh and body P and SV
waves, “Soil Dynamics and Earthquake Engineering”, Volume 66, November 2014,
Pages 220–230.
This chapter is part of and continuation of paper 1). Seismic ground motion of a layered
medium having an irregular topography has been studied by researchers from seismology and
earthquake engineering point of view over years. Aki & Larner (1970) applied analytical methods
to evaluate the site response on an irregular layered medium due to incident plane SH-waves.
Kohketsu (1987) extended the study to a multi-layered medium. Chen (1990, 1995) and Kohketsu
50
et al, (1991) used Aki-Larner method to generate synthetic seismograms for seismological study
of regional topography effects. Chen (1999) investigated the modal solutions and excitation of
Love waves in multilayered media with irregular interface.
For P-, SV-Waves and associated Rayleigh Waves, the analytical solutions are rare for
irregular layered medium because P-, SV-Waves cannot be decomposed in mathematical
formulation, and satisfactory of boundary conditions is often challenging. In the field of
seismology, an asymptotic solution of the fundamental Rayleigh mode, analytical formulas for
plane P- and SV-Waves in multi-layered medium was derived by Chen (1993, 1996). Vai et al,
(1999) presented a study using indirect boundary element method (IBEM) to simulate wave
propagation in 2-D irregular single-layered medium. Lee & Wu (1994a, b) used weighted-residual
method to solve an irregular canyon and obtained the site response spectral amplification.
51
3.2 Love Surface and Body SH Waves On and Below the Surface of an
Elastic Layered Media
Recall from Chapter 1, Figure 1.1 where we considered an N -Layered half-space with
Love Waves or Body SH Waves incident from the left (Report II, Figure 2.1):
Figure 3.1 N-layered half-space with Love Waves
For each of these l regular layer, with 1, , lN , the displacement of the Love surface
Waves in the layer take the form (in the anti-plane, y-component):
11
,: vv
l
l
ik x z
ll
ik x z
ll
C
C
ve
ve
(3.1)
They are respectively the upward and downward Love waves present in the
th
l layer. Here
()
()
kk
c
is the horizontal wave number of the waves at frequency and phase velocity
52
() cc . The term
ikx
e , which is also the same in each layer, is the horizontal component of the
waves, which together with the time harmonic term
it
e
, corresponds to the waves propagating in
the
ve x direction. The terms
l
y ik
e
are the vertical components of the waves with the
negative term propagating upwards
ve y and the positive term propagating downwards() y .
Here ()
ll l
is given by
1
1
1
2 22 2 2
2 2
11
l
l
l
l
kk
k
c
k
k
(3.2)
so that
l
k is the vertical wave number of the waves in the
th
l layer of the medium with shear
wave velocity
l
. In general, the wave velocities increase as one moves down through the layers,
so that
12 n
, with the semi-infinite half-space layer at the bottom having the highest
shear wave speed . With () cc the wave speed of the surface Love waves there, c ,
and the surface waves take the form:
ik x z
ikx k z
vC C ee
(3.3)
with
1
1
1
22 2 2 2
2 2
11
kk
k
c
k
k
(3.4)
is the complement of and is real, so that the term
ikx k y
e
in W
corresponds to a surface
wave term whose amplitude is exponentially decaying with depth in y below the surface. With
l
the shear wave speed in the
th
l layer of the medium, and () cc the Love wave speed and
also the (horizontal) phase velocity of the waves in each layer of the elastic media above the half
space (Figure 3.1), we can have
l
c or
l
c . If
l
c , the term ()
ll l
in (3.2) is real and both
waves
l
v
and
l
v
in (3.1) will correspond to harmonic plane waves. If, however,
l
c then, as from
(3.2):
1
2 2
1
l
l
c
is imaginary and
1
2 2
1
l
l
c
is real (3.5)
53
3.3 Love and Body SH Waves Incident On an Irregular Layered Elastic
Media
Figure 3.2 Irregular-Shaped N-layered half-space with Love Waves
The above section is on Love Waves propagating along an N-layered half-space where the
layers are perfectly flat. Consider next when this is not the case. Figure 3.2 here is a model where
the parallel layers are not perfectly flat at some (possibly finite) region, while it is flat as most
other parts. Without loss of generality, we will assume that each of these irregular surface at each
interface between the
th
l and
1
th
l layer, which extends from x to x can be represented
as an empirical curve ()
l
zhx at the interface. We assume that this curve will be the flat surface
l
zH almost everywhere, but will deviate from the flat surface within some finite region where it
is defined numerically by a set of points
,
ii
x z . The same curve can also be represented in polar
54
coordinates
ˆ
ˆ, r , with
ˆ
ˆ ˆ rr in a common coordinate system
ˆ ˆ , x z with origin at some
point
ˆ
O above the half space as shown.
With the Surface Love Waves
ll
v,v
or Body SH Waves at each layer l incident on these
irregular surfaces, additional scattered waves are generated, which can be represented by
(1)(1) (2)(2)
,,
0
ˆˆ () ( )cos
s
llnnl lnnl
n
vAHkrAHkr n
(3.6)
with both outgoing and incoming waves, for each layer 1, 2, , 1 lN , all except the last semi-
infinite layer. For the last semi-infinite layer , lN the scattered wave,
s s
N
vv
, takes the form
(1)
,
0
ˆ ()cos
ss
Nnn
n
vv A H kr n
(3.7)
with only outgoing waves satisfying Somerfield’s radiation condition at infinity.
The scattered waves, together with the free-field surface Love Waves, form the resultant
waves in the layered media. Writing
ff
ll l
vv v
as the free-field surface Love or Body SH waves
in the
th
l media, the resultant wave in the same media is
ffs
ll l
vv v , which, together, must satisfy
the following set of boundary conditions below (Lee and Wu, 1994a).
1) On the half-space surface,
0
() y hx , the resultant waves in the top layer (1) l must
together satisfy the half-space surface free-field stress condition:
11
1
11
0
ˆ ˆ
ff s
nt
vv
v
nn
(3.8a)
or
11
ˆˆ
s ff
vv
nn
(3.8b)
2) For 1, 2, , 1 lN , at the interface between the
th
l layer and the (1)
th
l layer below,
()
l
y hx , the resultant waves in these two layers must satisfy the continuity of
displacement and stress at the interface:
11 1
ff s ff s
ll l l l l
vv v v v v
(3.9a)
or
11
ssff ff
ll l l
vv v v
(3.9b)
55
1
1
11
1
ˆ ˆ
ˆˆ
ll
ll
ff s ff s
ll l l
ll
vv
nn
vv v v
nn
(3.10a)
or
11
11
ˆ ˆˆ ˆ
s s ff ff
ll l l
ll l l
vv v v
nn n n
(3.10b)
with the unknown wave functions on the left-hand side, the known functions on the right-hand
side. Here, in (3.8) and (3.10), ˆ n is the normal at the boundary point (, ) x y where ()
l
y hx , and
ˆ n
is the corresponding normal derivative of the wave functions given by:
sin
ˆ cos
ˆ
n
nrr
(3.11a)
where is the angle that the normal ˆ n makes with the radial vector, as shown in Figure 3.3 (Lee
and Wu, 1994a,b). For the free-field waves, which are available in rectangular coordinates, (3.11a)
can also be expressed as
ˆ
ˆ
xy
nn n
nxy
(3.11b)
where the normal ˆ n is expressed in rectangular coordinates as ˆˆ ˆ
xxyy
nne ne
; ˆ
x
e and ˆ
y
e are
respectively the unit vectors in the x- and y- directions.
Figure 3.3 Angle between radial vector and normal at a point
56
On the flat part of the interface between the interface of the
th
l and (1)
th
l layer, namely,
on the curve ()
l
y hx , where
l
yH exactly flat, since the Love Waves in between the layers already
satisfy the continuity of displacement and stress at the flat interface, only the scattered waves will
appear on the continuity equations at those points.
More precisely, (3.8b), (3.9b) and (3.10b) can be written as:
1
0 1
1
0
on an irregular surface point of ( ),
ˆ
ˆ
0 on a flat surface point of ( ) 0
ff
s
ff
v
yh x v
n
n
yh x
(3.12a)
1
1
on an irregular surface point of ( ),
0 on a flat surface point of ( )
ff ff
ll l ss ff
ll l
ll
vv y hx
vv v
yhx H
(3.12b)
1
1
1
1
and
on an irregular surface point of ( ),
=
0 on a flat surface point of
ss
ff ll
ll l
ff ff
ll
ll l
vv
nn
vv
yhx
nn
y
()
ll
hx H
(3.12c)
To solve this boundary-valued problem, one would have to find a wave function,
l
v , as
given in (3.6) for each layer 1, 2 , 1 lN , and as in (3.7) for the bottom semi-infinite layer
lN .
57
3.4 The Method of Weighted Residues (Moment) Method
This section will be the same as that already given in Lee and Wu (1994a, b) and is included
and summarized here for completeness. It is seen from the above sections that the wave functions
at each layer together have to satisfy the set of boundary conditions numerically at every point of
the interface given in (3.8), (3.9) and (3.10). Since the surfaces are now non-flat, a numerical
procedure has to be applied.
Harrington (1967) in his classical paper: “Matrix Methods for field Problems”, presented
a well-defined unified treatment of most of the existing numerical methods for the above
boundary-valued problems, which he called the “Moment Methods”. He applied the method to
Electromagnetic waves problems. Following Fenlon (1969), with applications to acoustic wave
problems, Lee and Wu (1994a, b) applied the method to elastic wave problems on arbitrary-shaped
canyons in elastic half-space. Another common name for the “Moment Method” is the “Method
of Weighted Residues”. Here is a very brief summary of the review of the method, following Lee
and Wu (1994a, b).
Assume that a general equation to be satisfied is of the form
f g L (3.13)
where L is a linear operator, f is an unknown function to be solved, and g is a given known
function. The boundary conditions in (3.8), (3.9) and (3.10) above are all of this form, with L a
linear combination of the identity and the derivative operators, f a sum or difference of the
unknown wave functions and their derivatives in adjacent media, and g the corresponding sum
or difference of the known free-filed Love Wave functions and their derivatives in adjacent media.
Each of the unknown wave functions f is here represented as a series of basic functions
01
,, , , ,
n
ff f
nn
n
f cf
(3.14)
58
with
n
c a sequence of unknown coefficients to be determined. The range of the indices of the
functions can either be infinite ( to ) or semi-infinite (0 to ). Since L is a linear
operator, (3.11) takes the form, in terms of residues, , to be set to zero:
0
nn
n
cf g
L (3.16)
next, we choose a scalar inner product , f g to be defined on any pair of functions in the domain
of L, and a set of weight functions
01
,, , , ,
n
ww w in the domain. We take the inner product of
the residues with each weight function and equate it to zero:
,,
,, 0
so , ,
mnn m
n
nn m m
n
nm n m
n
wcfgw
cfw gw
fw c gw
L
L
L
(3.17a)
or in matrix form:
mn n m
A cg (3.17b)
with
,
mn n m
A fw L , a matrix of infinite order, and
mm
g gw , is a known vector computed
from the free-field surface Love Waves or Body SH Waves. The matrix is composed of
21 N
set of boundary conditions in (3.12a), (3.12b) and (3.12c) for the set of
21 N coefficients from
the
21 N wave functions in (3.6) and (3.7) for the N Layered media.
Depending on the choice of the weight functions, these weight functions can also be
considered as a set of basis functions, orthogonal with respect to the inner product, and here the
residues are expanded in terms of these basis functions using the inner product and we set each
term of the expansion here to be zero.
59
Using the weighted residue method, the boundary conditions at each layer take the form:
1) Starting on the top layer, the half space surface, from (3.8b):
() () 1
11,
1,2
0
ˆ ()cos , ,
ˆ ˆ
ff
jj
nmn m
j
n
v
Hkr n w A w
nn
(3.18a)
or in matrix form, at the top surface of the first layer (surface of the half-space):
(1)
1, (1) (2)
11 (2)
0 1,
1
ˆ ˆ ()cos , ()cos ,
ˆˆ
,
ˆ
n
nm n m
n n
ff
m
A
Hkr n w H kr n w
A nn
v
w
n
(3.18b)
2) For the interface between the
th
l and (1)
th
l layer, for 1, 2, 3 l from (3.9b):
() ( ) () ()
,1 1,
1,2
0
1
ˆ ˆ ( ) cos , ( )cos ,
,
jjj j
nl m ln n l m ln
j
n
ff ff
ll m
Hkr n w A Hkr n w A
vv w
(3.19a)
() () ( ) ( )
,1 1 1,
1,2
0
1
1
ˆˆ ( )cos , ( )cos ,
ˆ ˆ
,
ˆ ˆ
jj j j
lnl mln l nl mln
j
n
ff ff
ll
ll m
H kr n wA H k r n wA
nn
vv
w
nn
(3.19b)
or in matrix form, at each interface between the
th
l and (1)
th
l layer, for 1, 2, 3 l :
() ()
1 ()
,
()
() ()
1,2
0 1,
11
1
1
1
ˆ ˆ ( )cos , ( ) cos ,
ˆ ˆ ()cos , ( )cos ,
ˆ ˆ
,
,
ˆˆ
jj
nl m n l m j
ln
j
jj
j
n ln
lnl m l nl m
ff ff
ll m
ff ff
ll
ll m
Hkr n w H k r n w
A
A
Hkr n w H k r n w
nn
vv w
vv
w
nn
(3.19c)
Note that for the interface between layer
1 N and the (bottom, last) semi-infinite layer N ,
(3.19a) and (3.19b) will have only the outgoing wave terms
(1)
,
,0,1,2
Nn
An and without the
incoming wave terms
(2)
,
,0,1,2
Nn
An
60
3.5 Numerical Implementation
This section will demonstrate the numerical procedure used to apply to the system of
complex equations derived in the last section, which used the method of weighted residues. In
studying the set of equations, it is observed that for an elastic half-space with (1) N layers, there
are (1) N interfaces in-between the layers, plus the topmost half-space with no elastic medium
above.
At this topmost half-space surface, there is one set of zero-stress equations (at 0 z , (3.8))
involving the wave coefficients on the topmost first layer. At each of the (1) N interfaces in-
between the layers, there are each two sets of the stress and displacement continuity equations (3.9)
and (3.10). There is thus a total of 2( 1) 1 2 1 NN set of equations. Each of the top (1) N
layers has two sets of waves, the upward and downward going, or the outgoing and incoming
waves (3.6). They are represented by Hankel Functions of the first and second kind, the outgoing
and incoming waves, respectively. On the other hand, the bottom-most semi-infinite layer has
only one set of waves, the downward going or outgoing waves (3.7). They are represented by
Hankel Functions of the first kind. This gives a total of 2( 1) 1 2 1 NN set of waves from
all the layers.
In summary, we have (2 1) N set of equations for (2 1) N set of waves. If each of such
(2 1) N set of waves has M terms, with M unknown coefficients, we will have a total of
(2 1) MN unknowns we need to solve. Ideally each set of waves has, M , or infinite
number of terms and there are (2 1) N number of equations. In reality, we have to truncate
to finitely many M terms for each set of waves, with the number M often dependent on the
frequency of the waves and the complexity of the equations. With M finite, we have a task of
solving the (2 1) MN set of complex equations for the (2 1) MN set of unknowns.
Numerically, we know that the Hankel Functions of both the first and second kinds are
both complex, and increase in magnitudes with increasing order. Thus if all the (2 1) MN
unknowns are put together in the (2 1) MN complex equations, one would have a very large
61
set of complex equations to solve, whose terms often have increasing magnitudes with increasing
M. It would be a numerically nightmare to solve them all at once.
Since each set of equations at each interface involves only waves at each side of the
interface, a simple, elegant numerical algorithm can be derived to allow each set of wave
coefficients at each media to be solved separately, making the problem much more simple
numerically. In other words, we are solving M complex equation in M unknowns at each step.
The following is a comprehensive description of the numerical procedure.
The idea is very simple. Starting from the top surface, where 0 z , we have the matrix
equation for the zero-stress boundary condition along the whole (regular and irregular) surface of
the half-space, in the form:
(1) * (2)
11 1 1 1
0
0
z
EA E A e
(3.20)
where from (3.18b):
1
E is the matrix with elements
(1)
1
ˆ ()cos ,
ˆ
nm
H kr n w
n
,
*
1
E
is similarly the matrix with elements
(2)
1
ˆ ()cos ,
ˆ
nm
Hkr n w
n
(1)
1
A
,
(2)
1
A
are respectively the vectors
(1) (1) (1)
10 11 12
,,, AAA
,
(2) (2) (2)
10 11 12
,,, AAA
and
1
e is the vector
1
,
ˆ
ff
m
v
w
n
Eliminating
(2)
1
A
, one has, at 0 z :
1
(2) * (1)
11 11 1
0
1
*(1)
11 1 1
(0) (0) (0)
z
AE EA e
EE A e
(3.21)
62
In the interface between the first and second layer, the continuity equations take the form:
1
1
(1) * (2)
11 1 1 1
(1) * (2)
22 2 2 2
zh
zh
EA E A e
EA E A e
(3.22a)
or
(1) * (2)
11 1 1 1 1 1 1
(1) * (2)
21 2 2 1 2 21
Eh A E h A e h
Eh A E h A e h
(3.22b)
for stress continuity.
Similarly, for the displacement continuity equations:
(1) * (2)
11 1 1 1 1 11
(1) * (2)
21 2 2 1 2 21
Dh A D h A d h
Dh A D h A d h
(3.23)
Using (3.21),
(2)
1
A
can be eliminated from the L.H.S. of the stress (3.22):
(1) * (2)
11 1 1 1 1 11
1
(1) * * (1)
11 1 1 1 1 1 1 1 11
(1)
11 1 1 1
() () ()
() () (0) (0) (0) ( )
() ()
Eh A E h A e h
Eh A E h E E A e e h
hA h
Ee
(3.24a)
where
1
**
11 11 1 1 1 1
1
**
11 11 1 1 1 1
() () () (0) (0)
() () () (0) (0)
hEhEh E E
heh Eh E e
E
e
(3.24b)
In exactly the same way, using (3.21),
(2)
1
A
can be eliminated from the L.H.S. of the
displacement (3.23), resulting in:
(1) * (2)
11 1 1 1 1 11
1
(1) * * (1)
11 1 1 1 1 1 1 1 1 1
(1)
11 1 1 1
() () ()
() () (0) (0) (0) ( )
() ()
Dh A D h A d h
Dh A D h E E A e d h
hA h
Dd
(3.25a)
63
where
1
**
11 11 1 1 1 1
1
**
11 11 1 1 1 1
() () () (0) (0)
() () () (0) (0)
hDhDh E E
hdh Dh E e
D
d
(3.25b)
The continuity matrix equations at the interface of the first and second layer now takes the form,
without the
(2)
1
A
terms:
(1)
11 1 1 1
(1) * (2)
21 2 2 1 2 21
() () hA h
EhA Eh A eh
Ee
(3.26a)
(1)
11 1 1 1
(1) * (2)
21 2 2 1 2 21
() () hA h
Dh A D h A d h
Dd
(3.26b)
In matrix form, (3.26a) and (3.26b) become:
* (1)
11 2 1
11 21 2 1 (1) 2
1 * (2)
11 21 2 1 2
11 2 1
()
()
()
()
he h
h Eh E h A
A
h Dh D h A hd h
e
E
D
d
(3.27)
which shows that both
(1)
2
A
and
(2)
2
A
of the coefficients of the waves in the second layer can
be expressed in terms of the single set of
(1)
1
A
coefficients of waves in the top layer. One can
thus write:
* (1)
11 2 1
11 21 2 1 (1) 2
1 * (2)
11 21 2 1 2 11 2 1
1
()
()
()
()
he h
h Eh E h A
A
h Dh D h A hd h
e
E
D
d
(3.28a)
so that such a transformation is obtained:
(1)
21 1 21 1 (1) 2
1
(2)
22 1 2
22 1
() ()
()
()
h h A
A
h A
h
t T
T
t
(3.28b)
Repeating this iteration at each interface of layers proceeding downwards one can start with the
continuity equations at interface between layer l and 1 l , as in (3.22b) and (3.23), at the
interface
l
zh :
64
(1) * (2)
(1) * (2)
11 11 1 1
ll l l l l ll
ll l l l l ll
Eh A E h A e h
Eh A E h A e h
(3.29a)
(1) * (2)
(1) * (2)
11 11 1
ll l l l l l l
ll l l l l ll
Dh A D h A d h
Dh A D h A d h
(3.29b)
As in (3.28b),
(1)
l
A
and
(2)
l
A
are to be expressed in terms of
(1)
1
A
:
(1)
1 1 (1)
1
(2)
2
2
() ()
()
()
ll ll l
ll l
ll
h h A
A
h A
h
t T
T
t
(3.30)
from which (3.29a) and (3.29b) can be expressed as, with the coefficients
(1)
1
A
of waves in the
top layer on the R.H.S.:
(1)
1
(1) * (2)
11 11 1
() ()
ll l l
ll l l l l ll
hA h
E hA E h A e h
+
Ee
(3.31a)
(1)
1
(1) (2)
11 11 11
() ()
ll l l
ll l l l l l
hA h
Dh A D h A d h
=
Dd
(3.31b)
or again in matrix form, as in (3.28a):
* (1)
1
11 (1) 1
1 * (2)
11 1
1
()
()
()
()
ll l l
ll ll l l l
ll ll L l l
ll l l
he h
h Eh Eh A
A
h Dh D h A
hd h
e
E
D
d
(3.32a)
resulting in a transformation, as in (3.28b), expressing the coefficients
(1)
1 l
A
and
(2)
1 l
A
for
layer 1 l to be expressed in terms of
(1)
1
A
from waves of the top layer.
(1)
1,1 1,1 (1) 1
1
(2)
1,2
1 1,2
() ()
()
()
ll ll
l
ll
l ll
h h
A
A
h
A h
t T
T
t
(3.32b)
65
The reason why we select to have the coefficients of all waves in each layer expressed in
terms of (one of the two) a set of coefficients of waves n the top layer is because, we know, for
surface waves, and possibly body waves, the top layer waves are more dominant.
The above described procedure will continue until the interface with the bottom semi-
infinite layer lN is reached, namely at
1 N
zh
. There the continuity equations will be different
from those in the previous interfaces, since the bottom semi-infinite layer will now have only one
set of waves, namely, the downward, outgoing waves. The stress and displacement continuity
equations now take the form, as in (3.29a) and (3.29b), with 1 lN , 1 lN :
(1) * (2)
11 1 1 1 1 11
(1)
11
NN N N N N NN
NN N N N
Eh A E h A e h
Eh A e h
(3.33a)
(1) * (2)
11 1 1 1 1 11
(1)
11
NN N N N N NN
NN N N N
Dh A D h A d h
Dh A d h
(3.33b)
As before, the coefficients
(1)
1 N
A
and
(2)
1 N
A
of the waves in the ( 1)
th
N layer can be
expressed in terms of
(1)
1
A
, that of the waves in the top layer
(1)
11 1 1 1
(1)
11
() ()
NN N N
NN N N N
hA h
Eh A e h
Ee
(3.34a)
(1)
11 1 1 1
(1)
11
() ()
NN NN
NN N N N
hA h
Dh A d h
Dd
(3.34b)
with new matrices
11
()
NN
h
E ,
11
()
NN
h
D and vectors
11
()
NN
h
e ,
11
()
NN
h
d in terms of
(1)
1
A
, the wave coefficients of the top layer, on the L.H.S. of the equations.
66
(3.32a) and (3.32b) are the final form of the matrix equations, from which the wave
coefficients
(1)
1
A
of the top layer and
(1)
A
N
in the bottom semi-infinite layer are to be related.
Normally, one will rewrite (3.32a), expressing
(1)
A
N
in terms of
(1)
1
A
, as:
1
(1) (1)
111 1 11 1
() ()
NNN N N NN NN
AEh h A h eh
Ee (3.35a)
And substituting it into (3.32b), which becomes a matrix equation with
(1)
1
A
of the top
layer as the only set of unknowns:
1
(1)
11 1 1 11 1
1
11 11 1 11
() ()
() ()
N N NN NN N N
N N NN N N NN N N
hDh Eh h A
Dh E h h d h h
DE
ed
(3.35b)
from which the coefficients
(1)
1
A
can now be evaluated, after which the coefficients of the
waves in all layers can be found.
67
3.6 The Diffracted Mode Shapes of Love and SH Body Waves
3.6.1 The Input Free-Field Waves
Recall from Report I where we considered N Layered half-space with Love Waves or
Body SH Waves incident from the left over a regular parallel layered media (Report I, Figure 2.1).
In a sub-section of Chapter 5 in Report I there we plotted the displacement mode shapes of Mode#1
to Mode#5 Love Surface Waves for a selected range of periods starting from 15 sec down to 0.04
sec (0.07Hz to 25Hz). This is followed by another sub-section of the mode shapes resulting from
Body SH Waves for a given incident angle. We actually take the Body SH Waves as the sixth
mode of waves at the parallel layered media.
Each of these six mode shapes at each period (frequency) will be used as the free-field
input waves onto the irregular parallel layered media to be studies here. In what follows, we will
take a rather simple two-layered media to illustrate the process. The following parameters are used
for the two-layered media:
Layer
Thickness
(km)
P-wave Speed
( , km/s)
S-wave Speed
( , km/s)
Density
( ,gm/cc)
1 1.38 1.70 0.98 1.28
2
6.40 3.70 2.71
Table 3.1 Two-Layer Velocity Model
They are the same first and last layers of the six-layered model given in Report I, except
the top layer is now 1.38 km thick. The computer Program, “Haskel.exe”, is again used to calculate
the phase velocities of each mode of Love waves, as in Report I, in the period range of periods
from 14.0 sec down to 0.04 sec for a total of 91 discrete period values.
Figure 3.4 is the input free-field mode shapes for Mode#1 Love Waves at four selected
period, T = 5.0, 1.5, 0.5 and 0.01 sec, or at frequency, f = 0.20, 0.67, 2.0 and 10.0 Hz. As before,
for Mode#1 waves, the mode shapes are available at all 91 pre-selected period values. It is noted
68
from the figures that, at all the periods shown, the wave amplitudes of the mode shapes are scaled
to have a maximum amplitude of 1 at the half-space surface, and that the amplitudes are practically
zero at and below the interface of the two layers, which is at 1.38km apart.
Figure 3.5 is the input free-field mode shapes for the corresponding Mode#2 Love Waves
at four selected period, T = 2.0, 1.0, 0.3 and 0.08 sec. For Mode#2 Love Waves, the modes are
available not at all 91 periods, but instead from T = 2.2 sec (Period#30, down from T=14.0 sec for
Mode#1). So the four periods plotted here are selected from a narrower range. As before, the
mode shapes are scaled to have a maximum amplitude of 1 starting from the surface 0 z . A
characteristic of the Mode#2 Love Waves is that the mode shape at each period now started to
decrease to have negative values, reaches a negative minimum before asymptotically approaching
zero. Again, as in Mode#1, the amplitudes are practically zero at and below the interface of the
two layers ( 1.38km) z .
Figure 3.6 is the input free-field mode shapes for the corresponding Mode#3 Love Waves
at four selected period, T = 1.0, 0.8, 0.4 and 0.08 sec. For Mode#3 Love Waves, the modes are
available again not at all 91 periods, but instead from T = 1.1 sec ( Period#40, down from T=14.0
sec for Mode#1). So the four periods plotted here are again selected from a narrower range. As
before, the mode shapes are scaled to have a maximum amplitude of 1 starting from the surface
0 z . A characteristic of the Mode#3 Love Waves is that the mode shape at each period now
started to decrease to have negative values, reaches a negative minimum, then cross the zero axis
back to have positive amplitudes, reaches a positive maximum, before asymptotically approaching
zero. Again, as in Mode#1 and Mode#2, the amplitudes are practically zero at and below the
interface of the two layers ( 1.38km) z .
We will expect the scattered and diffraction waves to be small at the interface. The same
observations and conclusions can be made for Mode #4 and #5 free-field Love Waves. The plots
have been omitted here.
69
Figure 3.4 Mode#1 Mode Shapes at f = 0.20, 0.67, 2.00 and 10.00 Hz
70
Figure 3.5 Mode#2 Mode Shapes at f = 0.50, 1.00, 3.33 and 12.50 Hz
71
Figure 3.6 Mode#3 Mode Shapes at f = 1.00, 1.25, 2.50 and 12.50 Hz
72
Finally, the free-field mode shapes for incident SH Body Waves are plotted next. Figure
3.7 is the input free-field mode shapes for the corresponding SH Body Waves with angle of
incidence of 60
o
with respect to the horizontal direction at four selected period, T = 4.0, 1.0,
0.4 and 0.10 sec, or at frequency, f = 0.25, 1.0, 2.5 and 10.0 Hz. As in the case of Mode#1 Love
Waves, the mode shapes are available at all 91 pre-selected period values. It is noted from the
figures that, at all the periods shown, the wave amplitudes of the mode shapes are again scaled to
have maximum amplitude of 1 at the half-space surface. Unlike the Love Waves, depending on
the period of the body waves, the waves do not depreciate to zero as you go deeper down from the
surface. The waves in the first layer are harmonic, oscillating at the given period of the waves
between the scaled maximum amplitude of 1 , and the oscillation increases with decreasing period,
or increasing frequency, as the figure shows.
Below the first layer, on the semi-infinite layer, it continues to oscillate, though at a higher
period and lower amplitude, but they are not depreciating to zero, since the body waves are
harmonic waves, harmonic in both the horizontal x and vertical z directions.
We thus will expect in the next sub-section that the diffracted waves from SH Body Waves
will behave differently from those from the Love Surface Waves.
73
Figure 3.7 SH Body Waves Mode Shapes at f=0.25, 1.00, 2.50 and 10.00 Hz
74
3.6.2 The Diffracted Mode Shapes
We next consider the case of irregularly shaped layered media superimposed onto the
parallel two-layered media studied here. We select a shallow, “almost-flat” ellipse with a ratio of
“Vertical minor axis/Horizontal major axis = 0.1” at both the half-space surface and the interface
of the two layers. The half-width or radius of the horizontal major axis is taken to be 1.0 km long
for both, or a major diameter of 2.0 km long. Thus the half-width or radius of the minor axis is
0.1 km deep, the case of a very shallow ellipse. Figure 3.8 is a sketch of such a two-layered elastic
media for cases of (i) Incident Love Surface Wwaves, and (ii) Incident Body SH Waves:
Figure 3.8 The Irregular Two-Layered Media with (i) Love Waves and (ii) SH Body Waves
The next six figures, Figure 3.9 to 3.14 are respectively the diffracted mode shapes for
Mode #1 to Mode#5 of Love Waves at the same two-layered irregular media. Each mode has mode
shapes plotted at four selected frequencies, at f = 13.33, 18.18, 20.0 & 25.0 Hz. These are
frequencies below period of 0.1 sec (or frequency beyond 10Hz) as it was found that since the
75
irregular part of the layer are almost flat, the long period waves overlooked the irregularities and
the diffracted waves resulted are small and insignificant.
Figure 3.9 is the diffracted mode shapes for Mode#1 Love Waves at the above stated four
selected frequencies. With the irregular almost-flat elliptic surfaces from 1.0 km x to
1.0 km x on both the half-space surface
0 km z and the surface of interface of the two media
1.38 km z , the diffracted mode shapes are plotted at equally-spaced intervals along x
from
2.0 km x to 2.0 km x at 0.1 km apart. The dashed line on the left side of each graph
represents the input free-field mode shapes propagating along the parallel-layered media from the
left arriving at the irregular surfaces. For all frequencies up to almost 15.0 Hz, the waves are
unaffected by the almost-flat irregular surfaces. At the irregular interfaces between the two media,
as pointed out in the previous section, as can also be seen by the dashed line on the left, the mode
shape amplitudes are already almost zero. That is thus not much diffraction to be expected.
As the frequencies of the waves increase from 15.0 to 25.0 Hz, as the next three graphs
shown, where the frequencies are respectively, f = 18.18, 20.0 and 25.0 Hz, it is seen that the Love
Waves do have some degrees of diffraction in the top elastic media, with such diffraction most
noticeable on the 25 Hz mode shape graph on the lower right corner graph.
Figures 3.10 through 3.14 are the corresponding diffracted mode shapes for Mode#2 to
Mode#5 Love Waves at the same four selected frequencies. As the mode number increases, as
observed in the previous section, both the mode shapes of the input free-field and that of the
diffracted Love Waves here went through the 1 M times of sign changes as they go down
vertically from the top surface, where "" M is the mode number. The diffraction are also more
complex as the mode number increases from Figure 3.9 through 3.14.
We will next work on multi-layered media with the layers being closer together. It will be
expected that these diffracted patterns will be more complicated as the number of layers increase,
and as the layers get closer and closer together.
76
Figure 3.9 Mode#1 Diffracted Mode Shapes at f = 13.33, 18.18, 20.00 & 25.00 Hz
77
Figure 3.10 Mode#2 (2-D) Diffracted Mode Shapes at f = 13.33, 18.18, 20.00 & 25.00 Hz
78
Figure 3.11 Mode#2 (3-D) Diffracted Mode Shapes at f = 13.33, 18.18, 20.00 & 25.00 Hz
79
Figure 3.12 Mode#3 Diffracted Mode Shapes at f = 13.33, 18.18, 20.00 & 25.00 Hz
80
Figure 3.13 Mode#4 Diffracted Mode Shapes at f = 13.33, 18.18, 20.00 & 25.00 Hz
81
Figure 3.14 Mode#5 Diffracted Mode Shapes at f = 13.33, 18.18, 20.00 & 25.00 Hz
82
Finally, Figure 3.15 is the graphs of the diffracted mode shapes corresponding to incident
plane SH Body Waves. As seen from Figure 3.7 above, the input free-field Body SH Waves,
unlike the Love Waves, are more oscillatory in both layers, without depreciating to zero at the
interface.
As with the cases of Love Waves, the irregular almost-flat elliptic surfaces are from
1.0 km x to 1.0 km x on both the half-space surface 0 km z and the surface of interface of
the two media 1.38 km z , the diffracted mode shapes are again plotted at equally-spaced
intervals along x from 2.0 km x to 2.0 km x at 0.1 km apart. As before, the dashed line on
the left side of each graph again represents the input free-field mode shapes of Body SH Waves
propagating at the parallel-layered media from below arriving at the irregular surfaces. The angle
of incidence of the body waves is 60
o
with respect to the horizontal. At all frequencies, the
mode shapes of the diffracted waves are very oscillatory and are of very different amplitudes than
those of the incident free-field waves.
As the frequencies of the waves increase from 15.0 to 25.0 Hz, as the next three graphs
shown, where the frequencies are respectively, f = 18.18, 20.0 and 25.0 Hz, it is seen that the Body
SH Waves undergo large degrees of diffraction in both the top elastic media and the bottom semi-
infinite media, more so on the top media. The amplification of the waves at those frequencies does
indeed get greater than two. Table 3.2 gives the amplification factors at various frequencies,
Frequency (Hz) 16.67 18.18 20.00 20.83 21.74 22.73 23.81 25.00
Amplification 2.61 2.56 2.58 2.61 2.63 2.57 2.70 2.57
Table 3.2 Amplification Factors at Various Frequencies
As in the case of Love Waves, we will next work on multi-layered media with the layers
being closer together. It will be expected that these diffracted patterns will be even much, much
more complicated as the number of layers increase, and as the layers get closer and closer together.
83
Figure 3.15 SH Body Waves Diffracted Mode Shapes at f = 13.33, 18.18, 20.00 & 25.00 Hz
84
3.7 Generalization to More Complex Layered Interfaces
3.7.1 More General Irregular Layered Media
The method we presented here in previous sections on the scattering and diffraction of
elastic waves by surface and subsurface topographies in a homogeneous, elastic half space for out-
of-plane Love surface waves can be further generalized to Love Waves in a more irregularly
layered, elastic half spaces. In this section, we will study cases of Love Waves in a model in which
the geometry of the surface topography and of the interface surfaces for one or several layers has
to be represented by different convex or concave surfaces. Such cases generate wave-scattered
and wave-diffracted motions, and they need to be approximated by the Hankel Functions with
different origins.
Figure 3.16 Irregularly Layered Media with Multiple Coordinate Systems
85
This is illustrated in Figure 3.16, in which, for example, the bottom interface of the second
layer (layer 2), is convex upward, thus creating a “hill-interface” within the layer. The waves in
such layers will best be represented in terms of their own coordinate systems, such as
22
(, ) x z in
Figure 3.16. In general, the waves in all layers can then each be represented by their own
coordinate systems and the addition theorem can be used to apply the boundary conditions at all
interfaces. Conceptually, the method of solution to such problems will be same as the one
presented in this paper, except that more model-specific modeling will be required to solve each
problem. We will show examples of such modeling in this section
3.7.2 Defining Two Origins on the Lth Layer
From Figure 3.17, We define two coordinates
,
L LL
Or above Lth -interface and
,
L LL
Or below L1th -interface with a distance of
L
D between two origins, the total
displacement of
L
W with respect to
L
O and
L
W with respect to
L
O as follows:
(1)
,
,()
L
in
LLLL Lnn LL
n
WW r A H kre
(3.36)
(1)
,
,()
L
in
L L LL Lnn LL
n
WW r A H kre
(3.37)
Transformation of coordinates from
L
O to
L
O or from
L
O to
L
O :
,,
L L LLLL LL
WW r W W r (3.38)
,
1
,
(),
(),
L
L
in
Lm n L L L L
m
in
Lm n L L L L
m
A Jkr e r D
A Hkre r D
(3.39)
For n ,
(1)
,,
()
Lm n m L L L n
n
A HkDA
(3.40)
,,
()
Lm n m L L Ln
n
A JkDA
(3.41)
86
Figure 3.17 Two Origins for Lth Layer
As in previous sections, applying Graf Addition Formula (Figure 3.18), the Hankel
Function can be expressed as:
1
1
1
() ( ) ,
()
() ( ) ,
L
L
L
in
mLL nm L L L L
m in
nLL
in
mLL nm L L L L
m
Jkr H kD e r D
Hkre
HkrJ kDe r D
(3.42)
87
thus,
(1)
,
1
,
1
,
,
1
,
()
() ( ) ,
() ( ) ,
() ,
() ,
L
L
L
L
L
in
Ln n L L
n
in
LnmLL nmLL L L
nm
in
LnmLLnmLL L L
nm
im
Lm m L L L L
m
im
Lm m L L L L
m
AH kr e
A Jkr H kD e r D
A HkrJ kD e r D
AJ kre r D
AH kre r D
(3.43)
where,
1
,,
()
Ln L n n m L L
n
A AH kD
(3.44)
,,
()
Ln Ln n m L L
n
A AJ kD
(3.45)
Figure 3.18 Transformation using Graf’s Addition Theorem
88
From
1 L
O
to
L
O , the distance between two origins is
L
H ; for L W and 1 L W to match the
boundary condition at L1th -interface,
11 (1) (1)
11 1 1
1
11
1
11
1
11
1
11
() 1 ( )
1()()
1()()
1( )( )
()( )
LL
L
L
L
n
in in
nL L n L L
nm
im
nm L L m L L
m
nm
im
nm L L m L L
m
nm
im
nm L L m L L
m
mn L L m L L
m
H k re H k re
Hk H J k re
Hk H J k re
Hk H J k re
Hk H J k r
L im
e
(3.46)
From
L
O to
1 L
O
, for L W and 1 L W to match the boundary condition at Lth -interface
11 LL
(3.47)
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
()( )
()
() ( )
() ( )
() ( )
()( )
L
L
L
L
L
im
nm L L m L L
m in
nLL
im
nm L L m L L
m
im
nm L L m L L
m
im
nm L L m L L
m
nm L L m L L
m
HkHJ kr e
Hkre
JkHH kr e
HkHJ kr e
JkHH kr e
HkHJ kr
1
1
1
1
() ( )
L
L
im
im
nm L L m L L
m
e
JkHH kr e
(3.48)
These are examples of transformations needed for all the waves to be transformed to one
origin at one common interface to satisfy the corresponding displacement and stress continuity
boundary conditions. The next subsections are results for an example.
89
3.7.3 The Case of a Two-Layer Canyon-Hill Interface
Next consider the case of irregularly shaped layered media superimposed onto the canyon-
hill interface studied here. We select a shallow, “almost-flat” ellipse with a ratio of “Vertical
minor axis/Horizontal major axis = 0.1” at both the half-space surface and the interface of the two
layers. The half-width or radius of the horizontal major axis is taken to be 1.0 km long for both,
or a major diameter of 2.0 km long. Thus the half-width or radius of the minor axis is 0.1 km deep,
the case of a very shallow ellipse. Figure 3.19 is a sketch of such a two-layered elastic media for
cases of incident Love Surface Waves:
Figure 3.19 The Irregular Two-Layered Media with Love Waves
The next twelve figures, Figure 3.20 to 3.31 are respectively the diffracted mode shapes
for Mode #1 to Mode#5 of Love Waves at the same two-layered irregular media. Each mode has
mode shapes plotted at eight selected frequencies, for example from Figure 3.20 in Mode #1 at f
= 1.00, 2.00, 3.33 & 5.00 Hz and Figure 3.21 (above frequency of 1 Hz) in Mode #1 at f = 6.67,
10.00, 12.50 & 13.33. These are frequencies below period of 1.0 sec as it was found that since the
irregular part of the layer are almost flat, the long period waves overlooked the irregularities and
the diffracted waves resulted are small and insignificant.
As stated, Figure 3.20 and 3.21 are the diffracted mode shapes for Mode#1 Love Waves at
the above stated eight selected frequencies. With the irregular almost-flat elliptic surfaces from
1.0 km x to 1.0 km x on both the half-space surface 0 km z and the surface of interface of
the two media 1.38 km z , the diffracted mode shapes are plotted at equally-spaced intervals
along x
from 2.0 km x to 2.0 km x at 0.1 km apart. The dashed line on the left side of each
90
graph represents again the input free-field mode shapes propagating along the parallel-layered
media from the left arriving at the irregular surfaces.
It is worthwhile to compare these figures of the canyon-hill model to that in Figure 3.9 and
3.14. Figure III.9 the diffracted mode shapes for Mode#1 Love Waves at four selected frequencies
for the canyon-canyon model. Recall that the waves of the canyon-canyon model were computed
using only one origin on top and the waves in the present canyon-hill model were computed with
multiple origins at each layer, where the origins are defined both on top and below interfaces of
the layers. The waves in the present mode are thus better represented and more realistic. It is seen
that as a result, the waves in Figure 3.21 and 3.22 for Mode#1 have shown more interference and
the resultant waves are more interactive at all eight frequencies for f =1.00 Hz to f =13.33 Hz.
Next Figures 3.22 through 3.31 are the corresponding diffracted mode shapes for Mode#2 to
Mode#5 Love Waves, with ten graphs in two pages for each mode, one graph per frequency. The
dashed lines on the left of each graph are again the input “free-field” Love Waves. Again the new
model with multiple origins used for each set of waves resulted in more interaction and diffraction
at each mode. Note that as the frequencies increase, the waves get more complicated. Also as the
mode number increase (from Mode#1 to Mode#5), the input Love Waves have more zero nodes,
and the resultant waves show significant interaction to the left and right and ve ve of the zero
nodes. The Mode#5 resultant waves and thus more complicated than that of the Mode#4 resultant
waves, all the way down to the Mode#1 waves. The contribution of each mode, though, has less
weight as the mode number increases, with Mode#1 having the greatest weight, and hence
contribution to the total waves.
These observations are consistent with the results of the canyon-canyon model in the
previous section, namely Figure 3.10 through 3.14 for Mode#2 to Mode#5 of the canyon-canyon
model. There, it was already noted that as the mode number increases, both the mode shapes of
the input Love Waves and that of the resultant, diffracted waves went through the 1 M times of
sign changes as they go down vertically for the top surface, where = 1 to 5 M , is the mode
number. Again, as pointed out, the diffraction gets more complex as the mode number increases
for Figure 3.22 to Figure 3.31.
91
The method developed in this section now enables us to compute analytical representations
of seismic-wave motions in irregular media, which can in general be modeled only numerically.
This will be particularly useful for verification and testing of the accuracy with which transmitting
boundaries in numerical algorithms are able to transmit the waves without spurious reflections.
92
Figure 3.20 Mode#1 Diffracted Mode Shapes at f = 1.00, 2.00, 3.33 & 5.00 Hz
93
Figure 3.21 Mode#1 Diffracted Mode Shapes at f = 6.67, 10.00, 12.50 & 13.33 Hz
94
Figure 3.22 Mode#2 (2-D) Diffracted Mode Shapes at f = 1.00, 2.00, 2.50 & 3.33 Hz
95
Figure 3.23 Mode#2 (3-D) Diffracted Mode Shapes at f = 1.00, 2.00, 2.50 & 3.33 Hz
96
Figure 3.24 Mode#2 (2-D) Diffracted Mode Shapes at f = 5.00, 6.67, 10.00 & 12.50 Hz
97
Figure 3.25 Mode#2 (3-D) Diffracted Mode Shapes at f = 5.00, 6.67, 10.00 & 12.50 Hz
98
Figure 3.26 Mode#3 Diffracted Mode Shapes at f = 1.43, 2.50, 3.12 & 3.57 Hz
99
Figure 3.27 Mode#3 Diffracted Mode Shapes at f = 5.00, 6.67, 10.00 & 11.11 Hz
100
Figure 3.28 Mode#4 Diffracted Mode Shapes at f = 2.00, 2.50, 3.33 & 5.00 Hz
101
Figure 3.29 Mode#4 Diffracted Mode Shapes at f = 5.88, 6.67, 9.09 & 10.00 Hz
102
Figure 3.30 Mode#5 Diffracted Mode Shapes at f = 2.00, 2.50, 3.12 & 5.00 Hz
103
Figure 3.31 Mode#5 Diffracted Mode Shapes at f = 5.88, 7.14, 9.09 & 10.00 Hz
104
Chapter 4
Diffraction Around an Irregular Layered Elastic
Media, II: Rayleigh and Body P, SV Waves
4.1 Introduction
This chapter is a continuation of our sequence of reports on waves in elastic layered
media:
Report I Synthetic Translational Motions of Surface Waves On or Below a Layered Media
Report II Diffraction Around an Irregular Layered Elastic Media, I: Love and Body SH
Waves - 1
Report III Diffraction Around an Irregular Layered Elastic Media, II: Rayleigh and Body P,
SV Waves - 2
which has now been summarized and appeared on two journal papers:
1) Lee, V.W., Liu, W.Y., Trifunac, M.D., Orbovi ć, N. Scattering and diffraction of
earthquake motions in irregular elastic layers, I: Love and SH waves, “Soil Dynamics
and Earthquake Engineering”, Volume 66, November 2014, Pages 125–134.
2) Lee, V.W., Liu, W.Y., Trifunac, M.D., Orbovi ć, N. Scattering and diffraction of
earthquake motions in irregular, elastic layers, II: Rayleigh and body P and SV waves,
“Soil Dynamics and Earthquake Engineering”, Volume 66, November 2014, Pages
220–230.
This chapter is part of and continuation of paper 2), we will consider the case of Rayleigh
and Body P-, SV-Waves on the same irregular layered elastic media. The method of weighted
residues, or the moment method, is again used in the analyses.
In this chapter, we describe a method for the computation of scattered and diffracted waves
for in-plane Rayleigh Surface Waves and Body P- and SV-Waves around a local irregularity in a
105
layered medium. By the method used and the nature of the model, the work presented here follows
closely the related study in which we examined the scattering and diffraction of Love Surface
Waves and Body SH Waves in Chapter 3.
A brief review of the related studies and of the weighted–residues method were presented
in Chapter 3, and it will not be repeated here. Instead, we will proceed straight to the presentation
of the present work, introducing additional references as we go along and as they are required by
the specific stages of the analysis.
106
4.2 Rayleigh and Body P, SV Waves On and Below the Surface of Elastic
Layered Media
We next consider again the case of an N -layered half space with Rayleigh and Body P,
SV Waves incident from the left. For each regular flat layer l , with l1, ,N , the P- and SV-
Wave potentials in the layer respectively take the form:
:
ll l l
,, ,
l
l
l
l
ik x a z
ll
ik x a z
ll
ik x b z
ll
ik x b z
ll
A
A
B
B
e
e
e
e
(4.1)
Figure 4.1: N-layered half-space with Rayleigh Waves
107
Those potentials represent respectively the upward and downward propagating waves in
the
th
l layer, ()
()
kk
c
is the horizontal wave number of the P- and SV-Waves at frequency
and phase velocity cc . The term
ikx
e , which is same in all layers, describes the horizontal
propagation of the waves, which together with the time-harmonic term
it
e
corresponds to waves
propagating in the ve x direction. The terms
l
ika z
e
for the P-Waves and
l
ikb z
e
for the SV-
Waves are, respectively, the corresponding vertical components of the waves. The ones with the
ve exponent are propagating upward ve y , and those with the ve exponent are
propagating downward ve y . Here,
ll l
aa ,c and
ll l
bb ,c are respectively defined
given by
l
l
l
l
1
1
1
2 22 2 2
2 2
l
l
1
1
1
2 22 2 2
2 2
l
l
kk
k
c
a1 1
k
k
kk
k
c
b1 1
k
k
(4.2)
so that
l
ka
and
l
kb
are the vertical wave numbers of the P- and SV-Waves in the
th
l layer of the
medium with longitudinal wave speed
l
a
and shear wave velocity
l
. In general, the wave
speeds increase as one goes down into the layers, so that
12 N1 N
, with the semi-
infinite half-space layer at the bottom having the highest shear wave speed . The same can be
said about the longitudinal wave speeds, so that
12 N1 N
, with
the longitudinal
wave speed of the semi-infinite medium furthest below being the highest. With cc the wave
speed of the surface Rayleigh Waves, c , and the surface waves take the form:
ik x az
ikx kaz
ik x bz
ikx kbz
AA
BB
ee
ee
(4.3)
108
where
1
1
22 2 1
2 2
2 2
1
1
1
22 2 2 2
2 2
kk
k
c
a1 1
k
k
kk
k
c
b1 1
k
k
(4.4)
and a , b are respectively the complements of
a
, b and both are real, so that the terms
kaz
e in
and
kbz
e in
both correspond to surface wave terms with amplitudes that are exponentially
decaying with depth below the surface.
With
l
the shear wave speed in the
th
l layer of the medium, and cc , the Rayleigh
Wave speed and also the (horizontal) phase velocity of the waves in each layer of the elastic media
above the half space (Figure 4.1), we can have
l
c
or
l
c
.
Next section, we will consider the case when the layers are not perfectly flat.
109
4.3 Rayleigh and Body P, SV Waves Incident on an Irregular Layered Elastic
Media
Figure 4.2 Irregular-Shaped N-layered half-space with Rayleigh Waves
Figure 4.2 here is a model where the parallel layers are not perfectly flat at some finite
region, while it is flat as most other parts. Without loss of generality, we will assume that each of
these irregular surface at each interface between the
th
l and 1
th
l layer, which extends from
x
to x can be represented as an empirical curve
()
l
zhx
at the interface. We assume
110
that this curve will be the flat surface
l
zH
almost everywhere, but will deviate from the flat
surface within some finite region where it is defined numerically by a set of points ,
ii
x z . The
same curve can also be represented in polar coordinates
ˆ
ˆ, r , with
ˆ
ˆˆ rr in a common
coordinate system
ˆ ˆ , x z with origin at some point
ˆ
O above the half space as shown.
As in the case of surface Love waves, with the surface Rayleigh waves potentials or Body
P, SV Wave potentials ,
ll
for P-waves, and ,
ll
for SV-Waves at each layer l incident on
these irregular surfaces, additional scattered wave potentials are generated, which can be
represented by
(1) (1) (2) (2)
,,
0
(1) (1) (2) (2)
,,
0
ˆ ˆ () ()cos
ˆˆ () ( )cos
ll
l l
s
llnn lnn
n
s
llnn lnn
n
AHkr A H kr n
B Hkr B H kr n
(4.5)
with both outgoing and incoming waves, for each layer 1, 2, , 1 lN , all except the last
semi-infinite layer. For the last semi-infinite layer, lN , the scattered wave,
s s
N
and
s s
N
respectively the P- and SV- potentials, take the form
(1)
,
0
(1)
,
0
ˆ ()cos
ˆ ()cos
l
l
ss
Nnn
n
ss
Nnn
n
A Hkr n
BH k r n
(4.6)
with only outgoing waves satisfying Somerfield’s radiation condition at infinity.
The scattered waves, together with the free-filed surface Rayleigh or Body P, SV Waves,
form the resultant waves in the layered media. Writing
ff
ll l
and
ff
ll l
respectively as the free-field surface Rayleigh or Body P, SV Waves in the
th
l media, the resultant
wave in the same media is
ffs
ll l
, and
ffs
ll l
, respectively for the P- and SV- Wave
potentials, which, together, must satisfy the following set of boundary conditions below (Lee and
Wu, 1994b).
111
1) On the half-space surface,
0
() yhx
, the resultant waves in the top layer (1) l must
together satisfy the half-space surface traction (stress) free boundary condition:
ˆˆ 0
rr
TTe Te
at
0
() y hx
(4.7)
or separating the contribution of Traction due to the free-field and scattered waves:
ffs
TT T
(4.8a)
gives, at
0
() yhx
:
s ff
rr
s ff
TT
TT
(4.8b)
2) For 1, 2, , 1 lN , at the interface between the
th
l
layer and the
(1)
th
l
layer below,
()
l
yhx
, the resultant waves in these two layers must satisfy the continuity of displacement
and stress at the interface, at
()
l
yhx
:
1
1
ll
ll
UU
TT
(4.9)
with
l
U
,
l
T
respectively the displacement and traction vector in layer l ,
1 l
U
,
1 l
T
the
corresponding ones in layer 1 l
. Separating the contribution of the displacement and traction
due to the free-field and scattered waves:
ffs
ll l
ffs
ll l
UU U
TT T
(4.10a)
gives, at
()
l
yhx
:
11
11
ssff ff
ll l l
s s ff ff
ll l l
UU U U
TT T T
(4.10b)
with the unknown wave functions on the left-hand side, the known functions on the right-hand
side. As in (4.5) above, both the displacement and traction vectors can at each layer be expressed
in the radial and angular components:
112
ˆ ˆ
ˆ ˆ
ll r l
r
ll r l
r
UU e U e
TT e T e
(4.11)
so that the displacement and traction vectors in (4.10b) can be separated into component forms, at
()
l
yhx
:
11
11
s s ff ff
ll l l
rr r r
ss ff ff
ll ll
UU UU
UU UU
(4.12a)
and
11
11
s s ff ff
ll l l
rr r r
ss ff ff
ll l l
TT T T
TT T T
(4.12b)
Here the radial and angular components of traction,
r
T
and
T
, can be expressed in terms of the
various components of stresses in cylindrical coordinates (Lee and Wu, 1994b):
cos sin
sin cos
rr r
r
T
T
(4.13)
where is the angle that the normal
ˆˆ ˆ ˆ ˆ cos sin
rr r
nne ne e e
makes with the radial
vector, as shown in Figure 4.3 (Lee and Wu, 1994a,b) here
Figure 4.3 Angle between radial vector and normal at a point
113
Expressions for the radial and angular components of the displacement vectors and stresses
can be expressed in terms of the corresponding wave potentials (Mow and Pao, 1973). They will
take the form, for the scattered wave potentials, at a point
ˆ ,(), rr on the irregular surface
(or at any point anywhere):
() () () ()
1,2 , 1 , 2
() () () ()
1,2 , 1 , 2
ˆ ˆ ( , ,) ( , ,)
ˆˆ (, , ) (, , )
l l
l l
sjj jj
lj lnr lnr
r
n
sjj jj
l j ln ln
n
UADnkrBDnkr
UADnkrBDnkr
(4.14)
where
()
1
ˆ (, , )
l
j
r
Dnkr
,
()
2
ˆ (, , )
l
j
r
Dnkr
are the corresponding radial displacements from the P-
and SV- scattered wave potentials, and
()
1
ˆ (, , )
l
j
Dnkr
,
()
2
ˆ (, , )
l
j
Dnkr
are their
corresponding angular displacements.
Similarly, the traction components at a point on the irregular surface
ˆ ,(), rr
with normal
ˆˆ ˆ ˆ ˆ cos sin
rr r
nne ne e e
take the form:
() () () ()
1,2 , 1 , 2
() () ( ) ( )
1,2 , 1 , 2
ˆˆˆˆ (, , , ) ( , , , )
ˆˆ ˆ ˆ (, , , ) ( , , , )
l l
l l
sjj jj
ljlnr lnr
r
n
sjj jj
ljln ln
n
TA nkrnB nkrn
TA nkrnB nkrn
TT
TT
(4.15)
where
() ()
11
ˆˆ (, , , )
l
jj
rr
nk r n
TT ,
() ( )
22
ˆˆ (, , , )
l
jj
rr
nk r n
TT are the corresponding radial
components of traction from the P- and SV- scattered wave potentials. Similarly
() ( )
11
ˆˆ (, , , )
l
jj
nk r n
TT ,
() ( )
22
ˆˆ (, , , )
l
jj
nk r n
TT are their corresponding angular stresses.
Using the weighted residue method, as in the case of Love Waves, the boundary
conditions at each layer take the form:
114
1) Starting on the top layer, the half space surface, from (4.8b) and (4.15):
1 1
1 1
( ) () () ()
11,2 1,1
1,2
0
( ) () () ()
11,2 1,1
1,2
0
ˆˆ ˆ ˆ (, , , ), (, , , ), ,
ˆˆ ˆ ˆ (, , , ), (, , , ), ,
jjjjff
rmnr mn m
r
j
n
jjjjff
mn mn m
j
n
nk r n w A nk r n w B T w
nk r n w A nk r n w B T w
TT
TT
(4.16)
2) For the interface between the
th
l and
(1)
th
l
layer, for 1, 2, 3 l from (4.12a) and
(4.14) for displacements:
1 1
() () ( ) ()
1,2 ,
1,2
0
() () () ()
11,2 1,
1
ˆˆ (, , ), ( , , ),
ˆ ˆ (, , ), ( , , ),
,
l l
l l
jjj j
r m ln r m ln
j
n
jjj j
r m ln r m ln
ff ff
ll m
rr
D nkr w A D nkr w B
Dnk r w A D nk r w B
UU w
(4.17a)
1 1
( ) () () ()
1,2 ,
1,2
0
() ( ) () ()
11,2 1,
1
ˆˆ (, , ), ( , , ),
ˆˆ (, , ), ( , , ),
,
l l
l l
jjj j
mln m ln
j
n
jjjj
ml n m l n
ff ff
ll m
Dnkr w A Dnkr w B
Dnk r w A Dnk r w B
UU w
(4.17b)
and from (4.12b) and (4.15) for stresses:
1 1
() () ( ) ()
1,2 ,
1,2
0
() () () ()
11,2 1,
1
ˆˆ ˆ ˆ (, , , ), (, , , ),
ˆ ˆ ˆ ˆ ( , ,, ), ( , ,, ),
,
l l
l l
jjjj
r m ln r m ln
j
n
jjjj
rmlnrmln
ff ff
ll m
rr
nkrnw A nkrnw B
nk rnw A nk rnw B
TT w
TT
TT
(4.18a)
115
() ( ) () ()
1,2 ,
1,2
0
1 1
() ( ) () ( )
12 1, 1,
1
ˆˆ ˆˆ (, , , ), (, , , ),
ˆ ˆ ˆ ˆ (, , , ), (, , , ),
,
l l
jjjj
mln m ln
j
n
l l
jjj j
mm ln l n
ff ff
m ll
nk r n w A nk r n w B
nk r n w A nk r n w B
TT w
TT
TT
(4.18b)
In matrix form, (4.17a), (4.17b), (4.18a), (4.18b) take the form:
() () () ()
12 1 2
() () () ()
12 2 2
() () () ()
1,2
12 1 2
() ()
12
(, ,) (, ,) ( 1, , ) ( 1, , )
(, ,) (, ,) ( 1, , ) ( 1, , )
(, , ) (, , ) ( 1, , ) ( 1, , )
(, , ) (
jj j j
rr r r
jj j j
jj j j
j
rr r r
jj
D l mn D l mn D l mn D l mn
D l mn D l mn D l mn D l mn
Tlmn T lmn Tl mn Tl mn
Tlmn T l
()
,
()
,
()
0 1,
() () ()
1, 12
1
1
1
1
,, ) ( 1, , ) ( 1, , )
,
,
,
,
j
ln
j
ln
j
n ln
j jj
ln
ff ff
ll m
rr
ff ff
ll m
ff ff
ll m
rr
ff ff
ll m
A
B
A
B mn Tl mn Tl mn
UU w
UU w
TT w
TT w
(4.19)
with properly defined matrix elements
()
1
(, , )
j
r
D lmn
,… and
()
1
(, , )
j
r
Tlmn
,… and so on.
The traction components of motions at each interface are computed from the corresponding
strain components of motions, which are available form Report III: “Synthetic Rotational Motions of
Surface Waves On or Below a Layered Media”.
Note that for the interface between layer 1 N and the (last) semi-infinite layer N , (4.17a),
(4.17b) and (4.18a), (4.18b) will have only the outgoing wave terms
(1) (1)
,,
,, 0,1,2
Nn N n
AB n and
without the incoming wave terms
(2) (2)
,,
,, 0,1,2
Nn N n
AB n
116
4.4 Numerical Implementation
This section will demonstrate the numerical procedure used to apply to the system of
complex equations derived in the last section, which used the method of weighted residues. In
studying the set of equations, it is observed that for an elastic half-space with
(1) N
layers,
there are
(1) N
interfaces in-between the layers, plus the topmost half-space with no elastic
medium above.
At this topmost half-space surface, there is one set of zero-stress equations (at 0 z ) (4.7)
involving the wave coefficients on the topmost first layer. At each of the (1) N interfaces in-
between the layers, there are each two sets of the stress and displacement continuity equations
(4.8b) and (4.9). There is thus a total of 2( 1) 1 2 1 NN set of equations. Each of the top
(1) N layers has two sets of waves, the upward and downward going, or the outgoing and
incoming waves (4.5). They are represented by Hankel Functions of the first and second kind, the
outgoing and incoming waves, respectively. On the other hand, the bottom-most semi-infinite
layer has only one set of waves, the downward going or outgoing waves (4.6). They are
represented by Hankel Functions of the first kind. This gives a total of 2( 1) 1 2 1 NN
set of waves from all the layers.
In summary, we have (2 1) N set of equations for (2 1) N set of waves. If each of
such (2 1) N set of waves has M terms, with M unknown coefficients, we will have a total of
(2 1) M N unknowns we need to solve. Ideally each set of waves has, M , or infinite
number of terms and there are (2 1) N number of equations. In reality, we have to truncate
to finitely many M terms for each set of waves, with the number M often dependent on the
frequency of the waves and the complexity of the equations. With M finite, we have a task of
solving the (2 1) M N set of complex equations for the (2 1) M N set of unknowns.
Numerically, we know that the Hankel Functions of both the first and second kinds are
both complex, and increase in magnitudes with increasing order. Thus if all the (2 1) M N
unknowns are put together in the (2 1) M N complex equations, one would have a very, very
117
large set of complex equations to solve, whose terms often have increasing magnitudes with
increasing M . It would be a numerically nightmare to solve them all at once.
Since each set of equations at each interface involves only waves at each side of the
interface, a simple, elegant numerical algorithm can be derived to allow each set of wave
coefficients at each media to be solved separately, making the problem much simple numerically.
In other words, we are solving M complex equation in M unknowns at each step. The following
is a comprehensive description of the numerical procedure.
Starting from the top surface, where 0 z , we have the matrix equation for the zero-stress
boundary condition along the whole (regular and irregular) surface of the half-space, in the form:
(1) * (2)
11 1 1 1
0
0
z
EC E C e
(4.20a)
where from (4.17b):
(1)
11
EE
is the matrix with elements defining the stress terms from the P- and SV- scattered
waves in the top layer with the Hankel Functions
(1)
1 n
H kr
and
(1)
1 n
H kr
,
*(2) (1)
11 1
E E conjug E
is the matrix corresponding to the stress terms in the top layer with
the Hankel Functions
(2)
1 n
H kr
and
(2)
1 n
H kr
,
(1)
1
C ,
(2)
1
C are respectively the two vectors
(1) (1) (1) (1) (1) (1)
10 10 11 11 12 12
,,,,,,
T
AB AB AB and
(2) (2) (2) (2) (2) (2)
10 10 11 11 12 12
,,,,,,
T
AB AB AB in the top layer, and
(4.20b)
1
e
is the vector of the free-field stresses in the top layer.
As in the case of Love and SH Body waves, eliminating
(2)
1
C
, one has, at 0 z :
1
(2) * (1)
11 11 1
0
1
*(1)
11 1 1
(0) (0) (0)
z
CE EC e
EE C e
(4.21)
118
In the interface between the first and second layer, the continuity equations take the form:
or
1
1
(1) * (2)
11 1 1 1
(1) * (2)
22 2 2 2
(1) * (2)
11 1 1 1 1 1 1
(1) * (2)
21 2 2 1 2 21
zh
zh
EC E C e
EC E C e
Eh C E h C e h
Eh C E h C e h
(4.22)
for stress continuity. Here
(1)
2
C ,
(2)
2
C are respectively the two corresponding vectors
(1) (1) (1) (1) (1) (1)
20 20 21 21 22 22
,,,,,,
T
AB AB AB
and
(2) (2) (2) (2) (2) (2)
20 20 21 21 22 22
,,,,,,
T
AB AB AB
in
the second layer, and
(1) * (2)
22 2 2
, EE E E
are respectively the matrices with elements
defining the stress terms from the P- and SV- scattered waves in the second layer with the Hankel
Functions of the first and second kinds, corresponding to the coefficients
(1)
2
C ,
(2)
2
C just as
how they are defined in the top first layer.
Similarly, for the displacement continuity equations between the first and second layer:
(1) * (2)
11 1 1 1 1 1 1
(1) * (2)
21 2 2 1 2 21
Dh C D h C d h
Dh C D h C d h
(4.23)
Using (4.21),
(2)
1
C
can be eliminated from the L.H.S. of the stress (4.22):
(1) * (2)
11 1 1 1 1 1 1
1
(1) * * (1)
11 1 1 1 1 1 1 1 1 1
(1)
11 1 1 1
() () ()
() () (0) (0) (0) ( )
() ()
Eh C E h C e h
Eh C E h E E C e e h
hC h
Ee
(4.24a)
where
1
**
11 11 1 1 1 1
1
**
11 11 1 1 1 1
() () () (0) (0)
() () () (0) (0)
hEhEh E E
heh Eh E e
E
e
(4.24b)
119
In exactly the same way,
(2)
1
C
can be eliminated from the L.H.S. of the displacement (4.23),
resulting in:
(1) * (2)
11 1 1 1 1 1 1
1
(1) * * (1)
11 1 1 1 1 1 1 1 1 1
(1)
11 1 1 1
() () ()
() () (0) (0) (0) ( )
() ()
Dh C D h C d h
Dh C D h E E C e d h
hA h
Dd
(4.25a)
where
1
**
11 11 1 1 1 1
1
**
11 11 1 1 1 1
() () () (0) (0)
() () () (0) (0)
hDhDh E E
hdh Dh E e
D
d
(4.25b)
The continuity matrix equations at the interface of the first and second layer now takes the
simplified form, without the
(2)
1
C terms:
(1)
11 1 1 1
(1) * (2)
21 2 2 1 2 21
() () hC h
E hC E h C eh
Ee
(4.26a)
(1)
11 1 1 1
(1) * (2)
21 2 2 1 2 21
() () hC h
Dh C D h C d h
Dd
(4.26b)
In matrix form, (4.25a) and (4.25b) become:
* (1)
11 2 1
11 21 2 1 (1) 2
1 * (2)
11 21 2 1 2 11 2 1
()
()
()
()
he h
h Eh E h C
C
h Dh D h C
hd h
e
E
D
d
(4.27)
which shows that both
(1)
2
C and
(2)
2
C of the coefficients of the waves in the second layer
can be expressed in terms of the single set of
(1)
1
C coefficients of waves in the top layer. One
can thus write:
* (1)
11 2 1
11 21 2 1 (1) 2
1 * (2)
11 21 2 1 2 11 2 1
1
()
()
()
()
he h
h Eh E h C
C
h Dh D h C hd h
e
E
D
d
(4.28a)
120
so that such a transformation is obtained:
(1)
21 1 21 1 (1) 2
1
(2)
22 1 2
22 1
() ()
()
()
h h C
C
h C
h
t T
T
t
(4.28b)
Repeating this iteration at each interface of layers proceeding downwards one can start with the
continuity equations at interface between layer l and (l+1), as in (4.24a), (4.24b), at the interface
l
zh
:
(1) * (2)
(1) * (2)
11 11 1 1
ll l l l l l l
ll l l l l ll
Eh C E h C e h
Eh C E h C e h
(4.29a)
(1) * (2)
(1) * (2)
11 1 1 1
ll l l l l l l
ll l l l l ll
Dh C D h C d h
Dh C D h C d h
(4.29b)
As in (4.23b), suppose by induction,
(1)
l
C
and
(2)
l
C
are to be expressed in terms of
(1)
1
C
:
(1)
1 1 (1)
1
(2)
2
2
() ()
()
()
ll ll l
ll l
ll
h h C
C
h C
h
t T
T
t
(4.30)
from which (4.24a), (4.24b) can be expressed as:
(1)
1
(1) * (2)
11 1 1 1
() ()
ll ll
ll l l l l l l
hC h
E hC E h C e h
Ee
(4.31a)
(1)
1
(1) (2)
11 1 1 11
() ()
ll l l
ll l l l l l
hC h
Dh C D h C d h
Dd
(4.31b)
or again in matrix form, as in (4.23a):
* (1)
1
11 (1) 1
1 * (2)
11 1
1
()
()
()
()
ll l l
ll ll ll l
ll ll L l l
ll l l
he h
h Eh Eh C
C
h Dh D h C
hd h
e
E
D
d
(4.32a)
121
resulting in a transformation, as in (4.23b), expressing the coefficients
(1)
1 l
C
and
(2)
1 l
C
for
layer (l+1) to be expressed in terms of
(1)
1
C
from waves of the top layer.
(1)
1,1 1,1 (1) 1
1
(2)
1,2
1 1,2
() ()
()
()
ll ll
l
ll
l ll
h h
C
C
h
C h
t T
T
t
(4.32b)
The reason why we select to have the coefficients of all waves in each layer expressed in terms of
(one of the two) a set of coefficients of waves n the top layer is because, we know, for surface
waves, and possibly Body Waves, the top layer waves are more dominant.
The above described procedure will continue until the interface with the bottom semi-
infinite layer lN is reached, namely at
1
N
zh
. There the continuity equations will be different
from those in the previous interfaces, since the bottom semi-infinite layer will now have only one
set of waves, namely, the downward, outgoing waves. The stress and displacement continuity
equations now take the form, as in (4.24a), (4.24b), with 1, 1 lN l N :
(1) * (2)
11 1 1 1 1 11
(1)
11
NN N N N N N N
NN N N N
Eh C E h C e h
Eh C e h
(4.33a)
(1) * (2)
11 1 1 1 1 11
(1)
11
NN N N N N NN
NN N N N
Dh C D h C d h
Dh C d h
(4.33b)
As before, the coefficients
(1)
1 N
A
and
(2)
1 N
A
of the waves in the
(1)
th
N
layer can be expressed
in terms of
(1)
1
A
, that of the waves in the top layer
(1)
11 1 1 1
(1)
11
() ()
NN NN
NN N N N
hC h
Eh C e h
Ee
(4.34a)
(1)
11 1 1 1
(1)
11
() ()
NN N N
NN N N N
hC h
Dh C d h
Dd
(4.34b)
122
with new matrices
11
()
NN
h
E ,
11
()
NN
h
D and vectors
11
()
NN
h
e
,
11
()
NN
h
d
in
terms of
(1)
1
C
, the wave coefficients of the top layer, on the L.H.S. of the equations.
(4.27a) and (4.27b) are the final form of the matrix equations, from which the wave
coefficients
(1)
1
C of the top layer and
(1)
N
C in the bottom semi-infinite layer are to be related.
Normally, one will rewrite (2.27a) , expressing
(1)
N
C
in terms of
(1)
1
C
, as:
1
(1) (1)
11 1 1 1 1 1
() ()
NNN N N N N NN
CEh h C h eh
Ee
(4.35a)
and substituting it into (4.27b), which becomes a matrix equation with
(1)
1
C of the top layer as
the only set of unknowns:
1
(1)
11 1 1 1 1 1
1
11 11 1 11
() ()
() ()
N N NN NN N N
NN NN N N NN N N
hDh Eh h C
Dh E h h d h h
DE
ed
(4.35b)
from which the coefficients
(1)
1
C can now be evaluated, after which the coefficients of the
waves in all layers can be found.
123
4.5 The Diffracted Mode Shapes of Rayleigh and P, SV Body Waves
4.5.1 The Input Free-Field Waves
Recall from Report II where we considered an N - Layered half-space with Rayleigh Waves
or Body P, SV Waves incident from the left over a regular parallel layered media (Report II, Figure
2.1). In a sub-section of Chapter 4 in Report II there we plotted the displacement mode shapes of
Mode#1 to Mode#5 Rayleigh Surface Waves for a selected range of periods starting from 15 sec
down to 0.04 sec (0.07Hz to 25Hz). This is followed by another sub-section of the mode shapes
resulting from body P, SV Waves for a given incident angle. We actually take the body P, SV
Waves as the eleventh and twelfth modes of waves at the parallel layered media.
Each of these mode shapes at each period (frequency) will be used as the free-field input
waves onto the irregular parallel layered media to be studies here. In what follows, we will take a
rather simple two-layered media to illustrate the process, as in the case of Love and body SH
Waves. The following parameters are used for the two-layered media:
Layer
Thickness
(km)
P-Wave Speed
( α, km/s)
S-Wave Speed
( β, km/s)
Density
( ρ, gm/cc)
1 1.38 1.70 0.98 1.28
2 ∞ 6.40 3.70 2.71
Table 4.1 Two-Layer Velocity Model
They are the same first and last layers of the six-layered model given in Report II, except
the top layer is now 1.38 km thick. The computer Program, “Haskel.exe”, is again used to calculate
the phase velocities of each mode of Love Waves, as in Report II, in the period range of periods
from 14.0 sec down to 0.04 sec for a total of 91 discrete period values.
Figure 4.4 is the input free-field mode shapes for Mode#1 Rayleigh Waves for both the
horizontal (x-) and vertical (z-) components of motions at four selected period, T = 5.0, 1.5, 0.5
and 0.15 sec, or at frequency, f = 0.20, 0.67, 2.0 and 6.67 Hz. As before, for Mode#1 waves, the
mode shapes are again available at all 91 pre-selected period values. It is noted from the figures
124
that, at all the periods shown, the wave amplitudes of the mode shapes are scaled to have a
maximum amplitude of 1 at the half-space surface, and that the amplitudes are practically zero at
and below the interface of the two layers, which is at 1.38km apart.
Figure 4.5 is the input free-field mode shapes for the corresponding Mode#2 Rayleigh
Waves for both the horizontal (x-) and vertical (z-) components of motions at four selected period,
T = 3.0, 1.5, 0.3 and 0.15 sec or at frequency, f = 0.33, 0.67, 3.33 and 6.67 Hz. For Mode#2
Rayleigh Waves, the modes are available not at all 91 periods, but instead from T = 4.6 sec
( Period#18, down from T=14.0 sec for Mode#1). So the four periods plotted here are selected
from a narrower range. As before, the mode shapes are scaled to have a maximum amplitude of 1
starting from the surface 0 z . A characteristic of the Mode#2 Love Waves is that the mode shape
at each period now started to decrease to have negative values, reaches a negative minimum before
asymptotically approaching zero. Again, as in Mode#1, the amplitudes are practically zero at and
below the interface of the two layers ( 1.38km) z .
Figure 4.6 is the input free-field mode shapes for the corresponding Mode#3 Rayleigh
Waves for both the horizontal (x-) and vertical (z-) components of motions at four selected period,
T = 2.4, 1.2, 0.24 and 0.15 sec or at frequency, f = 0.42, 0.83, 4.17 and 6.67 Hz. For Mode#3
Rayleigh Waves, the modes are available again not at all 91 periods, but instead from T = 2.6 sec
( Period#28, down from T=14.0 sec for Mode#1). So the four periods plotted here are again
selected from a narrower range. As before, the mode shapes are scaled to have a maximum
amplitude of 1 starting from the surface 0 z .
125
Figure 4.4 Mode#1 Rayleigh Mode Shapes at f = 0.20, 0.67, 2.00 and 6.67 Hz
126
Figure 4.5 Mode#2 Rayleigh Mode Shapes at f = 0.33, 0.67, 3.33 and 6.67 Hz
127
Figure 4.6 Mode#3 Rayleigh Mode Shapes at f = 0.42, 0.83, 4.17 and 6.67 Hz
128
A characteristic of the Mode#2 and #3 Rayleigh Waves is that the mode shapes for both
the horizontal and vertical components of motions at each period now started to decrease to have
negative values, reaches a negative minimum, then cross the zero axis back to have positive
amplitudes, reaches a positive maximum, before asymptotically approaching zero. Again, as in
Mode#1 and Mode#2, the amplitudes are practically zero at and below the interface of the two
layers(1.38km) z .
We will expect the scattered and diffracted waves to be small at the interface. The same
observations and conclusions can be made for Mode #4 and #5 free-field Rayleigh Waves. The
plots have been omitted here.
Finally, the free-field mode shapes for incident P, SV Body Waves are plotted next. Figure
4.7 is the input free-field mode shapes for the corresponding P- and SV- Body Waves with angle
of incidence of
60
o
with respect to the horizontal direction at two selected period, T = 1.0,
0.3sec., or at frequency, f = 1.0, 3.33 Hz. The top two graphs are for incident P-Waves, while the
bottom two graphs are for incident SV-Waves. As in the case of Mode#1 Rayleigh Waves, the
mode shapes are available at all 91 pre-selected period values for both incident P- and SV- Body
Waves. It is noted from the figures that, at the periods shown, the wave amplitudes of the mode
shapes are again scaled to have maximum amplitude of 1 at the half-space surface. Unlike the
Rayleigh Surface Waves, depending on the period of the body waves, the waves do not depreciate
to zero as you go deeper down from the surface. The waves in the first layer are harmonic,
oscillating at the given period of the waves between the scaled maximum amplitude of 1 , and
the oscillation increases with decreasing period, or increasing frequency, as the figure shows.
Below the first layer, on the semi-infinite layer, it continues to oscillate, though at a higher
period and lower amplitude, but they are not depreciating to zero, since the body waves are
harmonic waves, harmonic in both the horizontal x- and vertical z- directions.
We thus will expect in the next sub-section that the diffracted waves from P, SV Body
Waves will behave differently from those from the Rayleigh Surface Waves.
129
Figure 4.7 Mode#11,12 - Incident P, SV Body Waves Mode Shapes at f = 1.00, 3.33 Hz
130
4.5.2 The Diffracted Mode Shapes
We next consider the case of irregularly shaped layered media superimposed onto the
parallel two-layered media studied here. We select here the same irregular layered media model
as that for Love Surface Waves and Body SH Waves. It is a shallow, “almost-flat” ellipse with a
ratio of “Vertical minor axis / Horizontal major axis = 0.1” at both the half-space surface and the
interface of the two layers. The half-width or radius of the horizontal major axis is taken to be 1.0
km long for both, or a major diameter of 2.0 km long. Thus the half-width or radius of the minor
axis is 0.1 km deep, the case of a very shallow ellipse. Figure 4.8 is a sketch of such two-layered
elastic media for cases of
Figure 4.8 The Irregular Two-Lyered Media with (i) Rayleigh Waves and (ii) P, SV Waves
The next twelve figures, Figure 4.9 to 4.20 are respectively the diffracted mode shapes for
Mode #1 to Mode#5 of Rayleigh Waves at the same two-layered irregular media. Each mode have
mode shapes plotted at four selected frequencies, at f = 13.33, 18.18, 20.0 & 25.0 Hz. These are
frequencies below period of 0.1 sec (or frequency beyond 10Hz) as it was found that since the
irregular part of the layer are almost flat, the long period waves overlooked the irregularities and
the diffracted waves resulted are small and insignificant.
131
Figure 4.9 and 4.10 are the diffracted mode shapes respectively for the horizontal x-
component and vertical z-component motions of Mode#1 Rayleigh Waves at four selected periods
of T=4.0, 1.0, 0.5 0.30 sec, or frequencies of f = 0.25, 1.0, 2.0 & 3.33 Hz. With the irregular
almost-flat elliptic surfaces from 10 km x . to 10 km . x on both the half-space surface
0 km z and the surface of interface of the two media 1.38 km z , the diffracted mode shapes
are plotted at equally-spaced intervals along x from 2.0 km x to 2.0 km x at 01 km . apart.
The dashed line on the left side of each graph represents the input free-field mode shapes
propagating along the parallel-layered media from the left arriving at the irregular surfaces. The
shapes of the elliptic canyon on top and at the interface are also plotted with dashed lines, showing
how they have deformed from the regular elliptic shape.
For all periods above 1 sec. or for frequencies up to almost 1.0 Hz, the waves are unaffected
by the almost-flat irregular surfaces, as shown by the case of T=4.0s or f = 0.25Hz in the two
figures (4.9) and (4.10). At the irregular interfaces between the two media, as pointed out in the
previous section, as can also be seen by the dashed line on the left, the free-filed incoming mode
shape amplitudes are already almost zero. That is thus not much diffraction to be expected.
For periods below T=0.3s, or for frequencies above f = 3.33 Hz, the amplitudes of x-
component of Mode#1 Rayleigh Waves depreciate so fast that the waves are practically zero not
far below the half-space surface, and hence there is highly any diffracted waves resulted.
132
Figure 4.9 Mode#1 x-comp. Diffracted Mode Shapes at f = 0.25, 1.25, 2.00 & 3.13Hz
133
Figure 4.10 Mode#1 z-comp. Diffracted Mode Shapes at f = 0.25, 1.25, 2.00 & 3.13Hz
134
The cases for the higher modes of Rayleigh Waves are much different, however. Figures
4.11 through 4.20 are the corresponding diffracted mode shapes for Mode#2 to Mode#5 Rayleigh
Waves at the similar but extended four selected frequencies.
As stated above, the higher mode Rayleigh Waves, unlike the Mode#1 waves, are more
oscillatory in the top layer before decaying to almost zero at the interface with the second layer.
As the mode number increases, as observed in the previous section, both the mode shapes of the
input free-field and that of the diffracted Rayleigh Waves here went through the (M - 1) times of
sign changes as they go down vertically from the top surface, where “M” is the mode number.
Thus the diffracted mode shapes are also more oscillatory at the top layer. This behavior increases
with increasing frequency, unlike the mode#1 Rayleigh Waves. Further, such diffraction behavior
also gets more complex, for both the horizontal x- and vertical z-component motions, as the mode
number increases from Figure 4.9 through 4.20.
At present, we are able to do the calculations of the diffracted waves for the Rayleigh
Waves mode#2 to 5 up to f = 6.67Hz (period# 61, T = 0.15s). Calculations on frequencies beyond
this, up to the highest frequency of f = 25.0Hz (period# 91, T = 0.04s), will need more work on
the numerical procedure to ensure meaningful, computational results. This is what we are currently
working on.
We will then next work on multi-layered media with the layers being closer together. It
will be expected that these diffracted patterns will be more complicated as the number of layers
increase, and as the layers get closer and closer together.
135
Figure 4.11 Mode#2 (2-D) x-comp. Diffracted Mode Shapes at f = 1.25, 2.38, 3.57 & 7.14Hz
136
Figure 4.12 Mode#2 (3-D) x-comp. Diffracted Mode Shapes at f = 1.25, 2.38, 3.57 & 7.14Hz
137
Figure 4.13 Mode#2 (2-D) z-comp. Diffracted Mode Shapes at f = 1.25, 2.38, 3.57 & 7.14Hz
138
Figure 4.14 Mode#2 (3-D) z-comp. Diffracted Mode Shapes at f = 1.25, 2.38, 3.57 & 7.14Hz
139
Figure 4.15 Mode#3 x-comp. Diffracted Mode Shapes at f = 1.00, 2.08, 3.57 & 7.14Hz
140
Figure 4.16 Mode#3 z-comp. Diffracted Mode Shapes at f = 1.00, 2.08, 3.57 & 7.14Hz
141
Figure 4.17 Mode#4 x-comp. Diffracted Mode Shapes at f = 1.25, 2.08, 4.17 & 7.14Hz
142
Figure 4.18 Mode#4 z-comp. Diffracted Mode Shapes at f = 1.25, 2.08, 4.17 & 7.14Hz
143
Figure 4.19 Mode#5 x-comp. Diffracted Mode Shapes at f = 1.33, 2.00, 3.33 & 4.17Hz
144
Figure 4.20 Mode#5 z-comp. Diffracted Mode Shapes at f = 1.33, 2.00, 3.33 & 4.17Hz
145
Figure 4.21 Mode#11 P-Body Waves Diffracted Mode Shapes at f = 1.25 & 5.26Hz
146
Figure 4.22 Mode#12 SV-Body Waves Diffracted Mode Shapes at f = 1.25 & 5.56Hz
147
Finally, Figure 4.21 and 4.22 are respectively the graphs of the horizontal x- and vertical
z-component motions of diffracted mode shapes corresponding to incident plane P- and SV- Body
Waves. In each figure, the top two graphs are respectively for incident P- and incident SV-Waves
at period of T=1sec, or frequency of f=1Hz. The bottom two graphs are respectively for incident
P- and incident SV-Waves at period of T=0.3sec, or frequency of f=3.33Hz. As seen from Figure
4.7 above, the input free-field mode shapes of the Body P- and SV-Waves, unlike the Rayleigh
Waves, are more oscillatory in both layers, without depreciating to zero at the interface. The
oscillatory nature also increases significantly with frequencies.
Figure 4.21 and 4.22 as with the cases of the mode shapes of Rayleigh Waves in the
previous figures (Figures 4.9 to 4.20), the irregular almost-flat elliptic surfaces are from
1.0 km x to 1.0 km x on both the half-space surface 0 km z and the surface of
interface of the two media 1.38 km z , the diffracted mode shapes are again plotted at equally-
spaced intervals along x from 2.0 km x to 2.0 km x at 01 km . apart. Here, as before, the
dashed line on the left side of each graph again represents the input free-field mode shapes of Body
P- or SV- Waves propagating at the parallel-layered media from below arriving at the irregular
surfaces. The angle of incidence of the body waves is 60
o
with respect to the horizontal. At all
frequencies, the mode shapes of the diffracted waves are very oscillatory and are of very different
amplitudes than those of the incident free-field waves.
At the frequency of f=3.33Hz, it is seen that the diffracted waves are already highly
oscillatory and of large amplification relative to the free-field mode shapes. The amplification is
indeed much greater than two, as previously observed for body SH-Waves.
At present, as with the case of Rayleigh Waves, we are able to do the calculations of the
diffracted waves for the incident P- and SV-Body Waves up to f = 6.67Hz (period# 61, T = 0.15s).
Calculations on frequencies beyond this, up to the highest frequency of f = 25.0Hz (period# 91,
T = 0.04s).
148
4.6 Generalization to More Complex Layered Interfaces
4.6.1 More General Irregular Layered Media
Figure 4.23 Two Origins for Lth Layer
We define two coordinates from Figure 4.23
,
L LL
Or above Lth - interface and
,
L LL
Or below L1th - interface with a distance of
L
D between two origins: the potentials
of
L
and
L
with respect to
L
O , the potentials of
L
and
L
with respect to
L
O . At Lth -
interface
L1
,
L1
and
L 1
,
L 1
from L1th layer,
L
,
L
and
L
,
L
at Lth layer.
149
Use
L
O as the common origin,
L
O to
L
O . For waves at 1 Lth - interface,
L 1
,
L1
from
1 L
O
to
L
O ,
L1
,
L1
from
1 L
O
to
L
O from Figure 4.24.
Figure 4.24 Two Origins for (L-1)th Layer
150
4.6.2 The Case of a Two-Layer Canyon-Hill Interface
Next consider the case of irregularly shaped layered media superimposed onto the canyon-
hill interface studied here. We select a shallow, “almost-flat” ellipse with a ratio of “Vertical
minor axis/Horizontal major axis = 0.1” at both the half-space surface and the interface of the two
layers. The half-width or radius of the horizontal major axis is taken to be 1.0 km long for both,
or a major diameter of 2.0 km long. Thus the half-width or radius of the minor axis is 0.1 km deep,
the case of a very shallow ellipse. Figure 4.25 is a sketch of such a two-layered elastic media for
cases of incident Rayleigh Body waves; this is a simple example of the general cases of Figure
4.23.
Figure 4.25 The Irregular Two-Layered Media with Rayleigh Body Waves
The next twelve figures, Figure 4.26, 4.27 to 4.36, 4.37 are respectively the diffracted mode
shapes respectively for Mode #1 to Mode#5 of Rayleigh Waves at the same two-layered irregular
media. Each mode has mode shapes plotted at eight selected frequencies, for example from Figure
4.26 in Mode #1 (horizontal, x-) at f = 2.00, 3.33, 6.67 & 13.33 Hz and Figure 4.27 in Mode #1
(vertical, z-) at f = 2.00, 3.33, 6.67 & 13.33. These are frequencies below period of 1.0 sec as it
was found that since the irregular part of the layer are almost flat, the long period waves overlooked
the irregularities and the diffracted waves resulted are small and insignificant.
Figure 4.26 and 4.27 is the diffracted mode shapes for Mode#1 Rayleigh Waves at the
above stated eight selected frequencies. With the irregular almost-flat elliptic surfaces from
1.0 km x to 1.0 km x on both the half-space surface 0 km z and the surface of
interface of the two media 1.38 km z , the diffracted mode shapes are plotted at equally-spaced
151
intervals along x
from 2.0 km x to 2.0 km x at 0.1 km apart. The dashed line on the left
side of each graph represents the input free-field mode shapes propagating along the parallel-
layered media from the left arriving at the irregular surfaces. Figures 4.28, 4.29 through 4.36, 4.37
are the corresponding diffracted mode shapes for Mode#2 to Mode#5 Rayleigh Waves at the eight
selected frequencies.
It is worthwhile again to compare these ten figures of the canyon-hill model for Rayleigh
Waves to these in Figure 4.9, 4.10 to Figure 4.19, 4.20 of the canyon-canyon model in previous
section. Figure 4.9, 4.10 are respectively the horizontal (x-) and vertical (z-) components of the
diffracted mode shapes for Mode#1 Rayleigh Waves at selected frequencies for the canyon-canyon
model. Recall that, as in Chapter 3 of Love Waves, the waves at the canyon-canyon model were
also computed using only one origin on top. The waves in the present canyon-hill model, however,
were computed with multiple origins at each layer, with the origins defined both on top and below
the interfaces of the layers, respectively for waves diffracted for the top and bottom interfaces. It
is seen that the waves in Figure 4.26, 4.27 for Mode#1 of the canyon-hill model are different and
characteristic of the model studied, for all four frequencies presented.
Next Figure 4.28, 4.29 through 4.35, 4.36 are the corresponding horizontal (x-) and vertical
(z-) components of diffracted mode shapes for Mode#2 to Mode#5 Rayleigh Waves, with four
graphs for the x-components, four graphs for the z-components on two pages, one graph per
frequency. The dashed lines on the left of each graph are again the input “free-field” Rayleigh
Waves. The new model with multiple origins used for each set of waves resulted again as a more
realistic of the interactive natures of the waves.
The same observations can be made here for the five modes as that of Love Waves on the
previous chapter. As frequencies increases, both components of waves get more complicated.
Also as the mode number increases (from Mode#1 to Mode#5), the input Rayleigh Waves, as with
Love Waves, have more zero nodes (the number of zero nodes being are less than the mode
number). The resultant waves again show significant interaction to the left and right (+ve and -ve)
of the zero nodes. Mode#5 waves, as expected, are thus much more complicated that Mode#4
152
waves, and so forth down to Mode#1. Note that, as the mode number increases, their relative
amplitudes and contribution to the total waves decrease.
Again, these contributions are consistent with the results of the canyon-canyon model
earlier in this chapter. As the mode number increases, as observed in the previous section, both
the mode shapes of the input free-field and that of the diffracted Love Waves here went through
the 1 M times of sign changes as they go down vertically from the top surface, where "" M is
the mode number. The diffraction are also more complex as the mode number increases.
The method developed in this section now enables us to compute analytical representations
of seismic-wave motions in irregular media, which can in general be modeled only numerically.
This will be particularly useful for verification and testing of the accuracy with which transmitting
boundaries in numerical algorithms are able to transmit the waves without spurious reflections.
153
Figure 4.26 Mode#1 Diffracted Mode Shapes (x-) at f = 2.00, 3.33, 6.67 & 13.33 Hz
154
Figure 4.27 Mode#1 Diffracted Mode Shapes (z-) at f = 2.00, 3.33, 6.67 & 13.33 Hz
155
Figure 4.28 Mode#2 (2-D) Diffracted Mode Shapes (x-) at f = 3.13, 5.26, 6.67 & 9.09 Hz
156
Figure 4.29 Mode#2 (3-D) Diffracted Mode Shapes (x-) at f = 3.13, 5.26, 6.67 & 9.09 Hz
157
Figure 4.30 Mode#2 (2-D) Diffracted Mode Shapes (z-) at f = 3.13, 5.26, 6.67 & 9.09 Hz
158
Figure 4.31 Mode#2 (3-D) Diffracted Mode Shapes (z-) at f = 3.13, 5.26, 6.67 & 9.09 Hz
159
Figure 4.32 Mode#3 Diffracted Mode Shapes (x-) at f = 2.00, 2.78, 3.13 & 5.26 Hz
160
Figure 4.33 Mode#3 Diffracted Mode Shapes (z-) at f = 2.00, 2.78, 3.13 & 5.26 Hz
161
Figure 4.34 Mode#4 Diffracted Mode Shapes (x-) at f = 2.00, 2.50, 4.55 & 9.09 Hz
162
Figure 4.35 Mode#4 Diffracted Mode Shapes (z-) at f = 2.00, 2.50, 4.55 & 9.09 Hz
163
Figure 4.36 Mode#5 Diffracted Mode Shapes (x-) at f = 2.00, 2.50, 3.33 & 4.17 Hz
164
Figure 4.37 Mode#5 Diffracted Mode Shapes (z-) at f = 2.00, 2.50, 3.33 & 4.17 Hz
165
Chapter 5
Synthetic Scattered and Diffracted Time Histories of
Love and Rayleigh Waves
5.1 Introduction
This chapter is presented the synthetic scattered and diffracted time histories of Love and
Rayleigh Waves which has now been summarized and appeared on two journal papers:
1) Lee, V.W., Liu, W.Y., Trifunac, M.D., Orbovi ć, N. Scattering and diffraction of
earthquake motions in irregular elastic layers, I: Love and SH waves, “Soil Dynamics
and Earthquake Engineering”, Volume 66, November 2014, Pages 125–134.
2) Lee, V.W., Liu, W.Y., Trifunac, M.D., Orbovi ć, N. Scattering and diffraction of
earthquake motions in irregular, elastic layers, II: Rayleigh and body P and SV waves,
“Soil Dynamics and Earthquake Engineering”, Volume 66, November 2014, Pages
220–230.
5.2 Introduction in Love Wave
The scattered and diffracted Love Waves and SH Body-Wave mode shapes can next be used
to generate the synthetic out-of-plane transverse (y-) components of accelerations at all points on
and below the ground surface in the vicinity of the irregular layers. The input Love and Body SH
Waves to these diffracted waves are the out-of-plane waves generated in the regular layered media
of elastic half space. The complete description of this procedure is given in Todorovska et al.
(2013).
166
6.5, 8.0km, 6.0km, 0, 2
L
MR H s s
where
M = earthquake magnitude
R = epicentral distance
H = focal depth of earthquake
s = geologic site condition of the recording site, where
s = 0, alluvial site, s = 1, intermediate site, and s = 2, rock site
L
s = soil condition of the recording site, where
L
s = 0, rock soil,
L
s =1,
intermediate soil, and
L
s =2, soft soil type.
Table 5.1 Time-History Parameters
5.3 Synthetic Time Histories of Love Wave
The time history of the transverse out-of-plane motions at the top surface 0 z has been
generated by the SYNACC method (Trifunac 1971; Wong and Trifunac 1979; Lee and Trifunac
1985, 1987; Todorovska et al. 2013). The time histories at points 0 z , below the half-space
surface, have been generated by the recently generalized SYNACC program, which uses the mode
shapes below the half-space surface to construct the motions at depth. Because of the scattering
and diffraction, these out-of-plane mode shapes are now space dependent, and the acceleration-
time histories are different at every point on the half-space surface and at depth.
Figure 5.1 shows three synthetic displacement-time histories calculated in the two-layered
medium at 101 depths equally spaced from the layered half-space top surface (0) z to a depth
of 2km z , below the first layer which extends to a depth of 1.38 km z . The time histories are
at three vertical sets of points along the half-space surface, at 2.0km, x (1.0 km
to the left of
the left rim of the canyon), 0.0 km x (the center of the canyon), and 2.0 km x (1.0 km to the
right of the right rim of the canyon). The time histories at the left and right rims of the canyon are
167
all plotted at 0km, z the layered half-space surface, while the time histories at the middle of the
canyon start at 0.1 km z , the bottom of the canyon.
Figure 5.1 Horizontal, out-of-plane, synthetic displacements at x = -2, 0, and 2 km
The origin at the center of the canyon on the half-space surface is situated at an epicentral
distance of 8.0 km R from the earthquake source of focal depth 6.0 km H , which would
correspond to a hypocentral distance of
12
22
, DR H or 10.0 km D . The time history of
displacement at the half-space surface, 0km, z on the left at 2.0 km x , is thus at an epicentral
distance of 6.0 km R
and a hypocentral distance of 8.5km, 1.5km D closer. Similarly, the
time history of displacement at the half-space surface, 0km z , on the right at 2.0 km x is
thus at an epicentral distance of 10.0 km R
and a hypocentral distance of 11.7 km D , 1.7 km
further away. The new SYNACC program determines that the appropriate duration of the
displacement record should be just above 40 s. The depths of the time histories in the figure are
at intervals of 0.20 km apart. Of the 101 displacement-time histories, a few are plotted red, and
168
those are the time histories at the interface. In the middle graph, the top red curve is at the bottom
of the canyon, while the one below is at the elliptical interface.
It can be seen that the waves arrive earlier on the left side of the canyon compared with the
waves below the center of the canyon, while the waves on the right side of the canyon arrive later,
with each side differing by a few seconds. The amplitudes of the waves are similar but different
because of the scattering and diffraction. In all of the graphs, the strong motions are seen only in
the top layer, down to z = 1.38 km, which is the interface with half space. This is consistent with
the fact that the Love-Wave mode-shape amplitudes diminish exponentially in the half space.
5.4 The Fourier and Response Spectral Amplitudes of Love Wave
The time histories above show that the motions, because of scattering and diffraction, can
differ from point to point at points on or below the half-space surface in the vicinity of the
irregularly layered media. To further illustrate these differences, we will show the spectral
amplitudes of the motions at different points.
Figure 5.2 shows the response as well as the Fourier spectral amplitudes at four surface
points on the layered media. This is a typical Volume 3 plot, as in “routine digitization and data
processing” (Lee and Trifunac 1979) of strong-motion accelerograms, which we use to describe
spectral content in the strong ground motions in the form that is used in engineering design.
These spectral amplitudes show similar trends and yet are all different. They are spectral
plots for only four points of the irregularly layered media. Each point on top and below the layered
media, to the left and to the right of the irregular surface and interface, will all have its own spectral
amplitude. The pseudo relative velocity (PSV) spectra are shown for damping values of 0.0 and
0.20 of the critical value. The spectra for no damping are characterized by rapid oscillations
because the oscillator has no memory (Udwadia and Trifunac 1974; Gupta and Trifunac 1988).
The spectra for 20% damping are smooth because of the considerable memory of the oscillator
with large damping. These spectra show the clear differences in response at different surface
stations.
169
Figure 5.2 Response- and Fourier-amplitude spectra of surface motions
at x = -2, -1, 1, and 2km
170
Figure 5.3 Contours of Fourier amplitude spectra (FS) in the area surrounding the
inhomogeneous layer ( 2x 2 km and 0depth 2 km) at 12 periods: T = 0.15, 0.20, 0.30,
0.40, 0.50, 0.75, 1.00, 1.50, 2.00, 4.00, 5.00, and 7.50 s
The simplest way to show the variations of Fourier spectral (FS) amplitudes is to plot their
contours at selected periods. Figure 5.3 shows such contours in units of in/s, at 12 selected periods:
T = 0.15, 0.20, 0.30, 0.40, 0.50, 0.75, 1.00, 1.50, 2.00, 4.00, 5.00, and 7.50 s. The contour levels
are for the Fourier spectra in logarithmic scale, log (FS), in the range -1.0 – 2.25 in steps of 0.25,
so that the corresponding Fourier amplitudes in linear scale are in the range 0.10 in/s to over 100
in/s. As can be seen from the figure, the Fourier amplitudes are larger near the top medium of the
layer, with amplitudes varying from point to point in the range of around 1 in/s to above 100 in/s.
Below the top layer, z > 1.38 km, the motions are all below 1 in/s.
171
Figure 5.4 Contours of pseudo-relative velocity spectra (PSV) in the area surrounding the
inhomogeneous layer ( 2x 2 km and 0depth 2 km), at 12 periods: T = 0.15, 0.20, 0.30,
0.40, 0.50, 0.75. 1.00, 1.50, 2.00, 4.00, 5.00, and 7.50 s - and for 5% of critical damping
Figure 5.4 shows the corresponding contour plots of the PSV response spectral amplitudes
at 5% damping in units of in/s. Those contours show trends and ranges of amplitudes similar to
the Fourier-spectrum amplitudes in Figure 5.3.
5.5 Summary for Love Wave
We illustrated the wave propagation in and near an irregularly layered medium for the out-
of-plane transverse component of earthquake ground motions corresponding to the incoming Love
Surface Waves and Body SH Waves. Additional waves were generated at each layer as a result of
scattering and diffraction at the irregular half-space surface and irregular interfaces between the
172
layers. The out-of-plane scattered and diffracted shear waves were computed using the weighted-
residues method to satisfy the (out-of-plane shear) zero-stress boundary condition on the top half-
space surface and the continuity of (out-of-plane) stress and displacement at the interface surfaces
between adjacent layers.
The contour plots of the Fourier and response spectra illustrate how the motions vary from
point to point at or below the half-space surface and for different periods of motions. These waves
generated by scattering and diffraction will result in amplification and de-amplification at various
points on the layered medium depending upon the different irregular geometries.
In this paper, we illustrated the amplification of motions near inhomogeneity only for
translational components of motion. With spatial differentiation, it is now also possible to
calculate the amplification of rotational components of motion on ground surfaces and anywhere
inside the layers (Trifunac 2009). However, it will be necessary to study many different layer
geometries to identify the general trends for both translational and rotational components of strong
motion. These studies are beyond the scope of this work and will be presented in our future papers.
Previous work on the scattering and diffraction of elastic waves by surface and subsurface
topographies in a homogeneous, elastic half space can now be extended to an irregularly layered,
elastic half space. The method presented in this paper will be useful because more realistic
geometries can be modeled, which will permit synthesis of artificial earthquake motions for use in
design that are more realistic than the frequently used elastic half space, with or without the
uniformly parallel layers. The methodology presented in this paper can also be extended to the
cases of irregularly layered poroelastic media.
5.6 Introduction in Rayleigh Wave
The scattered and diffracted, in-plane Rayleigh and Body P and SV Wave mode shapes can
next be used to generate the synthetic, scattered and diffracted, in-plane radial (x-) and vertical (z-)
components of accelerations at all points on and below the surface in the vicinity of the irregularly
layered medium. The input Rayleigh and Body P and SV Waves to these diffracted waves are the
in-plane waves generated in the regularly layered media of elastic half space. The complete
173
description of the procedures for generation of these time histories is given in Todorovska et al.
(2013). As in Paper I, we will use the following parameters to generate the time-histories.
6.5, 8.0km, 6.0km, 0, 2
L
MR H s s
where
M = earthquake magnitude
R = epicentral distance
H = focal depth of earthquake
s = geologic site condition of the recording site, where
s = 0, alluvial site; s = 1, intermediate site; and s = 2, rock site
L
s = soil condition of the recording site, where
L
s = 0, rock soil;
L
s =1, intermediate soil; and
L
s =2, soft-soil type
Table 5.2 Time-History Parameters
5.7 Synthetic Time Histories of Rayleigh Wave
The time history of the radial and vertical in-plane motions at the top surface 0 z has
been generated by the SYNACC method (Trifunac 1971; Wong and Trifunac 1979; Lee and
Trifunac 1985, 1987; Todorovska et al. 2013). The time histories at points 0 z , below the half-
space surface, have been generated by the updated SYNACC program, which uses the mode
shapes below the half-space surface to construct the motions at depth. Because of the scattering
and diffraction, these out-of-plane mode shapes are now space dependent, and therefore the
acceleration time histories are different at every point on the half-space surface and at depth.
174
Figure 5.5 Horizontal, in-plane, synthetic displacements at x = -2.0, 0.0, and 2.0 km
Figure 5.6 Vertical, in-plane, synthetic displacements at x = -2.0, 0.0, and 2.0 km
175
Figures 5.5 and 5.6 show the horizontal (radial; x- component) and vertical (z- component)
motions for synthetic-displacement time histories calculated in the two-layered model at 101
depths equally spaced from the half-space surface (0) z to a depth of 2km z below the first
layer up to a depth of 1.38 km. z The time histories are plotted as three vertical lines, at
2.0 km x (on the left), 0.0 km x (in the center), and 2.0 km x (on the right) of the center
of the canyon. The time histories on the left and right sides of the canyon are all plotted from
0km, z which is the layered half-space surface, while the time histories in the middle of the
canyon start at 0.1 km, z which is the bottom of the canyon.
The origin at the center of the canyon on the half-space surface is situated at the epicentral
distance of 8.0 km R from an earthquake source at focal depth 6.0 km H , so that the
hypocentral distance is 10.0 km. D
The time history at the top and on the left is at 2.0 km, x
and thus it is at the epicentral distance of 6.0 km R
and a hypocentral distance of
8.5 km, 1.5 km D closer. Similarly, the time history at top right is at 2.0 km, x and thus it is
at an epicentral distance of 10.0 km R
and a hypocentral distance of 11.7 km, D or 1.7 km
further away.
In all of the figures, the strong motions are seen mainly in the top layer, at depths of up to
1.38 km. Further, in all of the figures, whether it is the radial x- component or the vertical z-
component, and whether it is the acceleration, velocity, or displacement motion, it can be seen that
the waves arrive earlier on the left side and later on the right side.
5.8 Fourier- and Response-Spectral Amplitudes
The time histories in Figure 5.5 show that the motions can differ appreciably from point to
point in the vicinity of the irregularly layered medium. To illustrate the variations of these motions
from an earthquake-engineering viewpoint, we will next show the pseudo relative velocity (PSV)
and Fourier spectral (FS) amplitudes at selected points.
Figures 5.7 and 5.8 show the plots of the PSV spectral amplitudes at several surface points
on the layered medium. The format of the plots we use is a typical “Volume 3” plot, which we
176
use in the routine digitization (Trifunac and Lee 1979) and data processing (Lee and Trifunac 1979)
of strong-motion accelerograms to describe the spectral content in the strong ground motions.
Overall, these spectral amplitudes are very similar, but their details are all different. The
spectra are plotted at four points on the irregularly layered media. Each point on top and below the
layered media, and to the left and right of the irregular surface and interface, will have its own
spectral amplitudes.
Figure 5.7 Response- and Fourier-amplitude spectra - horizontal components
Illustrating how the Fourier- and Response-spectral amplitudes change from point to point
from left to right and from top to bottom in the vicinity of inhomogeneity would take too many
177
pages of plots to show the amplitudes at many different points. However, a simple and convenient
way to accomplish this is to plot the contours of the Fourier- and Response-spectral amplitudes at
a sufficient number of selected periods inside the medium. Figures 9 and 10 are plots of this type,
which show the radial (x-) and vertical (z-) contour plots of the Fourier-spectral amplitudes in units
of in/s at 12 selected periods:
Figure 5.8 Response- and Fourier-amplitude spectra - vertical components: T= 0.15, 0.20,
0.30, 0.40, 0.50, 0.75, 1.00, 1.50, 2.00, 4.00, 5.00, and 7.50 s, at the corresponding frequencies: f
= 6.67, 5.00, 3.33, 2.50, 2.00, 1.33, 1.00, 0.67, 0.50, 0.25, 0.20, and 0.133 Hz.
178
The contours are plotted with the logarithms of the Fourier amplitudes, so that the contour
levels are also in log scale, with log (FS) ranging from -1.0 to 2.25 in steps of 0.25, and the Fourier
amplitudes in the linear scale ranging from 0.10 in/s to over 100 in/s. It can be seen that for all of
the 12 periods shown the Fourier amplitudes are larger in the top layer, with amplitudes varying
from point to point in the range from around 1 in/s to above 100 in/s. Below the top layer, z > 1.38
km, and the motions are typically smaller than 1 in/s.
Figure 5.9 Fourier-spectral amplitudes - horizontal components
There is one noticeable difference between Figure 5.9 and 5.10, on the one hand, for the
in-plane, radial- and vertical-component Fourier amplitudes here, and the corresponding
amplitudes in Figure 5.11 and 5.12, on the other hand, for the out-of-plane, transverse-component
Fourier amplitudes in Paper I. Comparing the figures here and in Paper I, it can be seen that in
179
each of the 12 periods plotted, the Fourier amplitude contours vary much more and are more
oscillatory inside the layer. Physically, the out-of-plane, transverse motions in Paper I are of only
one component and one type of motion - namely, the horizontal transverse component of motion,
which does not lead to model conversion anywhere in the model. In contrast, the in-plane radial
and vertical motions here are of two components and of two types of motions - namely, the
horizontal and vertical in-plane motions on the one hand, and the longitudinal and shear-wave
motions of different wave speeds on the other. Thus, the mode conversions, which are present for
all motions studied in this paper lead to more complicated contour plots, as shown in Figure 5.9
and 5.10.
Figure 5.10 Fourier-spectral amplitudes - vertical component
180
Figures 5.9 and 5.10 show the contour plots of in-plane radial and vertical components of
the PSV response-spectral amplitudes at 5% damping in units of in/s These plots show the same
trends and ranges of amplitudes as the Fourier amplitudes in Figures 5.9 and 5.10. Again,
comparing with the corresponding Figure 5.11 and 5.12 for the out-of-plane, transverse component
5% PSV amplitudes in Paper I, while the contour levels there and here are of comparable
amplitudes, at each of the 12 periods shown, the contour values here change more and more rapidly
from point to point.
Figure 5.11 PSV-spectral amplitudes - horizontal component, for 5% damping
181
Figure 5.12 PSV-spectral amplitudes - vertical component, for 5% damping
5.9 Summary for Rayleigh Wave
We analyzed the wave propagation in an irregularly layered medium for the in-plane radial
and vertical components of motion. With the incoming Rayleigh Surface Waves and Body P and
SV Waves, additional scattered and diffracted waves are generated by irregular interfaces between
the layers. The scattered and diffracted waves are computed using the weighted-residues method
to satisfy (1) the (in-plane) coupled, normal, and shear zero-stress boundary conditions on the top
half-space surface; and (2) the continuity of (in-plane) coupled stress and displacement at the
layered interface surfaces.
182
5.10 Conclusion
We have in this chapter illustrated the wave propagation in and near an irregularly layered
medium for
1) out-of-plane transverse component of earthquake ground motions corresponding to
incoming Love Surface Waves and Body SH-Waves, and
2) in-plane redial and vertical components of earthquake ground motions corresponding to
incoming Rayleigh Surface Waves and Body P- and SV- Waves.
In both cases, the presence of the irregular interfaces generated additional scattered and
diffracted waves. In each cases, these waves are computed using the weighted-residues method,
such that the following are satisfied:
1) The zero normal and shear stress boundary conditions at the top of half-space surface,
2) The continuity of stress and displacement at each layered interface surface.
In both cases, the contour plots of the Fourier and Response Spectra show that the out-of-
plane and in-plane components of motions vary from point to point at or below half-space surface
and they are different at different periods. They resulted in amplification and de-amplification at
various points in the layered media below that half-space surface, which was very dependent upon
the various irregular geometry.
This work is just the beginning. Much more studies and analyses are needed so that the
work can be applied and extended to more complicated geometries of a layered media. It is
worthwhile also to extend the work from elastic, homogeneous media to laterally inhomogeneous
or poroelastic layered media.
183
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Spectra, 14(1), 225-239, 1998.
M. D. Trifunac Synthetic Translational Motions of Surface Waves On or Below a Layered
Media, Report I, 2013. (M. D. Trifunac, W. Y. Liu and V. W. Lee; reports/papers to be finalized)
M. D. Trifunac Synthetic Translational Motions of Surface Waves On or Below a Layered
Media, Report II, 2013. (M. D. Trifunac, W. Y. Liu and V. W. Lee; reports/papers to be finalized)
M. D. Trifunac Synthetic Rotational Motions of Surface Waves On or Below a Layered
Media, Report III, 2013. (M. D. Trifunac, W. Y. Liu and V. W. Lee; reports/papers to be
finalized)
M. D. Trifunac Diffraction Around An Irregular Layered Elastic Media, I: Love and Body SH
Waves, Report IV-Part 1, 2013 (M. D. Trifunac, W. Y. Liu and V. W. Lee; reports/papers to be
finalized)
Abstract (if available)
Abstract
The purpose of the thesis aims to analyze the wave propagation responses in the irregular layered elastic media due to seismic waves. As a result of excitation by incoming seismic waves, the underground structures generate additional waves from diffraction and scattering. Such generated waves always result in amplification or de-amplification of the input waves, which may cause the deformation, distribution and concentration of stresses near the ground surface and the structures above. Thus, it is important to understand the theoretical aspects and the effects of such diffraction behavior. ❧ Lee & Liu (2013) published a simplified mathematical formulation consisting of Fourier half-range expansions, which does not require either Lee & Cao’s arc approximation or complicated integration by Lin et al. (2010), to re-examine the cylindrical canyon model. This new approach automatically satisfies the boundary conditions at the flat ground surface and reduces the number of equations in half. In other words, the computational efficiency for calculation increases 100 percent, compared to the previous method. The site amplification responses computed from the new exact solutions are more precise and can be applied to higher frequencies, compared to the results by Lee & Cao (1991). Two papers for demonstrating the feasibility of the Fourier half-range expression approach to irregular elastic layers in Love and SH Waves (2014), Rayleigh and Body P and SV Waves (2014) have been published and incorporated into Chapter 3, 4 and 5.
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University of Southern California Dissertations and Theses
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Asset Metadata
Creator
Liu, Wen-Young
(author)
Core Title
Diffraction of elastic waves around layered strata with irregular topography
School
Viterbi School of Engineering
Degree
Doctor of Philosophy
Degree Program
Civil Engineering
Publication Date
05/09/2017
Defense Date
07/25/2016
Publisher
University of Southern California
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Tag
diffraction,elastic waves,irregular topography,layered strata,OAI-PMH Harvest
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English
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Advisor
Lee, Vincent W. (
committee chair
), Johnson, Erik A. (
committee member
), Trifunac, Mihailo D. (
committee member
), Wang, Chunming (
committee member
), Wong, Hung Leung (
committee member
)
Creator Email
ray@rayliuassociates.com,wenyounl@usc.edu
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Tags
diffraction
elastic waves
irregular topography
layered strata