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University of Southern California Dissertations and Theses
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Two-dimensional weighted residual method for scattering and diffraction of elastic waves by arbitrary shaped surface topography
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Two-dimensional weighted residual method for scattering and diffraction of elastic waves by arbitrary shaped surface topography
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Content
Thesis
Two-Dimensional Weighted Residual Method for Scattering and
Diffraction of Elastic Waves by Arbitrary Shaped Surface Topography
Heather Brandow
A Thesis submitted for the partial fulfillment of the degree of Doctor of Philosophy
at the Department of Sonny Astani Civil and Environmental Engineering, Viterbi
School of Engineering, University of Southern California
August 2015
1
Table of Contents
Table
of
Contents
1
Acknowledgement
4
List
of
Figures
5
Chapters
I.
Introduction
11
I.1
History
of
Elastic
Wave
Propagation
11
I.2
Prior
Analytical
Studies
14
I.3
Prior
Numerical
Studies
19
I.4
Summary
for
Incident
P-‐wave
on
2-‐D
Semi-‐Circular
Canyon
of
Elastic
Half
Space.
The
Solution
of
the
wave
equation
with
zero
stress
on
the
half-‐space
surface.
(Lee
and
Liu
25
)
23
I.4.1
Semi-‐Circular
Canyon
Model
(Lee
and
Liu
25
)
23
I.4.2
The
Zero
Normal
and
Shear
Stress
Boundary
Condition
on
the
Half-‐
Space
Surface
(Lee
and
Liu
25
)
26
I.4.3
The
Zero
Normal
and
Shear
Stress
Boundary
Condition
on
the
Canyon
Surface
(Lee
and
Liu
25
)
27
I.5
Objective
29
II.
Weighted
Residual
Method
for
Diffraction
of
Plane
P-‐
Waves
in
a
2-‐D
Elastic
Half
Space
On
an
Almost
Circular
Arbitrary-‐Shaped
Canyon
32
II.1
Introduction
32
II.2
Model
33
II.3
Boundary
Conditions
for
the
Canyon
Surface
34
II.4
Application
of
Weighted
Residual
Method
36
II.5
Numerical
Solutions
36
II.6
Comparison
of
Results
to
Previous
and
Existing
Studies
37
II.7
The
Case
of
the
Semi-‐Circular
Canyon
42
II.8
Application
to
Other
Arbitrary
Shapes
44
II.8.1
Elliptical
Canyon
44
II.8.2
Trapezoidal
Canyon
54
II.9
Observations
64
II.10
Summary
65
2
III.
Weighted
Residual
Method
for
Diffraction
of
Plane
P-‐
Waves
in
a
2-‐D
Elastic
Half
Space
On
a
Shallow
Almost
Circular
Arbitrary-‐
Shaped
Canyon
67
III.1
Introduction
67
III.2
Model
68
III.3
Boundary
Conditions
for
the
Canyon
Surface
69
III.4
Application
of
Weighted
Residual
Method
70
III.5
Results
71
III.5.1
Circle
Segment
72
III.5.2
Elliptical
Canyon
78
III.5.3
Shallow
Trapezoidal
Canyon
83
III.5.4
Nurek
Dam
88
III.6
Observations
92
III.7
Summary
93
IV.
Weighted
Residual
Method
for
Diffraction
of
Plane
P-‐
Waves
in
a
2-‐D
Elastic
Half
Space
On
an
Irregular
Shaped
Alluvial
Valley
95
IV.1
Introduction
95
IV.2
Model
95
IV.3
Presence
of
Irregular
Alluvial
Valley
96
IV.4
Boundary
Conditions
for
the
Canyon
Surface
97
IV.5
The
Application
of
Weighted
Residual
Method
98
IV.6
Numerical
Solutions
99
IV.7
Results
100
IV.7.1
Application
of
Other
Arbitrary
Shaped
Irregular
Alluvial
Valleys
101
IV.7.2
Elliptical
Shaped
Alluvial
Valley-‐
Soft
Soils
102
IV.7.3
Trapezoidal
Alluvial
Valley-‐
Soft
Soils
112
IV.7.4
Trapezoidal
Alluvial
Valley-‐
Hard
Soils
118
IV.8
Observations
119
IV.9
Summary
120
V.
Conclusions
122
V.1
Purpose
122
V.2
The
Numerical
Method
122
V.3
The
Physics
of
the
Problem
123
V.4
Achieving
Good
Results
123
V.5
Arbitrary
Shaped
Canyons
123
V.6
Shallow
Arbitrary
Shaped
Canyons
124
V.7
Arbitrary
Shaped
Valleys
125
V.2
Programming
Challenges
125
3
VI.
Future
Work
126
VI.1
Model
126
VI.2
Presence
of
Irregular
Alluvial
Valley
127
VI.3
Boundary
Conditions
for
the
Interface
128
VI.4
Application
of
Weighted
Residual
Method
128
VI.5
Results
for
a
Semi-‐Circular
Moon
Shaped
Alluvial
Valley-‐Hard
Soil
129
VI.6
Results
for
a
Semi-‐Circular
Moon
Shaped
Alluvial
Valley-‐Medium
Soil
137
References
142
Appendix
150
4
Acknowledgement
I
would
first
like
to
thank
my
advisor
Dr.
Lee.
Without
his
tireless
effort,
patience,
tutoring,
and
optimism
I
would
not
have
finished
this
thesis
or
my
masters
degree.
And,
without
his
constant
encouragement
to
return
school
and
finish
my
doctorate,
I
would
not
be
here
today.
I
would
also
like
to
thank
my
committee
members
Dr.
Wellford,
Dr.
Anderson,
Dr.
Trifunac,
Dr.
Carlson,
and
Dr.
Soibelman
for
not
only
taking
the
time
to
give
me
positive
feedback
and
help
me
achieve
my
goals
but
also
for
taking
the
time
to
be
part
of
my
committee.
Lastly
I
would
like
to
thank
my
dad,
mom,
brother
and
nephew
for
their
love
and
support
throughout
this
long
process.
Dad,
you’ve
inspired
me
to
be
an
engineer
since
the
fourth
grade
when
you
helped
me
build
my
shake
table
for
the
science
fair.
You’ve
pushed
me
and
helped
me
succeed
in
every
goal
I
set
for
myself
and
for
that
I
am
eternally
grateful.
5
LIST OF FIGURES
I. Introduction
Fig. 1.1
Compressional or P-wave 12
Fig. 1.2
Shear or S-Wave 13
Fig 1.3
2-D Circular Cylindrical Canyon (Cao and Lee
2,5
) 17
Fig. 1.4
The Flat Surface Boundary represented as a curved surface
(Cao and Lee
2,5
)
17
Fig. 1.5
A Canyon Surrounded by an Elastic Half-Space Represented by
a Finite Element Mesh
20
Fig. 1.6
The Nurek Dam 22
II. Weighted Residual Method for Diffraction of Plane P-Waves in a 2-D Elastic
Half Space On An Almost Circular Arbitrary-Shaped Canyon
Fig. 2.1
Arbitrary-shaped with coordinates at the half-space 33
Fig. 2.2a
Traction components 34
Fig. 2.2b
Traction components 35
Fig.
2.3
Semi-circular canyon. 40
Fig.
2.4
Arbitrary-shaped semi-circular canyon x/a vs. Ux and Uy for
η=10, θ=60°, N
max
=112 (results from Lee and Liu 2014
4
).
41
Fig.
2.5
Weighted residual method recreated semi-circular canyon that
matches Lee and Liu’s results η = 10, θ = 60°, N
max
=126.
42
Fig.
2.6
Arbitrary-shaped elliptical canyons. 44
Fig.
2.7
Ellipse-shaped canyon η = 2, θ = 5°, 30°, 60° ,90°, h/a = 1.25
N
max
= 70
46
Fig.
2.8
Ellipse-shaped canyon η = 8, θ = 5°, 30°, 60°, 90°, h/a = 1.25
N
max
= 104
48
Fig.
2.9a
Ellipse-shaped canyon η = 2, θ = 5°, 30°, h/a = 1.5 N
max
= 60
50
Fig.
2.9b
Ellipse-shaped canyon η = 2, θ = 60°, 90°, h/a = 1.5 N
max
= 60
51
Fig.
2.10
Ellipse-shaped canyon η = 8, θ = 5°, 30°, 60°, 90°, h/a = 1.5
N
max
= 100
53
6
Fig.
2.11
Trapezoidal canyon 54
Fig.
2.12
Trapezoid-shaped canyon η = 2, θ = 5°, 30°, 60°, 90°, h = 1,
N
max
= 28 Slope 60°
55
Fig.
2.13
Trapezoid-shaped canyon η = 6, θ = 5°, 30°, 60°, 90°, h = 1,
N
max
= 38 Slope 60°
57
Fig.
2.14
Trapezoid-shaped canyon η = 2, θ = 5°, 30°, 60°, 90°, h = 1,
N
max
= 18 Slope 45°
60
Fig.
2.15
Trapezoid-shaped canyon η = 6, θ = 5°, 30°, 60°, 90°, h = 1,
N
max
= 28 Slope 45°
62
III. Weighted Residual Method for Diffraction of Plane P-Waves in a 2-D Elastic
Half-Space On A Shallow Almost Circular Arbitrary- Shaped Canyon
Fig. 3.1
Arbitrary-shaped model with coordinates above the half-space 68
Fig.
3.2a
Traction components 70
Fig.
3.2b
Traction components 70
Fig.
3.3
Circle segment model 72
Fig.
3.4
Shallow circular segment x/a vs. Ux and Uy for η = 2, θ = 5°,
30°, 60°, 90° N
max
= 20
74
Fig.
3.5
Shallow circular segment x/a vs. Ux and Uy for η = 6, θ = 5°,
30°, 60°, 90° N
max
= 60
76
Fig.
3.6
Shallow elliptical canyon model
78
Fig.
3.7
Shallow elliptical canyon x/a vs. Ux and Uy for η = 2, θ = 5°,
30°, 60°, 90° b/a = 0.25 N
max
= 22
79
Fig.
3.8
Shallow elliptical canyon x/a vs. Ux and Uy for η = 6, θ = 5°,
30°, 60°, 90° b/a = 0.25 N
max
=44
81
Fig.
3.9
Shallow trapezoidal canyon model
83
Fig.
3.10
Shallow trapezoidal canyon x/a vs. Ux and Uy for η = 2, θ = 5°,
30°, 60°, 90° h/a = 0.25 N
max
= 14 Slope 60°
84
Fig.
3.11
Shallow trapezoidal canyon x/a vs. Ux and Uy for η = 6, θ = 5°, 86
7
30°, 60°, 90° h/a = 0.25 N
max
= 48 Slope 60°
Fig.
3.12
Nurek Dam model
88
Fig.
3.13
Nurek Dam x/a vs. Ux and Uy for η=2, θ=5°, 0°, 60°, 90° N
max
= 30
89
Fig.
3.14
Nurek Dam x/a vs. Ux and Uy for η = 8, θ = 5°, 30°, 60°, 90°
N
max
= 56
91
IV. Weighted Residual Method for Diffraction of Plane P-Waves in a 2-D Elastic
Half-Space On An Irregular Shaped Alluvial Valley
Fig.
4.1
Irregular-shaped alluvial valley with coordinates at the half
space
95
Fig.
4.2
Semi-circular alluvial valley 100
Fig.
4.3
β1/β = 0.1, ρ1/ρ = 0.1, µ1/µ = 0.001 for an irregular valley and
an arbitrary semi-circle canyon η = 2, θ = 60°, and N
max
= 80
using the weighted residual method
101
Fig.
4.4
Elliptical-shaped alluvial valley
102
Fig.
4.5a
Ellipse alluvial valley b/a = 1, η = 2, θ = 5°, 30°, and N
max
=20
103
Fig.
4.5b
Ellipse alluvial valley b/a = 1, η = 2, θ = 60°, 90°, and N
max
=
20
104
Fig.
4.6
Ellipse alluvial valley b/a=1.25, η=2, θ= 5°, 30°, 60°, 90° , and
N
max
= 24
107
Fig.
4.7
Elliptical alluvial valley b/a = 1.5, η = 2, θ = 5°, 30°, 60°, 90° ,
and N
max
= 36
109
Fig.
4.8
Elliptical alluvial valley b/a = 1.5, η = 8, θ = 5°, 30°, 60°, 90°,
and N
max
= 64
111
Fig.
4.9
Trapezoidal alluvial valley 113
Fig.
4.10
Trapezoidal alluvial alley b/a = 1, η = 2, θ = 5°, 30°, 60°, 90°,
N
max
= 16, and slope 60°
114
Fig.
4.11
Trapezoidal alluvial valley b/a = 1, η = 8, θ = 5°, 30°, 60°, 90°,
N
max
= 48, and slope 45°
116
8
Fig.
4.12
Trapezoidal alluvial valley b/a = 1, η = 2, θ = 5°, 30°, 60°, 90°,
N
max
= 20, and slope 45°
118
VI. Future Work
Fig.
6.1
Semi-Circular Moon Shaped Alluvial Valley with Coordinates
at the Half-Space
126
Fig.
6.2a
Moon Shaped Valley η=2, H1/a =0.8, H2/a=1, N
max
=12, θ=5°,
30°
130
Fig.
6.2b
Moon Shaped Valley η=2, H1/a =0.8, H2/a=1, N
max
=12, θ=60°,
90°
131
Fig.
6.3a
3-Dimensional Moon Shaped Valley- x/a vs. U
x
for η=2, H1=1
H2=0.8, N
max
=12,θ=5°, 10°, 15°, 20°, 25°, 30°, 35°, 40°, 45°,
50°, 55°, 60°65°, 70°, 75°, 80°, 85°, 90°
134
Fig.
6.3b
3-Dimensional Moon Shaped Valley- x/a vs. U
y
for η=2, H1=1
H2=0.8, N
max
=12, θ=5°, 10°, 15°, 20°, 25°, 30°, 35°, 40°, 45°,
50°, 55°, 60°65°, 70°, 75°, 80°, 85°, 90°
135
Fig.
6.4a
Moon Shaped Valley η=2, H1/a=0.8, H2/a=1, N
max
=15, θ=5°,
30°
137
Fig.
6.4b
Moon Shaped Valley η=2, H1/a=0.8, H2/a=1, N
max
=15, θ=60°,
90°
138
Fig.
6.5a
3-D Moon Shaped Valley- x/a vs. U
x
for η=2, H1=1 H2=0.8,
N
max
=15, θ=5°, 30°, 60°, 90°
140
Fig.
6.5b
3-D Moon Shaped Valley- x/a vs. U
y
for η=2, H1=1 H2=0.8,
N
max
=15, θ=5°, 30°, 60°, 90°
140
Appendix
Fig.
A.1
Shallow Elliptical Canyon x/a vs. Ux and Uy for η=2, θ=5°,
30°, 60°, 90° b/a=0.5 N
max
=20
150
Fig.
A.2
Shallow Elliptical Canyon x/a vs. Ux and Uy for η=8, θ=5°,
30°, 60°, 90° b/a=0.5 N
max
=72
151
9
Fig.
A.3
Shallow Elliptical Canyon x/a vs. Ux and Uy for η=2, θ=5°,
30°, 60°, 90° b/a=0.75 N
max
=24
152
Fig.
A.4
Shallow Elliptical Canyon x/a vs. Ux and Uy for η=8, θ=5°,
30°, 60°, 90° b/a=0.75 N
max
=80
153
Fig.
A.5
Shallow Trapezoidal Canyon x/a vs. Ux and Uy for η=2, θ=5°,
30°, 60°, 90° h/a=0.25 N
max
=14 Slope 45°
154
Fig.
A.6
Shallow Trapezoidal Canyon x/a vs. Ux and Uy for η=6, θ=5°,
30°, 60°, 90° h/a=0.25 N
max
=52 Slope 45°
155
Fig.
A.7
Shallow Trapezoidal Canyon x/a vs. Ux and Uy for η=2, θ=5°,
30°, 60°, 90° h/a=0.5 N
max
=16 Slope 45°
156
Fig.
A.8
Shallow Trapezoidal Canyon x/a vs. Ux and Uy for η=6, θ=5°,
30°, 60°, 90° h/a=0.5 N
max
=60 Slope 45°
157
Fig.
A.9
Shallow Trapezoidal Canyon x/a vs. Ux and Uy for η=2, θ=5°,
30°, 60°, 90° h/a=0.5 N
max
=14 Slope 60°
158
Fig.
A.10
Shallow Trapezoidal Canyon x/a vs. Ux and Uy for η=6, θ=5°,
30°, 60°, 90° h/a=0.5 N
max
=52 Slope 60°
159
Fig.
A.11
Shallow Trapezoidal Canyon x/a vs. Ux and Uy for η=2, θ=5°,
30°, 60°, 90° h/a=0.75 N
max
=14 Slope 60°
160
Fig.
A.12
Shallow Trapezoidal Canyon x/a vs. Ux and Uy for η=6, θ=5°,
30°, 60°, 90° h/a=0.75 N
max
=58 Slope 60°
161
Fig.
A.13
Shallow Trapezoidal Canyon x/a vs. Ux and Uy for η=2, θ=5°,
30°, 60°, 90° h/a=0.75 N
max
=12 Slope 45°
162
Fig.
A.14
Shallow Trapezoidal Canyon x/a vs. Ux and Uy for η=6, θ=5°,
30°, 60°, 90° h/a=0.75 N
max
=62 Slope 45°
163
Fig.
A.15
Ellipse Alluvial Valley x/a vs. U
x
and U
y
for η=8, b=1 N
max
=60
β
1
/β= ½, ρ
1
/ρ=2/3, µ
1
/µ=0.1667
164
Fig.
A.16
Ellipse Alluvial Valley x/a vs. U
x
and U
y
for η=8, b=1.25
N
max
=64 β
1
/β= ½, ρ
1
/ρ=2/3, µ
1
/µ=0.1667
165
Fig.
A.17
Trapazoid Alluvial Valley x/a vs. U
x
and U
y
for η=8, N
max
=34, 166
10
Slope 60°, β
1
/β= ½, ρ
1
/ρ=2/3, µ
1
/µ=0.1667
Fig.
A.18
Trapazoid Alluvial Valley x/a vs. U
x
and U
y
for η=2, N
max
=20,
Slope 45°, β
1
/β= ½, ρ
1
/ρ=2/3, µ
1
/µ=0.1667
167
Fig.
A.19
Trapazoid Alluvial Valley x/a vs. U
x
and U
y
for η=2, N
max
=30,
Slope 60°, β
1
/β= 2, ρ
1
/ρ=3/2, µ
1
/µ=6
168
Fig.
A.20
Trapazoid Alluvial Valley x/a vs. U
x
and U
y
for η=8, N
max
=54,
Slope 60°, β
1
/β= 2, ρ
1
/ρ=3/2, µ
1
/µ=6
169
Fig.
A.21
Trapazoid Alluvial Valley x/a vs. U
x
and U
y
for η=8, N
max
=36,
Slope 45°, β
1
/β= 2, ρ
1
/ρ=3/2, µ
1
/µ=6
170
11
I Introduction
I.1 History of Elastic Wave Propagation
Earthquake ground shaking is a result of seismic waves travelling through the earth; just
like waves that cross a pond and rock a boat as they pass beneath. The pattern of waves
remains the same unless disturbed by an obstacle, the boat, causing the waves to bend
around the obstacle or reflect back from the obstacle. The terminology used to describe
the bending and reflecting of the wave is called “diffraction” and “scattering”. The
variation, severity and intensity of earthquake damage has been ascertained to be the
result of the diffraction and scattering of seismic waves as geologic features such as
canyons, valleys, hills, cavities, and wedges disrupt the waves. Understanding the seismic
wave phenomenon is an important part of understanding earthquake ground motions, the
damage from earthquakes, and the design of civil engineering projects to resist this
shaking.
The intensity and severity of ground shaking at a site depends on three factors: the
magnitude of the earthquake, the distance from the epicenter, and the geology at the site.
It has also been observed that the amplitude of the seismic waves will increase or
decrease depending if you are on top of a hill, in a basin, or at the edge of a basin. Basins
are typically low lying areas that are either partially or completely surrounded by areas
that are much higher. Many urban areas like Los Angeles are located in a basin. When
earthquake waves travel through these basins filled with soft, deep sedimentary layers
and alluvial deposits, the waves slow down and the amplitude of the wave increases in
order to carry the same amount of energy as the waves in the rock. These amplified
earthquake may be very damaging to more flexible structures like what happened in the
Sylmar earthquake where at the transition from the rock mantle to the softer basin soils,
more violent shaking and resulting building damage was observed.
The general methodology for wave propagation of linear elastic waves starts with the
one-dimensional wave equation in a linear elastic system which is defined by the
following equations:
12
c
2
∂
2
u
∂x
2
=
∂
2
u
∂t
2
(1)
where u is a function of two parameters and defined as u=u(x,t) and c is a constant, u is
the displacement that corresponds to the point at coordinate x at time t. In two
dimensions, u is defined as u=u(x,y,t). The two-dimensional wave equation for an
isotropic media is as follows:
c
2
∂
2
u
∂x
2
+
∂
2
u
∂y
2
⎛
⎝
⎜
⎞
⎠
⎟
=
∂
2
u
∂t
2
(2)
These equations are thus used to study body waves and surface waves: P-waves and S-
waves, as they travel through an unbounded (infinite), elastic homogeneous solid. These
body waves can further be defined as three types of body waves; incident plane P-waves,
incident plane SH-waves and incident plane SV-waves. The P-waves (primary wave)
are compressional/dilatational waves that cause volumetric change in the soil but no
shearing deformations is produced. Figure 1.1 shows that the P-wave particle motion is
parallel to the P-wave propagation.
Fig. 1.1. Compressional or P-Wave
The S-waves (shear waves) cause shearing deformations but produce no volumetric
deformations. The s-waves can be represented by two perpendicular components; SH-
wave and SV-wave. SH-waves produce particle motion that occurs in the horizontal
plane. SV-waves produce particle motion in the vertical plane. Figure 1.2 shows that the
direction of particle motion is perpendicular to the direction of s-wave propagation.
13
Fig. 1.2. Shear or S-Wave
There are some basic characteristics of these waves in a homogeneous elastic solid. An
incident wave hits a surface and reflects back as itself, along with waves of other types.
For example, when an incident P wave hits a horizontal flat surface, it produces reflected
P and SV waves. When an SV wave hits a flat surface, P and SV waves are reflected.
When an SH wave hits a flat surface only an SH wave is reflected. The wave amplitude
also decrease as the wave radiates away from the source due to the damping of the
materials the wave passes through in which energy is absorbed into the material and
spreading of the wave energy over the distance the wave travels which is referred to as
geometric spreading.
The Sylmar Earthquake damage proved why the scattering and diffraction of seismic
waves by surface topographies has been an important research topic in understanding the
variation of the ground motions at different sites. In 1957 Gutenberg (Gutenberg
13
)
studied earthquake damage and noted that the degree of damage was a function of the
structural integrity, the properties of the arriving elastic waves, and the soil at the site.
He stated that, “In general, appreciable differences in shaking may exist at sites only a
thousand feet apart. “ He recognized this because sites with softer or denser alluvium soil
had higher amplitudes and longer durations of shaking than the stiffer soil sites. In
Hudson’s 1972 paper (Hudson
15
) Hudson noted that in the 1971 San Fernando
Earthquake, local geology significantly affected the patterns of ground motions. He
“showed that the influence of irregular geological structure or topography may
overshadow the effects of local site conditions.” In 1983, Huang (Huang
14
) studied
ground motions from several different earthquakes. He concluded that the wave
amplitudes depend on the 3-dimensional configuration of the local geology and the
14
direction of the arriving seismic waves. Depending on the relative location of faulting
and nearby large-scale topography, the topography can shield some areas from ground
shaking or amplify other areas. An example of this phenomenon is when the topography
of mountains and valleys scatter the surface waves generated by the rupture on the fault,
leading to variations of waves reflected from the basin edges and wave resonance within
the basin. Thus the distribution and amplitude of the ground shaking and the resulting
damage varies with the local topography. All of this is important in understanding of the
structural response of the building at a particular site and the design requirement for that
building at that site.
In 2000, Field and the SCEC Phase III Working Group (Group
11
) presented an overview
on the extent of site effects on the probabilistic seismic hazard analysis (PSHA). In their
findings, they stated that earthquake site effects amplification or deamplification, have
been known for 200 years and recognized in past earthquakes: 1811-1812, 1818 New
Madrid, great Japan Earthquake 1891, the 1906 San Francisco Earthquake and the 1933
Long Beach Earthquake. In summary, the observed amplification of ground motions
within a basin vary up to a factor of two from the shallow to deepest part of the basin.
The influence of the basin edge and subsurface topography on the focusing and scattering
will cause a substantial variability in the wave patterns. The focusing of waves caused by
subsurface geology was observed in the Northridge Earthquake sequence in Santa
Monica, where during aftershock recordings just 650 meters apart, the peak recorded
accelerations differed up to a factor of 5. The amplified ground motions generally
correlate with the variability of structural damage observed after the main shock.
1.2 Prior Analytical Studies
Researchers have developed two major tactics to examine the phenomenon of diffraction
and scattering of waves; analytical methods (which rely upon the solution of the
equations of elastodynamics by analytical means) and numerical methods (such as finite
element, boundary element method, boundary integral equation method and finite
difference method).
15
The Analytical method mathematically solves for the scattering and diffraction of waves
in an infinite media and is ideal for studying the effects of parameters, such as canyons
width vs. wavelength, on the amplitude of motions. The analytical method is not
restricted to finite domains and is thus more appropriate for theoretically studying wave
problems. In the past, this method has allowed the researcher to analytically solve a
linearly-elastic or visco-elastic media with a simple geometric obstacle such as a circular
or elliptical canyon and hills. For a limited set of geometries, analytical solutions are an
effective method for studying the diffraction and scattering of waves for various physical
geometries because it allows the researcher to vary different parameters within the
problem.
The following papers studied the scattering and diffraction of seismic waves due to the
geometry of the surface topographies and the specific types of incident waves. These
studies have shown that analytical techniques can explain the geophysical observations
regarding the amplification of seismic waves as a function of site conditions.
In 1981 and 1984 Lee (Lee
18,20
) studied the effects of three-dimensional scattering and
diffraction of plane waves by a hemispherical canyon and filled hemispherical alluvial
valley in a homogeneous elastic half space. These properties effect the transmission of
waves and the diffraction of waves across the boundary, posing additional boundary
conditions. With a general angle of wave incidence and a series solution, Lee studied the
ground motion near the canyon and valley. The nature of ground motion is found to
depend on key parameters such as (1) the angle of incidence, amplitude and type of
incident waves, (2) a dimensionless wave number frequency related to the ratio of the
diameter of the canyon to the wavelength of the incident waves, and (3) the ratio of
longitudinal to transverse wave speeds in the valley and the half-space, (4) the ratio of the
respective shear moduli, and (5) the ratio of the respective longitudinal wave speeds. The
displacement amplitude and phases for the nearby ground-surface show significant
departure from the uniform half-space motions. The angles of incidence determine the
overall trends of displacement amplitude on the nearby ground surface, relative to the
nature of the ground motion and the properties of the half-space. For olique incidence,
16
considerable amplification is observed in the incoming wave side of the spherical canyon
and a prominent shadow zone is realized the outgoing wave side of the spherical canyon.
In 1974, Trifunac and Wong (Trifunac and Wong
36
) analyzed the two-dimensional
scattering and diffraction of plane SH waves by a semi-elliptical canyon in elliptical
coordinates. Their results depended on two key parameters, the angle of incidence, and
the ratio of the canyon width to the wavelength of incident SH waves. Their results
showed that the short incident waves swiftly change the surface displacement amplitude
from one point to another.
In 1989 Cao and Lee (Cao and Lee
3
) studied the scattering of plane SH Waves by a
circular cylindrical canyon in an elastic half space. They found that the surface
amplification depended on three significant parameters (1) the angle of incidence, (2) the
ratio of canyon depth to its half-width, and (3) the dimensionless frequency solved by the
exact series solution.
In 1989 and 1990 Cao and Lee (Cao and Lee
2,5
) studied the scattering and diffraction of
SH and P waves on the same canyon using a series of Bessel function expansions. In the
paper, “Scattering and Diffraction of plane P waves by Circular Cylindrical Canyons
With Variable Depth-to-Width Ratios” (Cao and Lee
5
), they showed an approximate
method to solve the incident P wave on a Semi-Circular Canyon, Figure 1.3. In their
paper it showed that it is easy to apply the boundary conditions at the canyon surface but
it was difficult to solve the boundary conditions on the flat surface without making other
approximations that result in a complex set of equations. They developed an approximate
method where the flat surface is assumed to become a circular surface with a large
enough radius to approximately equal a flat surface, shown in Figure 1.4.
17
Fig. 1.3. 2-D Circular Cylindrical Canyon
Fig. 1.4. The Flat Surface boundary represented as a curved Surface
Therefore as the radius of the large circle approaches infinity this approximate method
transforms the flat surface into that of a circular canyon in the half space.
From the Boundary Conditions, and the transformation between the coordinate systems
(r
1
, θ
1
) and (r
2
, θ
2
) Cao and Lee were able to set up a system of equations in order to
solve for the Unknowns, and the displacements. It is a very complex problem with an
incident wave producing two sets of reflected waves.
18
Lee (Lee
19
) then expanded this work in 1990 with the study of a semi-parabolic
cylindrical canyon. He used an exact analytic series solution to show that the surface
topography has significant effects on the incident waves and the pattern of surface
displacement amplitudes.
In 1991 Lee and Todorovska (Lee and Todorovska
29
) developed an analytical method to
study the steady-state diffraction and scattering of mono-chromatic Rayleigh (surface)
waves by a shallow circular canyon. In this study they varied the shape of the canyon
from very shallow to semi circular and they looked at canyon motions excited by
Rayleigh waves with different wavelengths.
In 1992 Karl and Lee (Karl and Lee
16
) studied the scattering and diffraction of plane SV
waves by underground, circular, cylindrical cavities in an elastic half-space. They studied
the angle of incidence, when it exceeded the critical angle, of the plane SV waves. The
result showed surface waves being generated and the resulting surface displacement
amplitude and phases that deviate from that of a uniform half space. In 1993 Karl and
Lee (Karl and Lee
17
) expanded on this paper to consider the diffraction of elastic P waves
by the same tunnel.
In 2009, Lee, Liang and Luo (Lee, Liang and Luo
26
) used the auxiliary function method
to derive an exact analytical solution for diffraction of plane SH waves by a semi-circular
cavity in a half-space. They also used the method to compare the diffraction around a
semi-circular cavity in a half-space with that from a whole-circular cavity.
In 1996 Lee and Sherif (Lee and Sherif
28
) studied the scattering of a plane shear
horizontal (SH) or source emitted (SH) waves incident to an elastic wedge shaped
medium, with a circular canyon at its vertex. The results studied parameters including
the angle of the wedge, frequency of the incident wave to explain amplification of
seismic waves as a function of site conditions.
In 1991 Lee and Todorovska (Lee and Todorovska
30
) studied shallow circular alluvial
valley, with variable depth to width rations for incident elastic SH waves.
19
In 2001 Lee, Liang, and Yan (Lee, Liang, and Yan
22
) studied the scattering of plane P
waves by a circular arc layered alluvial valley, using a Fourier-Bessel series expansion
technique.
In 2002 Lee, Liang, and Zhang (Lee, Liang, and Zhang
23
) studied the arc layered alluvial
canyon again but for SH waves. All of these papers show that the amplitudes of the
surface displacements depend on incident wave angles and the geometry and properties
of the canyon and alluvial fill.
In 2004 Lee, Liang, and Yan (Lee, Liang, and Yan
21
) studied the scattering of plane SH
waves by a circular-arc hill with the same tunnel. Their solution was derived using the
Fourier-Bessel series expansion and auxiliary functions techniques for the stress and
displacement boundary conditions. They studied the effects of a hill with a tunnel and
showed that there will be significant effects on the resultant ground motion and stress
concentrations around the surface of the hill and tunnel.
In 2004 Lee, Luo, and Liang (Lee, Luo, and Liang
24
) also studied the diffraction of anti-
plane SH waves by a semi-circular cylindrical hill with an inside concentric semi-circular
tunnel, using the cylindrical wave function expansion. Numerical solutions were
approximated by the truncation of the infinite equations.
In 2005 Lee, Liang and Zhang
(Lee, Liang and Zhang
25
) studied the scattering of Plane P
waves by a semi-cylindrical hill. The results showed that the existence and geometry
with or without a tunnel has significant effect on the ground motion nearby.
I.3 Prior Numerical Solutions
The second tactic that researchers have used to study seismic waves is the numerical
method which includes the Finite Element Method (FEM), Boundary element method
(BEM), and boundary integral equation method (BIEM) and Finite Difference Method
(FD). FEM is a technique used to model complex systems using discrete elements to
20
approximate the soil foundations and the building. This model can then be analyzed
using numerical techniques to solve for the dynamic response to earthquake ground
motions. Researchers have been implementing the Finite Element Method (FEM) to
solve soil problems since the 1960s because it is ideal for analyzing finite bodies with
inhomogeneous material properties. Two and three-dimensional problems have been
addressed by developing a FEM mesh to define the soil geometry, taking into account the
physical characteristics such as a canyon, foundation of a building, arbitrary shaped
obstacle or geometric irregularities. Figure 1.5 represents a finite element mesh which
represents a canyon surrounded by an elastic half-space with boundary elements that
attempt to replace the true infinite media.
Figure 1.5 A Canyon Surrounded by an Elastic Half-Space Represented by a Finite
Element Mesh
The two major drawbacks in trying to study the wave scattering problem utilizing finite
elements are the computer demands as the mesh grows in size and complexity and the
challenges of addressing the finite domain of the mesh which must approximate an
infinite domain. The finite element matrices are typically banded (elements are only
locally connected) and the storage requirements for the system matrices typically grow
linearly with the problem size. The Boundary Elements are utilized to represent effects of
the infinite half space.
21
JT Chen used the null-field boundary integral equation method (BIEM) to study problems
such as the SH-wave diffraction by a semi-circular hill, scattering of Plane SH- waves by
multiple circular arc valleys, and the SH- wave scattering by a semi-elliptical hill, just to
name a few (Chen
6,7,8
). He found that the BIEM method was superior to the BEM
method for its five features, (1) free of calculating principal values, (2) exponential
convergence, (3) elimination of boundary-layer effect, (4) meshless, and (5) well posed
system.
In 2012, Gicev and Trifunac (Gicev and Trifunac
12
) used the finite difference model, a
numerical method for approximating the solutions to differential equations using finite
difference equations to approximate derivatives, to analyze the response of a two-
dimensional model of a building on a rectangular, flexible foundation in nonlinear soil.
They confirmed that seismic energy is absorbed in the soil prior to entering the building
thus reducing earthquake damage because the soil can absorb much larger volumes with
nonlinear deformations than any base isolator.
In 1980 Dravinski (Dravinski
10
) studied the diffraction of elastic waves by an alluvial
valley of arbitrary shape in an elastic soil medium. He wanted to bridge the gap between
the numerical method and the analytic method so he analyzed the problem using the
source method and the plane-strain method.
In 1993 and 1994, in two papers, Lee and Wu (Lee and Wu
31,32
) examined the scattering
and diffraction of plane SH waves, P waves, and SV waves by canyons of arbitrary
shapes in an elastic half space such as the Nurek Dam in Figure 1.6.
22
Fig. 1.6. Nurek Dam Model.
They defined the scattered wave potentials as a combination of both the sine and the
cosine functions, and the origin was defined above the half-space for the case of the
shallow canyon problem. In addition, they defined two sets of scattered P- and scattered
SV- wave potentials (a total of four sets of waves) so as to satisfy four sets of boundary
conditions, two on the half-space surface, and two on the canyon. Their solution for the
wave field used the method of weighted residuals which allowed them to define the shape
without using one of the six coordinate systems defined in the analytical method. Their
solutions were then checked against the analytic solutions of Cao and Lee (Cao and
Lee
2,5
) for accuracy. This thesis will simplify Lee and Wu’s weighted residual method
using Lee and Liu’s new method which is presented in the next section.
All the previous cases shown above are for half-space diffraction problems for
topographies that are geometrically simples and finite in dimensions. This thesis expands
these studies for the case of the scattering and diffraction of seismic waves from an
arbitrary shaped surface topography, in a two-dimensional elastic half-space subjected to
an incident Plane P- wave. This problem is a boundary value problem of wave diffraction
in an elastic half-space, since the wave diffraction occurs around the boundary of the
topography. The exact infinite series solution by the method of cylindrical wave function
expansion is presented in this study to solve the boundary conditions at the surface of the
half-space and the boundaries of the topography. The analytical infinite series solutions
23
are determined using the wave function expansion. Using this an approximate solution is
derived by truncating the infinite series, solving it numerically, and determining
convergence of the solution.
I.4 Summary for Incident P-Wave on 2-D Semi-Circular Canyon on an Elastic
Half-Space. The Solution of the Wave Equations with Zero-Stress on the Half-Space
Surface (Lee and Liu
27
)
Researchers continue to study the effects of scattering and diffraction of waves on two-
dimensional canyons in an elastic, isotropic, and homogeneous medium. These studies,
which assist researchers to understand earthquake ground motions in and around
topographic features, initially, addressed incident SH waves. (Trifunac
37
, Wong and
Trifunac
38
, Moeen-Vaziri and Trifunac
35
) In solving SH-waves, the method of images,
which assumes equal and opposite scatter waves upon reflection, has been used (Moeen-
Vaziri and Trifunac
34
). Because of the mode conversion, an incident P-wave, which
produces both a reflected P- and SV-wave, is more complex. Thus, the P-wave cannot be
solved using the method of images. Thus in 2014 Lee and Liu (Lee and Liu
27
) developed
a new analytical approach to solving the scattering and diffraction for an incident P- wave
on a 2-D Semi-Circular Canyon on an Elastic Half-Space.
I.4.1 Semi Circular Canyon Method (Lee and Liu
27
)
Fig. 1.7. 2-D Semi-Circular Canyon (Lee and Liu
27
)
Lee and Liu’s work encompasses the following model shown in Figure 1.7 which
represents a semi-circular canyon in an elastic half-space (y > 0). The geometry of the
24
canyon is transformed from the rectangular coordinate system into a cylindrical
coordinate system with the same origin at O, shown as follows:
(3)
.
The seismic wave is the incident plane P-wave defined by the potential , with
incidence angle θ
α
. The incident angle is measured with respect to the horizontal x-axis.
The waves have a circular frequency ω = 2πf, a longitudinal-wave velocity α, and a
transverse-wave velocity β.
The half-space is elastic, isotropic, and homogeneous, with the following material
properties: Lame constants λ and µ and mass density ρ. The longitudinal-wave velocity α
and transverse-wave velocity β are:
(4)
.
Therefore, the constants k
α
= ω/α (P-wave number) and k
β
= ω/β (SV-wave number) can
be determined.
Plane longitudinal (P) waves enter the half-space at angle θ
α
, resulting in displacements
and a propagation in the x-y plane. The incident plane P-wave is characterized by a
potential
φ
i
and is defined in the x-y plane as:
. (5)
x= rsinθ
y= rcosθ
r= x
2
+ y
2
θ = tan
−1
x
y
⎛
⎝
⎜
⎞
⎠
⎟
φ
i
α =
λ+2µ
ρ
β =
µ
ρ
φ
i
= e
ik
α
xcosθ
α
−ysinθ
α
( )−iωt
25
The time factor is exp(-iωt), where and t
is the time coordinate. The time factor is
removed from the equation because we are only interested in studying the amplitude of
the waves as a function of the canyon geometry and not the time aspect ωt, which
describes the time-dependent
oscillation of the wave due to the mode conversion along
the half-space surface at y = 0, the incident P-wave produces a reflected plane P-wave
with P-wave potential , reflection angle θ
α
, reflected plane SV-wave with a SV-wave
potential , and reflection angle θ
β
.
The mode conversion takes place in order to
satisfy the stress-free boundary conditions τ
yy
= 0 and τ
xy
= 0. The reflected P- and SV-
wave potentials are derived in Achenbach’s paper (Achenbach
1
).
The reflection angles θ
α
and θ
β
,
are calculated using Snell’s Law, for α > β implies θ
α
> θ
β
:
.
(6)
The equation for the reflected P-wave and SV-wave potentials defined in the x-y plane as
Reflected Plane P-Waves (7)
Reflected Plane SV-Waves
.
The reflection coefficients K
1
and K
2
were derived by Cao and Lee and are defined as
(Cao and Lee
5
)
(8)
.
The free-field waves are unaffected by the canyon and become a combination of the input
P-wave and reflected P- and SV-waves. The free-field P-wave potential
φ
ff
and the free-
field SV-wave potential
ψ
ff
are as follows:
(9)
i= −1
φ
r
ψ
r
sinθ
α
α
=
sinθ
β
β
φ
r
= K
1
e
ik
α
xcosθ
α
+ysinθ
α
( )
ψ
r
= K
2
e
ik
β
xcosθ
β
+ysinθ
β ( )
K
1
=
sin2θ
α
sin2θ
β
−κ
2
cos
2
2θ
β
sin2θ
α
sin2θ
β
+κ
2
cos
2
2θ
β
⎛
⎝
⎜
⎞
⎠
⎟
K
2
=
−sin2θ
α
cos2θ
β
sin2θ
α
sin2θ
β
+κ
2
cos
2
2θ
β
⎛
⎝
⎜
⎞
⎠
⎟
φ
ff
=φ
i
+φ
r
= e
ik
α
xcosθ
α
−ysinθ
α
( )
+K
1
e
ik
α
xcosθ
α
+ysinθ
α
( )
26
.
This new method presents a modified and more exact solution of the approximate method
used by in Cao and Lee (Cao and Lee
5
), for an incident P wave, in order to solve the
zero-stress boundary condition on the half-space surface.
Lee and Liu showed that on the half space, the sine terms and the cosine terms are not
independent of each other, and the theory that the {sin nθ} and {cos nθ} terms by
themselves are orthogonal in the half-space, where 0 ≤ θ ≤ π, our new scattered wave
potentials takes the form:
φ
s
= A
n
H
n
(1)
(k
α
r)sinnθ
n=0
∞
∑
(10)
ψ
s
= C
n
H
n
(1)
(k
β
r)sinnθ
n=0
∞
∑
(11)
Therefore, the scatter wave potentials expressed as sine functions can also be expressed
in the form of the cosine function if the sine function is expressed as a cosine series, as
follows:
φ
s
= H
m
(1)
(k
α
r)A
m
sinmθ = H
m
(1)
(k
α
r)
ε
n
π
s
mn
A
m
m=1
n+m=odd
∞
∑
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
n=0
∞
∑
m=1
∞
∑
cosnθ
(12)
ψ
s
= H
m
(1)
(k
β
r)C
m
sinmθ = H
m
(1)
(k
β
r)
ε
n
π
s
mn
C
m
m=1
n+m=odd
∞
∑
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
n=0
∞
∑
m=1
∞
∑
cosnθ
(13)
This expansion of sin mθ into a cosine series is accomplished because the function sin
mθ is an odd function and the range is the half range 0 to π
I.4.2 The Zero Normal and Shear Stress Boundary Condition on the Half-Space
Surface (Lee and Liu
27
)
The zero normal stress boundary condition at the half-space surface is:
ψ
ff
=ψ
r
= K
2
e
ik
β
xcosθ
β
+ysinθ
β ( )
27
σ
θ θ=0,π
= 0
(14)
τ
rθ θ=0,π
= 0
(15)
The first boundary condition is automatically satisfied by changing the SV wave potential
ψ
s
to be in terms of cosine Equation 13 before it’s substituted into Equation 14 to satisfy
the boundary condition for σ
θ,
keeping the P-wave equation in the same form as seen in
Equation 12. Which creates stresses in terms of sin functions only, Equation 16, which
automatically satisfies the zero stress boundary condition at the half-space surface since
sin π = sin 0 = 0.
σ
θ
= E
21
(3)
(n,n,k
α
r)A
n
−
m=0
m+n=odd
∞
∑
E
22
(3)
(n,n,k
β
r)
ε
n
π
C
m
s
mn
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
sinnθ
n=1
∞
∑
(16)
The second boundary condition is automatically satisfied by changing the P-wave
potential to be in terms of cosine, Equation 12 so that when it is substituted back into the
τ
rθ
, it will only be in terms of sine and the SV-wave potential stays the same Equation 13.
Which creates stresses in terms of sin functions only, Equation 17, which automatically
satisfies the zero stress boundary condition at the half-space surface since sin π = sin 0 =
0.
τ
θr
= − E
41
(3)
(m,n,k
α
,r)A
m
s
mn
ε
n
π
+E
42
(3)
(n,n,k
β
r)C
n
m=0
m+n=odd
∞
∑
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
n=0
∞
∑
sinnθ
(17)
I.4.3 The Zero Normal and Shear Stress Boundary Conditions on the Canyon
Surface (Lee and Liu
27
)
At the canyon surface the P- and SV- wave potentials are a combination of both the free-
field waves and the scattered waves. This combination must satisfy the zero-stress
boundary conditions at the surface of the canyon.
28
ϕ =ϕ
ff
+ϕ
s
ϕ
ff
= J
n
(k
α
r) a
n
sinnθ+b
n
cosnθ ( )
∑
ϕ
s
= A
n
H
n
(1)
(k
α
r)sinnθ
∑
ψ =ψ
ff
+ψ
s
ψ
ff
= J
n
(k
β
r) c
n
sinnθ+d
n
cosnθ ( )
∑
ψ
s
= C
n
H
n
(1)
(k
β
r)sinnθ
∑
(18)
The boundary conditions that need to be satisfied at the canyon surface are:
σ
r r=a
=σ
r
ff
+σ
r
s
r=a
=0
τ
rθ r=a
=τ
rθ
ff
+τ
rθ
s
r=a
=0
(19)
The stress-free boundary conditions at the canyon surface, r = a, gives, for 0 ≤ θ ≤ π:
σ
r
s
r=a
=−σ
r
ff
r=a
(20)
and using the knowledge that orthogonality of the {sin nθ} function along the half-space
[0,π],
sinmθsinnθdθ =
π
2
0
0
π
∫
,
m=n
m≠n
cosmθsinnθdθ =S
nm
0
π
∫
,
m+n=odd
m+n=even= 0
(21)
the normal stress equation due to the free-field and scatter waves becomes:
π
2
E
11
(3)
(n,n,k
α
a)A
n
+ E
12
(3)
(m,m,k
β
a)s
nm
C
m
=S
ff
(n)
m=1
m+n=odd
∞
∑
(22)
S
ff
(n)=−
π
2
E
11
(1)
(n,n,k
α
a)a
n
−E
12
(1)
(n,n,k
β
a)d
n
( )
− E
11
(1)
(m,m,k
α
a)b
m
+E
12
(1)
(m,m,k
β
a)c
m
( )
s
nm
m=1
m+n=odd
∞
∑
(23)
The zero shear stress boundary condition at the canyon surface, r = a, gives, for 0 ≤ θ ≤ π:
29
τ
rθ
s
r=a
=−τ
rθ
ff
r=a
(24)
Applying orthogonality of the sine function just like what was done for the case of the
normal stress results in the following equations:
E
41
(3)
(m,m,k
α
a)s
nm
A
m
+
π
2
E
42
(3)
(n,n,k
β
a)C
n
=t
ff
(n)
m=1
m+n=odd
∞
∑
(25)
t
ff
(n)=
π
2
E
41
(1)
(n,n,k
α
a)b
n
−E
42
(1)
(n,n,k
β
a)c
n
( )
− E
41
(1)
(m,m,k
α
a)a
m
+E
42
(1)
(m,m,k
β
a)d
m
( )
s
nm
m=1
m+n=odd
∞
∑
(26)
Equation 22 satisfies the boundary condition for normal stress and 25 satisfies the
boundary conditions for shear stress, therefore representing a set of 2n complex equations
with {An} and {Cn} as the unknowns. The {An} represents the coefficients for the P-
wave potential and the {Cn} represents the coefficients for the SV-wave potential. A
finite number of terms provides an approximate solution.
The displacement amplitudes can now be determined since the coefficients of the P and
SV wave potentials were found in the above section. These displacement amplitudes
which occur on the surface of the half space and within the canyon are important in
studying the variability of ground motions in the vicinity of the canyon. Lee and Liu’s
results are used as a comparison for validity of the methods developed in Chapter 2.
I.5 Objective
This thesis is an extension of prior research on the scattering and diffraction of elastic
waves due to topographic effects such as canyons and valleys. Section 1.4 summarized
the analytical results obtained by Lee and Liu for the Scattering and diffraction of an
incident P-wave on 2-D Semi-Circular Canyon. It also showed that this analytical
method could be used to study Semi-Circular Canyons with various angles of incidence,
different ratios of canyon depth to its half-width and different dimensionless frequencies.
The analytical method has its limitations in that the wave equation can only analyze a
30
regular shape surface defined by one of the six coordinate systems- rectangular,
cylindrical, elliptical, parabolic, spherical, and spheroidal.
This thesis builds upon the prior analytical work by Lee and Liu by using a more versatile
numerical method, the weighted residual method which allows for higher wave
frequencies to be studied. The weighted residual method has the advantage over the
analytical method in that it allows the shape of the canyon and valley to be defined by an
arbitrary shape without defining a coordinate system. An arbitrary shape is more realistic
because nothing in nature is regular. The weighted residual method commonly known to
electrical engineers as the moment method is a powerful method. It allows for Electrical
engineers to solve problems with amplitudes in the range of kilo Hertz and Mega and
Giga hz unlike civil engineers who use Hz. The weighted residual method accomplishes
this task by taking the weighted residue integral along the arbitrary shape and integrating
it, summing up the residuals by expanding it along the weight function. This arbitrary
shape canyon or valley will be defined and the boundary conditions on that canyon will
be defined by the traction vectors perpendicular to the canyon. This method opens many
opportunities for research on different canyon and valley surfaces.
In the following chapters, the diffraction of Plane P- Waves in a 2-D Elastic Half Space
will be presented for different cases that will be compared to prior research. In Chapter II
the Weighted Residual Method is applied to arbitrary-shaped canyons which are almost
circular, semi-circular, elliptical and trapezoidal with a half-width and depth ratio r/a
approximately equal to one and with the cylindrical (polar) coordinate system located at
the half-space of the canyon. The ratio of r/a can vary around the canyon as a function of
the canyon shape. In Chapter III the Weighted Residual Method is applied to “shallow”
arbitrary shaped canyons, circle segment, elliptical, trapezoidal and Nurek Dam where
the depth ratio r/a is less than one, noting that r/a is approximately equal to one at the
half-space boundary. In order to study the shallow canyons the origin of the coordinate
system was moved above the half-space surface to minimize problems with convergence
when the coordinate system was located at the half-space for the “shallow” canyons
causing the Hankel function to produce very large potentials thus causing convergence
problems. With the coordinate system being moved above the half-space and maintaining
31
the ratio r/a approximately equal to one, the convergence problems are minimized. In
Chapter IV the Weighted Residual Method is applied to an Irregular Shaped Alluvial
Valley for a semi-circular, elliptical and trapezoidal for soft soils with a wave speed ratio
β
1
/β < 1 and hard soils for a wave speed ratio β
1
/β >1.
Chapter II, III, IV have demonstrated that this method of weighted residuals works and
my results compare favorably to prior research. All of the results are presented in the
subsequent chapters.
I want to continue my work in this thesis after my defense so I’ve started preliminary
work which is discussed in Chapter V. The specific application that is discussed is the
moon shaped irregular valley which has the possibility of more closely modeling real
world applications. In the Northridge Earthquake, it was observed that the sloping
thickness alluvial layer in Sherman Oaks created basin edge effects because the
earthquake came in from a side that was thick and went into a thinner and thinner layer,
resulting in amplified ground waves. My research has developed the methodology that
can potentially analytically study this observed phenomena.
32
II WEIGHTD RESIDUAL METHOD FOR DIFFRACTION OF PLANE P-
WAVES IN A 2-D ELASTIC HALF-SPACE ON AN ALMOST CIRCULAR
ARBITRARY-SHAPED CANYON
II.1 Introduction
This chapter analyzes the diffraction of an in-plane P-waves in an elastic half-space by
arbitrary-shaped canyons using the weighted residual method. In it, it presents a solution
for any arbitrary shaped canyons where the depth of the canyon is approximately half the
width of the canyon, such as a semi-circle, ellipse or trapezoid.
Section 1.4 summarizes the results from the paper by Lee and Liu (Lee and Liu
19
)
analyzed, the harmonic motion induced by an incident P-wave for a two- dimensional
diffraction around a semi-circular canyon in an elastic half-space using an analytic
solution to satisfy the zero-stress boundary conditions. In past approaches, numerical
approximations of geometry and/or wave functions were made to satisfy the half-space
boundary condition using wave functions that are a function of both sine and cosine. (Cao
and Lee
2,3
)
This chapter expands on Lee and Liu’s new theory that the P- and SV-cylindrical wave
functions can be defined by the sine function only, by defining an arbitrary-shaped
canyon with harmonic motion induced by an incident P-wave. In this study, the weighted
residual method is applied for the solution of the wave function for the arbitrary-shaped
canyon. Unlike Lee and Wu’s weighted residual method (Lee and Wu 1994a,b
17,18
) which
defined the scattered wave potentials as a combination of both the sine and the cosine
functions, and resulted in two sets of scattered P- and scattered SV-waves, (a total of four
sets of waves) so as to satisfy four sets of boundary conditions, two on the half-space, and
two on the canyon, this new method will present a much simpler solution. The coordinate
system in this chapter is located at the half-space surface, which allows for arbitrary-
shaped canyons to be solved. This new simplified method uses only one set of scattered
P- and SV- waves, using the method of Lee and Liu (Lee & Liu 2014
19
) to automatically
satisfy the free-stress boundary conditions at the half-space surface. Using this improved
33
weighted residual method, the results for Lee and Liu’s semi-circle were verified and new
results for an ellipse, trapezoid, and rectangle are presented.
II.2 Model
Fig. 2.1. Arbitrary-shaped with coordinates at the half-space.
The model for the canyon has no restrictions on shape other than the surface of the
canyon must be continuous and defined by a sequential number of points whose polar
coordinates have an increasing value of θ. In Fig. 2.1, the two-dimensional, arbitrary-
shaped canyon in an elastic half-space (y > 0) is defined. Each point will have an (x,y)
coordinate and once transformed into polar coordinates, each point will have an (r, θ)
location. The geometry of the canyon is transformed from the rectangular coordinate
system into a cylindrical coordinate system with the same origin at O, shown as follows:
(27)
.
The figure encompasses the following: an incident plane P-wave defined by the potential
, with incidence angle θ
α
. The incident angle is measured with respect to the
horizontal x-axis. The waves have a circular frequency ω = 2πf, a longitudinal-wave
velocity α, and a transverse-wave velocity β.
x= rsinθ
y= rcosθ
r= x
2
+ y
2
θ = tan
−1
x
y
⎛
⎝
⎜
⎞
⎠
⎟
φ
i
34
Building on Lee and Liu’s (Lee and Liu
19
) theory presented in Chapter 1 which solves the
boundary conditions on the half-space surface, the free field and scattered wave
potentials in the half space take the form of:
(28)
.
(29)
.
II.3 Boundary Conditions for the Canyon Surface
Boundary conditions on the canyon surface must be satisfied for the free-field and
scattered wave potentials. To satisfy these boundary conditions for no force on the
canyon surface, the traction components—radial T
r
, and angular T
θ
—on the surface are
computed (see Fig. 2a). Both traction components must satisfy the condition that there is
zero stress as derived in the papers by Lee and Wu (Lee and Wu1994
17,18
). The equations
for the traction components (30) are derived from the stress Eq. (31):
(30)
.
Fig. 2.2a. Traction components.
φ
ff
=φ
i
+φ
r
= e
ik
α
xcosθ
α
−ysinθ
α
( )
+K
1
e
ik
α
xcosθ
α
+ysinθ
α
( )
ψ
ff
=ψ
r
= K
2
e
ik
β
xcosθ
β
+ysinθ
β ( )
φ
s
= A
n
H
n
(1)
(k
α
r)sinnθ
n=0
∞
∑
ψ
s
= C
n
H
n
(1)
(k
β
r)sinnθ
n=0
∞
∑
T
r
=σ
r
cosα +τ
rθ
sinα = 0
T
θ
=σ
θ
sinα +τ
rθ
cosα = 0
35
Fig. 2.2b. Traction components.
In Fig. 2b, the angle α is the angle between the radial vector
and the normal vector
at each point of the surface that is measured positive in the counter-clockwise direction
from the radial vector. The equations for the stresses are defined as (Mao and Pow
1973
11
):
(31)
.
The stresses are the in-plane stresses induced by the incident P-wave and
reflected P- wave and SV-wave potentials. These stresses are calculated directly from Eq.
31 and by substituting the free-field potentials in Eq. 28.
The stresses due to the scattered waves are calculated by substituting the
equations for the potentials of Eq. 31 into Eq. 29. and shown as:
ˆ r ˆ n
σ
r
= λ∇
2
φ+2µ
∂
2
φ
∂r
2
+
∂
∂r
1
r
∂ψ
∂θ
⎛
⎝
⎜
⎞
⎠
⎟
⎡
⎣
⎢
⎤
⎦
⎥
σ
θ
= λ∇
2
φ+2µ
1
r
∂φ
∂r
+
1
r
∂
2
φ
∂θ
2
⎛
⎝
⎜
⎞
⎠
⎟
+
1
r
1
r
∂ψ
∂θ
−
∂
2
ψ
∂r∂θ
⎛
⎝
⎜
⎞
⎠
⎟
⎡
⎣
⎢
⎤
⎦
⎥
τ
rθ
=µ
1
r
∂
∂θ
∂φ
∂r
+
1
r
∂ψ
∂θ
⎛
⎝
⎜
⎞
⎠
⎟
+
∂
∂r
1
r
∂φ
∂θ
−
∂ψ
∂r
⎛
⎝
⎜
⎞
⎠
⎟
−
1
r
1
r
∂φ
∂θ
−
∂ψ
∂r
⎛
⎝
⎜
⎞
⎠
⎟
⎡
⎣
⎢
⎤
⎦
⎥
τ
rθ
i+r
,σ
r
i+r
,σ
θ
i+r
σ
r
s
,τ
rθ
s
,σ
θ
s
36
Therefore, on the canyon, the boundary condition of zero stress that must be satisfied on
the surface of the arbitrary shape are a combination of the free-field and scattered stresses
(Lee and Wu
17,18
):
(32)
.
II.4 Application of Weighted Residual Method
The functions that define the stresses are an infinite summation. Therefore, the traction
equations are an infinite summation. An approximate solution would use a finite
summation with N terms and thus 2N unknowns, A
n
and C
n
, n = 1 to N. The procedure
for solving these two equations, traction Eq. 32, is a special case of the method of
moments defined in Roger Harrington’s paper, “Matrix Methods for Field Problems”
(Harrington
4
). A set of weighting functions, w
1
, w
2
, w
3
, …w
N
, in the range of 0 to π, is
defined and applied to the traction terms. This results in 2N equations that require
integration from 0 to π. Applying the method of weighted residuals, the weighting
function is chosen as:
.
(33)
For m=1,2….N, the function is
(34)
.
Since sin(mθ) equals zero when m = 0, Eq. 34 represent 2N pairs of equations with 2N
unknowns, A
n
and C
n
, for n=1, 2,….N.
II.5 Numerical Solutions
The equations for traction, Eq. 34, form a set of complex simultaneous equations with
unknowns A
n
and C
n
, and coefficient matrix with terms a
ij
,, c
ij
and constant terms r
i
,
shown as:
T
r
s
+T
r
i+r
=0
T
θ
s
+T
θ
i+r
=0
w
m
= sin mθ
T
r
s
( )
N
w
m
dθ = −
θ
∫
T
r
i+r
( )
N
w
m
dθ
θ
∫
T
θ
s
( )
N
w
m
dθ =
θ
∫
− T
θ
i+r
( )
N
w
m
dθ
θ
∫
37
(35)
The terms of the coefficient matrix are determined by substituting the stress equations for
the scattered waves (Eq. 29) into the traction equations T
r
and T
θ
,, (Eq. 30) and
multiplying each equation by the weighting function (Eq. 33). Each term requires
integration along the boundary of the canyon from 0 to π. The equations are integrated
using the Gaussian quadrature method. The surface of integration—the canyon surface—
is divided into 400 segments defined by 401 points. Each segment is subdivided again by
the 10-point Gauss-Legendre integration. This integration technique is highly accurate
when the integrand is very smooth, which is the case for the sine and cosine functions, as
the method converges much more quickly than in other integration schemes. (Willam
H.Press
6
)
The constant terms r
1
to r
2N
are calculated from tractions computed from the free-field
waves. These free-field stresses are substituted into the equations for traction (Eq. 30)
and then multiplied by the weighting function (Eq. 33), and integrated along the
boundary from 0 to π.
The set of 2N complex equations are solved for the unknowns A
1
to A
N
, and C
1
to C
N
.
II.6 Comparison of Results to Previous and Existing Studies
From the above analysis, which determined the coefficient A
n
and C
n
of the P- and SV-
wave potentials, the displacement amplitudes can now be determined. These
displacement amplitudes of interest occur on the surface of the half-space and within the
canyon surface. The amplitudes are important in studying the variability of ground
motions in the vicinity of a canyon.
a
11
... a
1N
c
11
... c
1N
. .
. .
. .
. .
a
2N,1
... a
2N,N
c
2N,1
... c
2N,N
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
⎥
⎥
A
1
.
A
n
C
1
.
C
n
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
=
r
1
.
.
.
.
r
2N
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
⎥
⎥
38
The free-field displacements are calculated by substituting the incident- and reflected-
wave potentials into Eq. 36:
u
x
ff
=
∂φ
∂x
+
∂ψ
∂y
u
y
ff
=
∂φ
∂y
−
∂ψ
∂x
(36)
The scatter-wave displacements are calculated by substituting the P- and SV-wave
potentials into Eq. 37:
u
r
s
=
∂φ
∂r
+
1
r
∂ψ
∂θ
u
θ
s
=
1
r
∂φ
∂θ
−
∂ψ
∂r
(37)
The resulting displacements for the scatter waves are in the following form:
(38)
(39)
.
In order to transform the scatter wave displacements
u
r
,
u
θ
in polar coordinates to
displacements
u
x
and
u
y
in rectangular coordinates, a transformation is used. The
transformation is as follows:
.
(40)
u
r
s
=
1
r
A
n
sinnθE
71
(3)
+C
n
cosnθE
72
(3)
⎡
⎣
⎤
⎦
u
θ
s
=
1
r
A
n
cosnθE
81
(3)
+C
n
sinnθE
82
(3)
⎡
⎣
⎤
⎦
E
72
(3)
=nH
n
(1)
(k
β
r)
E
71
(3)
= k
α
rH
n−1
(1)
(k
α
r)−nH
n
(1)
(k
α
r) ⎡
⎣
⎤
⎦
E
81
(3)
=nH
n
(1)
(k
α
r)
E
82
(3)
= − k
β
rH
n−1
(1)
(k
β
r)−nH
n
(1)
(k
β
r) ⎡
⎣
⎤
⎦
u
x
u
y
⎧
⎨
⎪
⎩
⎪
⎫
⎬
⎪
⎭
⎪
=
cosθ
1
−sinθ
1
sinθ
1
cosθ
1
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
u
r
u
θ
⎧
⎨
⎪
⎩
⎪
⎫
⎬
⎪
⎭
⎪
39
The total displacement amplitudes are a linear combination of the scattered and the free-
field waves, and the result of the real (Re) and imaginary (Im) parts
(41)
.
The phase angle of the points on the canyon and the surface of the half-space can be
calculated by using
. (42)
These two-dimensional displacement amplitudes, horizontal and vertical , and
the phase angle are plotted vs. the dimensionless horizontal distance x/a for a specific
dimensionless frequency η and an angle of incidence
θ
α
. The following plots use the
dimensionless frequency parameters η:
, (43)
where a is the radius of the canyon, λ
β
is the wave length of the shear wave at frequency
ω, β is the shear wave speed,κ
β
is the wave number, f is the cyclic frequency defined as ω
= 2πf, with a Poisson ratio of ν = 0.25. To verify the validity of this numerical method,
each plot was generated using an increasing value of N
max
—the total number of
equations—until convergence was achieved. The figures have a depth-to-half-width ratio
of h/a=1 and their displacement amplitudes on the half-space surface and the surface of
the canyon (r
1
=a) are plotted along the horizontal x-axis in the interval -4 ≤ x/a ≤ 4. The
point x/a = -1 corresponds to the left rim of the canyon, x/a = 0 to the bottom, and x/a = 1
to the right rim. The incident P-waves are assumed to arrive from the left (x/a < 0) in all
cases. The displacement amplitudes were computed and compared with the results
obtained by the closed-form analytic solutions presented in Lee and Liu(Lee and Liu
19
) as
shown in Fig. 4, which can be compared to the results shown in Fig.5.
u
x
= Re
2
(u
x
ff
+u
x
s
)+Im
2
(u
x
ff
+u
x
s
)
( )
1/2
u
y
= Re
2
(u
y
ff
+u
y
s
)+Im
2
(u
y
ff
+u
y
s
)
( )
1/2
φ = tan
−1
Im(u)/Re(u) [ ]φ = tan
−1
Im(u)/Re(u) [ ]
u
x
( ) u
y ( )
η=
2a
λ
β
=
k
β
a
π
=
ωa
πβ
40
Fig. 2.3. Semi-circular canyon.
41
Fig. 2.4. Arbitrary-shaped semi-circular canyon x/a vs. Ux and Uy for η=10, θ=60°,
N
max
=112 (results from Lee and Liu
19
).
42
x-component displacement y-component displacement
x/a x/a
Fig. 2.5. Weighted residual method recreated semi-circular canyon matches Lee and
Liu’s results η=10, θ=60°, N
max
=126.
In Fig. 2.5, the results are shown for the circular canyon with a coordinate system at the
half-space for θ
α
= 60, η = 10, and N
max
=112 using the weighted residual method verified
by the results of Lee and Liu (Lee and Liu
19
). The results shown in Fig. 2.5 can be
compared with the results in Fig. 2.4 for the incident P-wave on a 2D semi-circular
canyon, which was computed using Lee and Liu’s analytic solution with the matrix
equations of order from N = 104 - 112. The results matched Lee and Liu’s results using N
= 112 terms.
II.7 The Case of the Semi-Circular Canyon
The results for the case of the semi-circular canyon can also be compared theoretically by
matching the boundary condition equations derived from the weighted residual method
with those derived from Lee and Liu’s exact closed-form solution (Lee and Liu
19
). For
the incident P-wave, the total waves are a combination of the free-field waves and
scattered waves as follows:
φ
ff
= J
n
(k
α
a)(a
n
cosnθ+b
n
sinnθ)
n=0
∞
∑
φ
ff
= J
n
(k
α
a)(a
n
cosnθ+b
n
sinnθ)
n=0
∞
∑ (44)
(45)
φ
s
= A
n
H
n
(1)
(k
α
r)sinnθ
n=0
∞
∑
φ
s
= A
n
H
n
(1)
(k
α
r)sinnθ
n=0
∞
∑ (46)
0"
1"
2"
3"
4"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
ψ
ff
= J
n
(k
β
a)(c
n
sinnθ+d
n
cosnθ)
n=0
∞
∑
Angle
of
Incidence=60°
43
. (47)
This assumes that it is provided that the constants a
n
, b
n
, c
n
, d
n
are defined as follows:
, (48)
and taking into consideration the stress-free boundary condition at the surface of the
semi-circular canyon at r = a,
(49)
,
(50)
for the case of a semi-circular canyon, α=0. The orthogonality of the sine function gives
(51)
for m + n odd, and for m + n even s
nm
= 0. Therefore, the two boundary condition
equations reduce down to
(52)
.
(53)
Simplifying even further, Eq. 52 becomes
,
(54)
and Eq. 53 becomes
ψ
s
= C
n
H
n
(1)
(k
β
r)sinnθ
n=0
∞
∑
a
n
= sinnθ
α
(K
1
−1)ε
n
i
n
b
n
= cosnθ
α
(K
1
+1)ε
n
i
n
c
n
= sinnθ
β
K
2
ε
n
i
n
d
n
= cosnθ
β
K
2
ε
n
i
n
σ
r
s
cosα +τ
rθ
s
sinα
( )
θ
∫
sinmθdθ = σ
r
ff
cosα +τ
rθ
ff
sinα
( )
θ
∫
sinmθdθ
σ
θ
s
sinα +τ
rθ
s
cosα
( )
θ
∫
sinmθdθ = σ
θ
ff
sinα +τ
rθ
ff
cosα
( )
θ
∫
sinmθdθ
sinnθsinmθ =
π
2
0
π
∫
cosnθsinmθ =s
mn
0
π
∫
π
2
E
11
(3)
A
n
+ E
12
(3)
m+n
odd
∞
∑
s
mn
C
n
=−
π
2
E
11
(1)
a
n
−E
12
(1)
d
n
( )
− E
11
(1)
b
n
+E
12
(1)
c
n
( )
m+n
odd
∞
∑
s
mn
π
2
E
42
(3)
C
n
+ E
41
(3)
m+n
odd
∞
∑
s
mn
A
n
=−
π
2
E
41
(1)
b
n
−E
42
(1)
c
n
( )
− E
41
(1)
a
n
+E
42
(1)
d
n
( )
m+n
odd
∞
∑
s
mn
π
2
E
11
(3)
A
n
+ E
12
(3)
m+n
odd
∞
∑
s
mn
C
n
= s
ff
(n)
s
ff
(n)=−
π
2
E
11
(1)
a
n
−E
12
(1)
d
n
( )
− E
11
(1)
b
n
+E
12
(1)
c
n
( )
m+n
odd
∞
∑
s
mn
44
. (55)
Equations 54 and 55 are two boundary conditions that are identical to those derived in
Lee and Liu’s exact solution for a semi-circular canyon. Therefore, with the choice of the
sine function as the weighting function, the weighted residual methods resulted in the
exact closed-form solution for the case of a semi-circular canyon in an elastic half-space.
II.8 Application to Other Arbitrary Shapes
This new methodology to solving the arbitrary-shaped canyon is applicable to any
arbitrary-shaped canyon. This section of the paper will look at an elliptical-shaped
canyon and a trapezoidal-shaped canyon. For each shape, the number of equations, N
max
,
increases until convergence is reached or the problem becomes numerically unstable. The
procedure that has been applied in this paper, the method of moments using weighting
functions, is an approximate technique and its accuracy is a function of convergence as
the infinite series are truncated to a finite number of unknown terms: A
1
….A
n
and
C
1
….C
n
. Three factors that affect the results are the number of terms, the stability of the
Bessel functions, and integration. The cases of shallow canyons will be described and
presented in the next paper.
II.8.1 Elliptical Canyon
Fig. 2.6. Arbitrary-shaped elliptical canyon.
π
2
E
42
(3)
C
n
+ E
41
(3)
m+n
odd
∞
∑
s
mn
A
n
=t
ff
(n)
t
ff
(n)=−
π
2
E
41
(1)
b
n
−E
42
(1)
c
n
( )
− E
41
(1)
a
n
+E
42
(1)
d
n
( )
m+n
odd
∞
∑
s
mn
45
The results for an elliptical canyon of depths b/a = 1.25 and 1.5, as shown in Fig. 2.6, are
presented in this section. The incident p-waves are assumed to arrive from the front side
of the canyon for all cases. The results are presented in figures that show the x- and y-
displacement amplitudes for an angle of incidence of θ = 5°, 30°, 60°, and 90°. The case
for θ = 30° is for an oblique incident P-wave and the case of θ= 90° is for a vertical
incident P-wave. The x- and y-displacement amplitudes are plotted in the figures on the
horizontal axis from x/a = -4 to x/a = +4. For convenience, each graph has three areas of
interest: the front side (in which the waves arrive first x/a=-4 to x/a=-1), the back side
(from x/a = +1 to x/a = +4), and the surface of the canyon (from x/a =-1 to x/a =+1). The
results for h = 1.25 are shown in Figs. 2.7 and 2.8 and the results for b =1.5 are shown in
Figs. 2.9 and 2.10:
46
x-component displacement y-component displacement
x/a x/a
Fig. 2.7. Ellipse-shaped canyon b/a = 1.25, η= 2, θ = 5°, 30°,60°,90°, N
max
= 70.
Figure 2.7 shows the results for an elliptical canyon of depth b/a=1.25, a dimensionless
frequency of η = 2, and an incident angle of θ = 5°, 30°, 60°, and 90°. For the graphs
0"
1"
2"
3"
4"
5"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
5"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
5"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
5"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
5"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
5"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
5"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
5"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
Angle
of
Incidence=5°(Hor)
Angle
of
Incidence=30°
Angle
of
Incidence=60°
Angle
of
Incidence=90°(Ver)
47
with incident angle θ= 5°, the x-component displacement amplitudes slightly oscillate
about the free-field amplitude on the front side of the canyon and produce a shadowy
behavior along the back side of the canyon. Within the canyon surface, both the x- and y-
displacement amplitudes are oscillatory, producing a spike on the back side rim of the
canyon at x/a = +1. The y-component displacement amplitudes slightly oscillate about the
free-field amplitude on the front side of the canyon. On the back side of the canyon, the
oscillations are very steady, showing a shadowy behavior. Within the canyon surface, the
displacement amplitudes are oscillatory with a spike at the rims of the canyon at x/a = +1.
For the graphs with incident angle θ= 30°, the x-component displacement amplitudes
tend to oscillate about the free-field amplitude on the front side of the canyon, but
gradually oscillate and produce a shadowy behavior along the back side of the canyon.
Within the canyon surface both the x- and y-displacement amplitudes are highly
oscillatory and produce a spike on the front side and back side rim of the canyons. The y-
component displacement amplitudes tend to oscillate about the free-field amplitude on
the front side of the canyon, and produce a shadowy behavior along the back side of the
canyon. Within the canyon surface, the displacement amplitudes are highly oscillatory
with spikes at the rims of the canyon x/a = +1 and x/a = -1. For the graphs with incident
angle θ= 60°, the x- and y-component displacement amplitudes exhibit the same behavior
as the graphs for incident angle 30°, with slightly smaller amplitudes and small corner
spikes. For the graphs with incident angle of θ = 90°, the x- and y-component
displacement amplitudes are symmetric about the origin 0. The free-field amplitudes
exhibit decaying oscillatory amplification on the front side of the canyon and shadowy
behavior on the back side of the canyon. Within the canyon surface, the displacement
amplitudes are slightly oscillatory with spikes at both rims of the canyon for
displacement x/a = +1 and x/a = -1 for the x-component displacement amplitude.
48
x-component displacement y-component displacement
x/a x/a
Fig. 2.8. Ellipse-shaped canyon b/a = 1.25, η = 8, θ = 5°, 30°,60°,90°, N
max
= 104.
Figure 2.8 shows the results for an elliptical canyon of depth b/a = 1.25, dimensionless
frequency η = 8, and incidence angles of θ= 5°, 30°, 60°, and 90°. For the graphs with
0"
1"
2"
3"
4"
5"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
5"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
5"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
5"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
5"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
5"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
5"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
5"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
Angle
of
Incidence=5°(Hor)
Angle
of
Incidence=30°
Angle
of
Incidence=60°
Angle
of
Incidence=90°(Ver)
49
incident angle θ = 5°, the x-component displacement amplitudes oscillate about the free-
field amplitude on the front side of the canyon and produce a shadowy behavior along the
back side. Within the canyon surface, both the x- and y-displacement amplitudes are
oscillatory, producing spikes on the front side of the canyon rim at x/a = -1 and the back
side rim at x/a = +1. The y-component displacement amplitudes oscillate about the free-
field amplitude on the front side and the back side of the canyon. The oscillations are
very steady, showing a shadowy behavior. Within the canyon surface, the displacement
amplitudes are oscillatory with spikes at the rims of the canyon at x/a = +1 and x/a = -1.
For the graphs with t incident angle θ = 30°, the x-component displacement amplitudes
tend to rapidly oscillate about the free-field amplitude on the front side of the canyon, but
gradually oscillate on the back side of the canyon. Within the canyon surface, both the x-
and y-displacement amplitudes are highly oscillatory and only produce a spike on the
back side rim of the canyon at x/a = +1. The y-component displacement amplitudes tend
to oscillate about the free-field amplitude on the front side of the canyon. On the back
side of the canyon, the oscillations are very steady, showing a shadowy behavior. Within
the canyon surface, the displacement amplitudes are highly oscillatory with spikes at the
canyon rims at x/a = +1 and x/a =-1. These spikes are due to the sharp corners, which
result in a change of curvature. For the graphs with incident angle θ= 60°, the x- and y-
component displacement amplitudes exhibit the same behavior as the graphs for the
incident angle of 30° with slightly smaller amplitudes and no corner spike. For the graphs
with incident angle θ = 90°, the x- and y-component displacement amplitudes are
symmetric about the origin 0. The free-field amplitudes exhibit decaying oscillatory
amplification on the front side of the canyon and a shadowy behavior on the back side of
the canyon. Within the canyon surface, the displacement amplitudes are slightly
oscillatory with spikes at both rims of the canyon for displacement of x/a = +1 and x/a =
-1.
50
x-component displacement y-component displacement
x/a x/a
Fig. 2.9a. Ellipse-shaped canyon b/a = 1.5 η = 2, θ = 5°, 30°, N
max
= 60.
0"
2"
4"
6"
8"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
!4#
!3#
!2#
!1#
0#
1#
2#
3#
4#
!4# !2# 0# 2# 4#
!4#
!3#
!2#
!1#
0#
1#
2#
3#
4#
!4# !3# !2# !1# 0# 1# 2# 3# 4#
0"
2"
4"
6"
8"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
!4#
!3#
!2#
!1#
0#
1#
2#
3#
4#
!4# !3# !2# !1# 0# 1# 2# 3# 4#
!4#
!3#
!2#
!1#
0#
1#
2#
3#
4#
!4# !3# !2# !1# 0# 1# 2# 3# 4#
Angle
of
Incidence=5°
Angle
of
Incidence=30°
Phase
θ=30°
Phase
θ=5°
51
x-component displacement y-component displacement
x/a x/a
Fig. 2.9b. Ellipse-shaped canyon b/a = 1.5 η = 2, θ = 60°,90°, N
max
= 60.
Figures 2.9a,b show the results for an elliptical canyon of depth b/a=1.5, a
dimensionless frequency of η=2, and an incidence angle of θ = 5°, 30°, 60°, and 90°.
0"
2"
4"
6"
8"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
!4#
!3#
!2#
!1#
0#
1#
2#
3#
4#
!4# !3# !2# !1# 0# 1# 2# 3# 4#
!4#
!3#
!2#
!1#
0#
1#
2#
3#
4#
!4# !3# !2# !1# 0# 1# 2# 3# 4#
0"
2"
4"
6"
8"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
!4#
!3#
!2#
!1#
0#
1#
2#
3#
4#
!4# !3# !2# !1# 0# 1# 2# 3# 4#
!4#
!3#
!2#
!1#
0#
1#
2#
3#
4#
!4# !3# !2# !1# 0# 1# 2# 3# 4#
Angle
of
Incidence=60°
Angle
of
Incidence=90°
Phase
θ=90°
P
Phase
θ=60°
52
The displacement amplitudes tend to look similar along the free-field surface of the front
side of the canyon and back side of the canyon
as seen in Fig. 2.7 for the ellipse with b =
1.25. Figure 2.9 shows significant spikes at the rims of the canyons for an angle of
incidence of θ=60°.
In Fig. 2.9, the phase diagrams are shown alongside the corresponding displacement
amplitudes. All phase diagrams have been scaled by π and shifted arbitrarily to have a
zero phase angle at x/a = 0.
53
x-component displacement y-component displacement
x/a x/a
Fig. 2.10. Ellipse-shaped canyon b/a = 1.5, η = 8, θ = 5°, 30°,60°,90°, N
max
= 100.
Fig. 2.10. shows the results for elliptical canyon depth b/a=1.5, dimensionless frequency
η = 8, and angles of incidence θ = 5°, 30°, 60°, and 90°. The displacement amplitudes
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
Angle
of
Incidence=5°
Angle
of
Incidence=30°
Angle
of
Incidence=60°
Angle
of
Incidence=90°
54
tend to look similar along the free-field surface of the front side of the canyon and back
side of the canyon as those seen in Fig. 2.8 for the ellipse with h = 1.25. Figure 2.10
shows significant spikes at the rims of the canyons and much less oscillatory behavior
within the surface of the canyon, producing shadow-like behavior.
II.8.2 Trapezoidal Canyons
Fig. 2.11. Trapezoidal canyon.
The results for the trapezoidal canyon, as shown in Fig. 2.11, are presented in this
section. The trapezoidal canyon is defined by h/a = 1, with sloping sides of 60° and 45°
measured from the horizontal axis. The incident P-waves are assumed to arrive from the
front side of the canyon for all cases. Each figure shows the x- and y-displacement
amplitudes for angles of incidence θ= 30°, 60°, and 90°. The case for θ= 30° is for an
oblique incident P-wave, and the case of θ= 90° is for a vertical incident P-wave. The x-
and y-displacement amplitudes are plotted in the figures on the horizontal axis from x/a =
-4 to x/a = +4. For convenience, each graph has three areas of focus: the front side- from
x/a = -4 to x/a - 1, the back side-from x/a + 1 to x/a = + 4, and the surface of the canyon
from x/a = -1 to x/a + 1.
55
x-component displacement y-component displacement
x/a x/a
Fig. 2.12. Trapezoid-shaped canyon h/a = 1, η = 2, θ = 5°, 30°, 60°, 90°, N
max
= 28
Slope 60°.
0"
1"
2"
3"
4"
5"
6"
)4" )3" )2" )1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
5"
6"
)4" )3" )2" )1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
5"
6"
)4" )3" )2" )1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
5"
6"
)4" )3" )2" )1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
5"
6"
)4" )3" )2" )1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
5"
6"
)4" )3" )2" )1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
5"
6"
)4" )3" )2" )1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
5"
6"
)4" )3" )2" )1" 0" 1" 2" 3" 4"
Angle
of
Incidence=5°
Angle
of
Incidence=30°
Angle
of
Incidence=60°
Angle
of
Incidence=90°
56
Figure 2.12 shows the results for a trapezoidal canyon with depth of h/a = 1,
dimensionless frequency η = 2, a slope of 60°, and incidence angles of θ = 30°, 60°, and
90°. The x- and y-displacement amplitudes are plotted in the figures on the horizontal
axis from x/a = -4 to x/a = +4. For the graphs with incident angle θ= 5°, the x-component
displacement amplitudes gradually oscillate about the free-field amplitude on the front
side of the canyon and produce a shadowy behavior along the back side of the canyon.
Within the canyon surface, both the x- and y-displacement amplitudes are oscillatory,
producing clustered spikes at the canyon rims. The y-component displacement
amplitudes oscillate about the free-field amplitude on the front side of the canyon. On the
back side of the canyon, oscillations are very steady and show shadowy behavior. Within
the canyon surface, the displacement amplitudes are oscillatory with clustered spikes at
the canyon rims. For the graphs with incident angles θ = 30° and θ = 60°, similar trends
are observed for the x-component displacement amplitudes and y-component
displacement amplitude with larger amplitude-clustered spikes on the front sloping side
of the canyon (x/a=-1 to x/a=-0.5) and on the back sloping side of the canyon (x/a=+0.5
to x/a=+1). For the graphs with incident angle θ= 90°, the x-component displacement
amplitudes are symmetric about the origin. They exhibit a shadowy behavior about the
free-field amplitudes on the front and backside of the canyon. Within the canyon surface,
the displacement amplitudes are highly oscillatory with very large clustered spikes on the
sloping sides. The y-component displacement amplitudes are symmetric about the origin
and exhibit a shadowy behavior along the free-field amplitudes on the front and back side
of the canyon. Within the canyon surface, displacement amplitudes are highly oscillatory
with clustered spikes on the sloping sides.
57
x-component displacement y-component displacement
x/a x/a
Fig. 2.13. Trapezoid-shaped canyon h/a = 1, η = 6, θ = 5°, 30°, 60°, 90°, N
max
= 38
Slope 60°.
0"
2"
4"
6"
8"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
Angle
of
Incidence=5°
Angle
of
Incidence
θ=30°
Angle
of
Incidence=60°
Angle
of
Incidence=90°
58
Figure 2.13 shows the results for a trapezoidal canyon of depth h/a=1, dimensionless
frequency η = 6, a slope of 60°, and incidence angles θ= 30°, 60°, and 90°. The x- and y-
displacement amplitudes are plotted in the figures on the horizontal axis from x/a = -4 to
x/a = +4. For the graphs with an incident angle of θ= 5°, the x-component displacement
amplitudes oscillate about the free-field amplitude on the front side of the canyon, and
produce a shadowy behavior along the back side of the canyon. Within the canyon
surface, both the x- and y-displacement amplitudes are oscillatory, producing clustered
spikes at the canyon rims. The y-component displacement amplitudes oscillate about the
free-field amplitude on the front side of the canyon. On the back side of the canyon, the
oscillations are very steady, showing a shadowy behavior. Within the canyon surface, the
displacement amplitudes are oscillatory with clustered spikes at the rims of the canyon.
For the graphs with incident angle θ = 30°, the x-component displacement amplitudes
tend to rapidly oscillate about the free-field amplitude on the front side of the canyon but
gradually oscillate on the back side of the canyon. Within the canyon surface, the
displacement amplitudes are highly oscillatory with clustered spikes on the front-sloping
side (x/a =-1 to x/a =-0.5) and on the back-sloping side of the canyon (x/a=+0.5 to
x/a=+1). The oscillations are much more oscillatory than the ones observed for deep
elliptical canyons. The y-component displacement amplitudes tend to oscillate about the
free-field amplitude on the front side of the canyon. On the back side, the oscillations are
very steady, showing a shadowy behavior. Within the canyon surface, the displacement
amplitudes are highly oscillatory with small clustered spikes at the front-sloping side of
the canyon and larger clustered spikes at the back-sloping side. The oscillations are at a
higher frequency than the ones observed for the deep elliptical canyons. The graphs with
incident angle θ= 60° display the same trends as the ones for incidence angle θ = 30°
except without the spikes at the sloping sides of the trapezoidal canyon. For the graphs
with incident angle θ = 90°, the x-component displacement amplitudes are symmetric
about the origin and exhibit no amplitude change about the free-field amplitudes on the
front and back side of the canyon. Within the canyon surface, the displacement
amplitudes are highly oscillatory with very large clustered spikes on the sloping sides.
The y-component displacement amplitudes are symmetric about the origin and exhibit a
shadowy behavior along the free-field amplitudes on the front and back side of the
59
canyon. Within the canyon surface, displacement amplitudes are highly oscillatory with
clustered spikes on the sloping sides.
60
x-component displacement y-component displacement
x/a x/a
Fig. 2.14. Trapezoid-shaped canyon h/a = 1, η = 2, θ = 5°, 30°, 60°, 90°, N
max
= 18
Slope 45°.
0"
2"
4"
6"
8"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
\"
0"
2"
4"
6"
8"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
Angle
of
Incidence=5°
Angle
of
Incidence=30°
Angle
of
Incidence=60°
Angle
of
Incidence=90°
61
Figure 2.14 shows the results for a trapezoidal canyon of depth h/a=1, dimensionless
frequency η = 2, a slope of 45°, and angles of incidence θ= 5°, 30°, 60°, and 90°. For the
graphs with incident angle θ = 5°, the x-component displacement amplitudes gradually
oscillate about the free-field amplitude on the front side of the canyon and produce a
shadowy behavior along the back side of the canyon. Within the canyon surface both the
x- and y-displacement amplitudes are oscillatory. The y-component displacement
amplitudes exhibit a very steady shadowy behavior on the front and back side of the
canyon. Within the canyon surface, the displacement amplitudes are oscillatory. For the
graphs with the incident angles of θ = 30° and θ = 60°, the x- and y-component
displacement amplitudes exhibit the same trends as those seen for the incident angle θ =
5°. For the graphs with incident angle θ = 90°, the x-component displacement amplitudes
are symmetric about the origin and tend to exhibit a shadowy behavior along the free-
field amplitudes on the front side and back side of the canyon. Within the canyon surface,
the displacement amplitudes are highly oscillatory with very large clustered spikes at the
rims of the canyon at x/a =+1 and x/a =-1. The y-component displacement amplitudes are
symmetric about the origin and they also exhibit a shadowy behavior about the free-field
amplitudes on the front side and back side of the canyon. Within the canyon surface, the
displacement amplitudes are highly oscillatory with clustered spikes at the center of the
canyon.
62
x-component displacement y-component displacement
x/a x/a
Fig. 2.15. Trapezoid-shaped canyon h/a = 1, η = 6, θ = 5°, 30°, 60°, 90°, N
max
= 28
Slope 45°.
0"
2"
4"
6"
8"
10"
12"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
12"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
12"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
12"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
12"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
12"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
12"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
12"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
Angle
of
Incidence=5°
Angle
of
Incidence=30°
Angle
of
Incidence=60°
Angle
of
Incidence=90°
63
Figure 2.15 shows the results for a trapezoidal canyon of depth h/a = 1, dimensionless
frequency η = 6, slope of 45°, and angles of incidence θ= 5°, 30°, 60°, and 90°. For the
graphs with an incident angle of θ = 5°, the x-component displacement amplitudes
oscillate about the free-field amplitude on the front side of the canyon and produce a
shadowy behavior along the back side of the canyon. Within the canyon surface, both the
x- and y-displacement amplitudes are oscillatory, producing clustered spikes on the rims
of the canyon rim. The y-component displacement amplitudes oscillate about the free-
field amplitude on the front side of the canyon. On the back side of the canyon, the
oscillations are very steady, showing a shadowy behavior. Within the canyon surface, the
displacement amplitudes are oscillatory with clustered spikes that peak at the center of
the canyon. For the graphs with incident angle θ = 30°, the x-component displacement
amplitudes slightly oscillate about the free-field amplitudes on the front side of the
canyon and exhibit a shadowy behavior along the back side of the canyon. Within the
canyon surface, the displacement amplitudes are highly oscillatory, oscillating much
more rapidly than the ones observed for the trapezoidal canyons with a slope of 60°. The
y-component displacement amplitudes exhibit a shadowy behavior about the free-field
amplitudes on the front side and along the back side of the canyon. Within the canyon
surface, the displacement amplitudes are highly oscillatory. For the graphs with incident
angle θ = 60°, the x-component displacement amplitudes rapidly oscillate about the free-
field amplitudes on the front side of the canyon, but gradually oscillate on the back side
of the canyon. Within the canyon surface, the displacement amplitudes are highly
oscillatory with large clustered spikes at the rims of the canyon at x/a = -1 and x/a = +1.
The y-component displacement amplitudes gradually oscillate about the free-field
amplitudes on the front side of the canyon and along the back side of the canyon. Within
the canyon surface, the displacement amplitudes are highly oscillatory. For the graphs
with the incident angle θ = 90°, the x-component displacement amplitudes are symmetric
about the origin, and they tend to exhibit a shadowy behavior along the free-field
amplitudes on the front side and the back side of the canyon. Within the canyon surface,
the displacement amplitudes are highly oscillatory with very large clustered spikes at the
canyon rims of x/a =+1 and x/a =-1. The y-component displacement amplitudes are
symmetric about the origin, and they also exhibit a shadowy behavior about the free-field
64
amplitudes on the front side and around the back side of the canyon. Within the canyon
surface, the displacement amplitudes are highly oscillatory with clustered spikes at the
center of the canyon.
II.9 Observations
Results of the current research have been presented for various shaped canyons: semi-
circular, elliptical, and trapezoidal. These results have been presented in Figs. 2.7 – 2.15
and the following general observations can be made:
(1) This is the first time that Lee and Liu’s redefined cylindrical-wave function has
been used to satisfy the zero stress-boundary condition along the half-space for
arbitrary-shaped canyons. The application of the weighted residual method for the
solution of the wave function allows for a solution to non-circular arbitrary-
shaped canyons.
(2) This approach was applied to a semi-circular canyon with η = 10, N
max
= 112, and
θ = 60°. The results were compared to those of Lee and Liu
4
and shown to match
exactly. The same amplification of surface displacements, as seen in previous
studies, is also demonstrated in these results.
(3) This approach was applied to elliptical-shaped canyons with varying b/a
(depth/width) ratios of 1.25 and 1.5. The resulting plots for the displacement
amplitudes are for elliptical canyons with a higher frequency range than those for
semi-circular canyons in previous studies.
(4) The elliptical canyon with b/a = 1.25 produced total displacement amplitudes on
both the x- and y- components for η = 8, which were highly oscillatory around the
free-field surface on the front side of the canyon, and produced a more shadowy
behavior along the back side of the canyon. Within the canyon, there were
oscillations with spikes at the rims of the canyon at x/a = -1 and x/a = +1. The
oscillations along the surface became more rapid as η increased. Increasing the
elliptical depth to b = 1.5 decreased the oscillations within the canyon surface and
produced larger spikes with a greater magnitude at the front-side canyon rim at
x/a = -1.
65
(5) This approach was then applied to the trapezoidal canyon. The total displacement
amplitudes of both the x- and y-components on the half-space showed a trend of
being oscillatory on the front side of the canyon around the free-field amplitudes.
On the back side of the canyon, the x-component amplitudes were oscillatory
about the free field, but the y-component amplitudes produced a shadowy
behavior around the free field. Both the x- and y-components are highly
oscillatory about the canyon surface.
(6) At the trapezoidal canyon rims, spikes in the displacement amplitudes were
observed. The spikes are consistent with the spikes that were observed for the
semi-circular and elliptical canyons. These spikes tend to increase in amplitude at
higher dimensionless frequencies.
(7) Changing the slope of the trapezoidal canyon walls from 60° to 45° demonstrated
that the amplitudes of both the x- component and y-component appeared sensitive
to the angle of incidence and the slope of the canyon sides. The amplification of
the x-component amplitudes and y-component amplitudes changed from the
canyon sides to the canyon bottom. The degree of the spikes of the canyon rims
also changed.
(8) For the trapezoidal canyon, larger values of η (greater than 6) caused convergence
problems. Further study is necessary.
(9) This method provides good results for arbitrary-shaped canyons in which the
radius to the canyon surface is approximately equal to a. The solutions became
numerically unstable as the radius to the canyon boundary became small, because
the Bessel functions caused numerical problems.
II.10 Summary
The two-dimensional diffraction of incident P-waves around an arbitrary-shaped canyon
in an elastic half-space is presented in this Chapter. The scattered wave potentials for the
resulting P-waves and S-waves are defined by an infinite series of terms with Hankel
functions and only sine terms. Using the zero stress-boundary conditions, a solution is
made using the weighted residual method to create a set of simultaneous equations for the
unknown coefficients for a finite number of terms for the series.
66
The solution is applied to the traditional semi-circular canyon to verify the results and
show convergence with a finite number of terms. The method is then applied to canyons
of other shapes to demonstrate the versatility of the methodology. This paper
demonstrates that good results can be achieved in deep elliptical (b>=a) canyons with
frequencies as high as η = 8. However, the results become unstable for the deep elliptical
canyons with η > 8 and for the shallow canyons (b< a). Good results were achieved for
the trapezoidal canyons of slopes 60 and 45.
67
III WEIGHED RESIDUAL METHOD FOR DIFFRACTION OF PLANE P-
WAVES IN A 2-D ELASTIC HALF-SPACE ON A SHALLOW ALMOST
CIRCULAR ARBITRARY-SHAPED CANYON
III.1 Introduction
This chapter is a continuation of the diffraction of in-plane P-waves in an elastic half-
space by arbitrary-shaped canyons using the weighted residual. In it, it presents a solution
for the diffraction of in-plane P-waves by any shallow arbitrary shaped canyons where
the depth of the canyon can be significantly less than half the width of the canyon, such
as a circle segment, ellipse or trapezoid.
The application of the weighted residual method for the diffraction of plane P-waves in a
two-dimensional (2-D) elastic half-space was developed and presented in Chapter II. The
results of Chapter II demonstrated the validity of the method when compared to prior
research, but more importantly, showed the versatility of the method to address arbitrary-
shaped canyons, rather than simple circular- and elliptical-shaped canyons. In Chapter II,
the P- and SV-cylindrical wave functions were defined by the sine function, and the
weighted residual method was used to obtain the solution of the wave function for
arbitrary-shaped canyons. The coordinate system was located at the half-space surface,
which allowed for arbitrary-shaped canyons to be solved. As the canyons became
shallow, the solutions became numerically unstable because the Bessel functions blew up
as the radius to the canyon boundary (r/a) became small.
Expanding on the procedure used in Chapter II, the x-y origin is moved above the half-
space, an amount d, allowing for shallow canyons of arbitrary shapes to be solved for
incident P-waves. The new location for the origin above the half-space avoids small
values of r/a.
68
III.2 Model
Fig. 3.1. Arbitrary-shaped model with coordinates above the half-space.
In Fig. 3.1, the 2D arbitrary-shaped canyon in an elastic half-space (y > 0) is defined.
Each point will have an (x,y) coordinate and, once transformed into the cylindrical
coordinate system, will have a location of (r, θ). The geometry of the canyon is
transformed from the (x
1
, y
1
) rectangular-coordinate system at origin O
1
and moved
above the half-space with a new rectangular-coordinate system (x, y) and origin O. The
original coordinate system (x
1
, y
1
) with origin O
1
is represented by the following
geometry:
x
1
= r
1
sinθ
1
y
1
= r
1
cosθ
1
r
1
= x
1
2
+ y
1
2
(56)
θ
1
= tan
−1
x
1
y
1
⎛
⎝
⎜
⎞
⎠
⎟
.
The new coordinate system (x, y) with origin O is represented by the following geometry:
x= rsinθ
(57)
69
y= rcosθ
r= x
2
+ y
2
θ = tan
−1
x
y
⎛
⎝
⎜
⎞
⎠
⎟
.
The two systems are related to one another by the following formulas:
x= x
1
(58)
y= y
1
+d
.
Figure 3.1 encompasses the following: an incident plane P-wave, defined by the potential
φ
i
, with an incidence angle θ
α
. The incident angle is measured with respect to the
horizontal x-axis. The waves have circular frequency ω = 2πf, longitudinal wave
velocity α, and transverse wave velocity β.
Using the theory of Lee and Liu from Chapter I and with the new location of the
coordinate system, the free-field waves are unaffected by the canyon and became a
combination of the input P-wave and reflected P- and SV-waves. The free-field P-wave
potential
φ
ff
and the free field SV-wave potential
ψ
ff
in terms of the new coordinate
system (x,y) are:
φ
ff
=φ
i
+φ
r
= e
ik
α
xcosθ
α
−ysinθ
α
( )
+K
1
e
ik
α
xcosθ
α
+ysinθ
α
( )
(59)
ψ
ff
=ψ
r
= K
2
e
ik
β
xcosθ
β
+ysinθ
β ( )
.
Both of these potentials satisfy the zero-stress boundary conditions along the half-space
surface. The scattered wave potentials are defined by sine terms (sin nθ):
φ
s
= A
n
H
n
(1)
(k
α
r)sinnθ
n=0
∞
∑ (60)
ψ
s
= C
n
H
n
(1)
(k
β
r)sinnθ
n=0
∞
∑ .
III.3 Boundary Conditions for the Canyon Surface
Building on the weighted residual method presented in Chapter II, the boundary
conditions on the canyon surface must be satisfied for the free-field and scattered wave
70
potentials. To satisfy these boundary conditions, traction is computed for no force on the
surface of the canyon, Figures 3.2a and 3.2b.
Fig. 3.2a. Traction components.
Fig. 3.2b. Traction components.
On the canyon, the boundary condition of zero stress that must be satisfied on the surface
of the arbitrary shape are (Lee and Wu
5,6
)
T
r
s
+T
r
i+r
=0
(61)
T
θ
s
+T
θ
i+r
=0
.
III.4 Application of Weighted Residual Method
Applying the method of weighted residuals to solves for the unknowns, A
n
and C
n
, for n
= 1, 2,….N, the weighting function is:
71
w
m
= sin mθ
.
(62)
For m = 1,2….N, the function is:
T
r
s
( )
N
w
m
dθ = −
θ
∫
T
r
i+r
( )
N
w
m
dθ
θ
∫
(63)
T
θ
s
( )
N
w
m
dθ =
θ
∫
− T
θ
i+r
( )
N
w
m
dθ
θ
∫
.
The equations (Eq. 63) is solved by substituting the free-field waves (Eq. 59) and
scattered waves (Eq. 60) and free-field waves into the stress equations (Eq. 31) and then
into the traction equations T
r
and T
θ
,, (Eq. 61) and multiplying each equation by the
weighting function (Eq. 62). Each term requires integration along the boundary of the
canyon from 0 to π. Since the coordinate system is moved about the half-space surface
integrating from 0 to π would cause integration along the half-space from + ∞ to - ∞.
The integration is thus approximated for a finite distance on each side of the canyon,
thus, for a small angle θ > 0 and θ < π. The equations are integrated using the Gaussian
quadrature method. The surface of integration—the canyon surface—is divided into 400
segments defined by 401 points. Each segment is subdivided again by the 10-point
Gauss-Legendre integration. This integration technique is highly accurate when the
integrand is very smooth, which is the case for the sine and cosine functions, as the
method converges much more quickly than in other integration schemes. (Willam
H.Press
6
)
The set of 2N complex equations are solved for the unknowns A
1
to A
N
, and C
1
to C
N
.
III.5 Results
From the above analysis that determined the coefficient A
n
and C
n
of the P- and SV-wave
potentials, the displacement amplitudes can now be determined. These two-dimensional
displacement amplitudes are calculated from the free-field and scatter waves using the
equations presented in Chapter II, horizontal
u
x
( ) and vertical
u
y ( ) , are plotted vs. the
dimensionless horizontal distance x/a for a specific dimensionless frequency η and an
angle of incidence
θ
α
, with a dimensionless frequency parameters η, and where a is the
72
radius of the canyon, λ
β
, is the wave length of the shear wave at frequency ω, β is the
shear wave speed, κ
β
is the wave number, f is the cyclic frequency defined as ω = 2πf,
and the Poisson ratio of ν = 0.25.
This improved methodology is used to solve any shallow arbitrary-shaped canyon: a
circle segment, a shallow elliptical-shaped canyon, a shallow trapezoidal-shaped canyon,
and the Nurek Dam. For each shape, the number of equations, N
max
, is increased until
convergence is reached or the problem becomes numerically unstable. The procedure that
has been applied in this paper, method of moments using weighting functions, is an
approximate technique and its accuracy is a function of convergence as the infinite series
are truncated to a finite number of unknown terms A
1
….A
n
and C
1
….C
n
. Three factors
that affect results are the number of terms, the stability of the Bessel functions, and
integration.
III.5.1 Circle Segment
Fig. 3.3. Circle segment model.
The results for a circle segment of depth h/a = 0.5, as shown in Figure 3.3, are presented
in this section. The incident P-waves are assumed to arrive from the front side of the
canyon for all cases. The results are presented in figures that show the x- and y-
displacement amplitudes for angles of incidence θ= 5°, 30°, 60°, and 90°. The case for θ
73
= 30° is for an oblique incident P-wave and the case of θ= 90° is for a vertical incident P-
wave. The x- and y-displacement amplitudes are plotted in the figures on the horizontal
axis from x/a = -4 to x/a = + 4. For convenience, each graph has three areas of focus: the
front side (where the waves arrive first x/a=-4 to x/a=-1), the back side (from x/a=+ 1 to
x/a =+4), and the surface of the canyon (from x/a=-1 to x/a=+1).
74
x-component displacement y-component displacement
x/a x/a
Fig. 3.4. Shallow circular segment h/a=0.5, η = 2, θ = 5°, 30°, 60°, 90° N
max
= 20.
Figure 3.4 shows the results for a circle segment of depth h/a=0.5, for dimensionless
frequency η = 2, and for angles of incidence θ=5°, 30°, 60°, and 90°. For the graphs with
0"
1"
2"
3"
4"
5"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
5"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
5"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
5"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
5"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
5"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
5"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
5"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
Angle
of
Incidence=90°
Angle
of
Incidence=60°
Angle
of
Incidence=30°
Angle
of
Incidence=5°
75
incident angle θ= 5°, the x-component displacement amplitudes oscillate about the free-
field amplitudes on the front side of the canyon, and exhibit a more shadowy behavior
along the back side of the canyon. Within the canyon surface, the waves are slightly
oscillatory. The y-component displacement amplitudes show the same trend but the
oscillations increase. For the graphs with incident angles of θ = 30° and 60°, the x- and y-
component displacement amplitudes exhibit the same behavior as the graphs for the
incident angle θ = 5° with slightly larger amplitudes. For the graphs with incident angle θ
= 90°, the x-component displacement amplitudes are symmetric about the origin 0. The
free-field amplitudes exhibit a shadowy behavior on the front side and on the back side of
the canyon. Within the canyon surface, the displacement amplitudes are highly
oscillatory with spikes at the canyon rims. The y-component displacement amplitudes are
also symmetric about the origin. The free-field amplitudes oscillate about the front side
and the back side of the canyon. Within the canyon surface, the displacement amplitudes
are highly oscillatory with minor spikes at the rims of the canyons.
76
x-component displacement y-component displacement
x/a x/a
Fig. 3.5. Shallow circular segment h/a=0.5, η = 6, θ = 5°, 30°, 60°, 90° N
max
= 60.
Figure 3.5 shows the results for a circle segment of depth h/a = 0.5, for dimensionless
frequency η = 6, and angles of incidence θ = 5°, 30°, 60°, and 90°. For the graphs with
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
Angle
of
Incidence=5°
Angle
of
Incidence=30°
Angle
of
Incidence=60°
Angle
of
Incidence=90°
77
incident angle θ = 5°, the x-component displacement amplitudes rapidly oscillate about
the free-field amplitudes on the front side of the canyon and gradually oscillate on the
back side of the canyon. Within the canyon surface, the waves are highly oscillatory with
spikes at the canyon rims at x/a=-1 and x/a = +1. The y-component displacement
amplitudes show the same trend. For the graphs with incident angle θ = 30°, the x-
component displacement amplitudes rapidly oscillate about the free-field amplitudes on
the front side of the canyon, and gradually oscillate on the back side of the canyon.
Within the canyon surface, the waves are highly oscillatory with spikes at the canyon
rims at x/a =-1 and x/a = +1.The y-component displacement amplitudes show the same
trend, but the oscillations increase. For the graphs with incident angle θ = 60°, the x- and
y-component displacement amplitudes exhibit the same behavior as the graphs for
incident angle 30° with slightly smaller amplitudes. For the graphs with incident angle θ
= 90°, the x-component displacement amplitudes are symmetric about the origin 0. The
free-field amplitudes exhibit a shadowy behavior on the front side and the back side of
the canyon. Within the canyon surface, the displacement amplitudes are highly
oscillatory with spikes at the canyon rims. The y-component displacement amplitudes
are also symmetric about the origin. The free-field amplitudes rapidly oscillate on the
front side and on the back side of the canyon. Within the canyon surface, the
displacement amplitudes are highly oscillatory, with minor spikes at the canyon rims.
78
III.5.2 Elliptical Canyon
Fig. 3.6. Shallow elliptical-canyon model.
The results for three cases of a shallow elliptical canyon, as shown in Figure 3.6, are
presented in this section. The first case is for an ellipse with depth b/a = 0.25 and a
coordinate system located at d/a = 0.75 above the origin. Case II is for b/a = 0.5 and d/a =
0.5 and Case III is for b/a = 0.75 and d/a = 0.25 The incident P-waves are assumed to
arrive from the front side of the canyon for all cases. The results are presented in the
figures which show the x- and y-displacement amplitudes for angles of incidence θ = 30°,
60°, and 90°.
79
x-component displacement y-component displacement
x/a x/a
Fig. 3.7. Shallow elliptical-shaped canyon b/a = 0.25, η = 2, θ = 5°, 30°,60°,90°, N
max
= 22.
0"
1"
2"
3"
4"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
Angle
of
Incidence=90°
Angle
of
Incidence=60°
Angle
of
Incidence=30°
Angle
of
Incidence=5°
80
Figure 3.7 shows the results for a shallow elliptical canyon of depth b/a = 0.25,
dimensionless frequency η = 2, and angles of incidence θ = 5°, 30°, 60°, and 90°. For the
graphs with incident angle θ = 5°, the x-component displacement amplitudes gradually
oscillate about the free-field amplitudes on the front side of the canyon, but exhibit more
shadowy behavior along the back side of the canyon. Within the canyon surface, the
displacement amplitudes are highly oscillatory. The y-component displacement
amplitudes show similar trends. For the graphs with incident angle θ = 30°, the x- and y-
component displacements show similar trends as those seen for incident angle θ = 5°,
except large clustered spikes are forming at the center of the canyon. For the graphs with
incident angle θ = 60°, the x- and y-component displacement amplitudes exhibit the same
behavior as the graphs for incident angle of 30°, except that within the canyon surface, a
smaller cluster of spikes form at the very center of the canyon. For the graphs with
incident angle θ = 90°, the x-component displacement amplitudes are symmetric about
the origin 0. The free-field amplitudes exhibit a shadowy behavior on the front side and
on the back side of the canyon. Within the canyon surface, the displacement amplitudes
are highly oscillatory with a cluster of spikes forming at the center of the canyon. The y-
component displacement amplitudes are also symmetric about the origin. The free-field
amplitudes are oscillatory on the front side of the canyon and on the back side of the
canyon. Within the canyon surface, the displacement amplitudes are highly oscillatory.
81
x-component displacement y-component displacement
x/a x/a
Fig. 3.8. Shallow elliptical-shaped canyon b/a = 0.25, η = 6, θ = 5°, 30°,60°,90°, N
max
= 44.
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
\"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
Angle
of
Incidence=5°
Angle
of
Incidence=30°
Angle
of
Incidence=60°
Angle
of
Incidence=90°
82
Figure 3.8 shows the results for a shallow elliptical canyon of depth b/a = 0.25,
dimensionless frequency η = 6, and angles of incidence θ= 5°, 30°, 60°, and 90°. For the
graphs with incident angle θ= 5°, the x-component displacement amplitudes rapidly
oscillate about the free-field amplitudes on the front side of the canyon, but gradually
oscillate on the back side of the canyon. Within the canyon surface, the displacement
amplitudes are highly oscillatory. The y-component displacement amplitudes oscillate
about the free-field amplitudes on the front side of the canyon. On the back side of the
canyon, the oscillations are very steady. Within the canyon surface, the displacement
amplitudes are more oscillatory. For the graphs with incident angle θ = 30°, the x-
component displacement amplitudes rapidly oscillate about the free-field amplitudes on
the front side of the canyon, but gradually oscillate on the back side of the canyon.
Within the canyon surface, the displacement amplitudes are highly oscillatory. The y-
component displacement amplitudes oscillate about the free-field amplitudes on the front
side of the canyon. On the back side of the canyon, the oscillations are very steady.
Within the canyon surface, the displacement amplitudes are more oscillatory. For the
graphs with incident angle θ= 60°, the x- and y-component displacement amplitudes
exhibit the same behavior as the graphs for the incident angle of 30°, except that within
the canyon surface, a cluster of spikes form at the very center of the canyon. For the
graphs with the incident angle θ = 90°, the x-component displacement amplitudes are
symmetric about the origin 0. The free-field amplitudes exhibit a shadowy behavior on
the front side of the canyon and on the back side of the canyon. Within the canyon
surface, the displacement amplitudes are highly oscillatory with a cluster of spikes
forming at the center of the canyon. The y-component displacement amplitudes are also
symmetric about the origin. The free-field amplitudes are oscillatory on the front side and
on the back side of the canyon. Within the canyon surface, the displacement amplitudes
are highly oscillatory.
83
III.5.3 Shallow Trapezoidal Canyons
Fig. 3.9. Shallow trapezoidal-canyon model.
The model for the trapezoidal canyon, shown in Figure 3.9, is defined by a base of a = 1,
a depth of h, a coordinate system located above the origin at d/a and with sloping sides of
60° and 45° measured from the horizontal axis. The depth varies from h/a = 0.25 to 0.75.
The incident P-waves are assumed to arrive from the front side of the canyon in all cases.
Each figure shows the x- and y-displacement amplitudes for an angle of incidence θ= 5°,
30°, 60°, and 90°.
84
x-component displacement y-component displacement
x/a x/a
Fig. 3.10. Shallow trapezoidal Canyon h/a = 0.25, η = 2, θ = 5°, 30°, 60°, 90°, N
max
=
14 Slope 60°.
0"
1"
2"
3"
4"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
Angle
of
Incidence=90°
Angle
of
Incidence=60°
Angle
of
Incidence=30°
Angle
of
Incidence=5°
85
Figure 3.10 shows the results for a trapezoidal canyon of depth h/a = 0.25, dimensionless
frequency η = 2, slope of 60°, and angles of incidence θ = 5°, 30°, 60°, and 90°. For the
graphs with the incident angle θ= 5°, the x-component displacement amplitudes tend to
slightly oscillate about the free-field amplitudes on the front side of the canyon, but
exhibit a more shadowy behavior along the back side of the canyon. Within the canyon
surface, the displacement amplitudes are slightly oscillatory. The y-component
displacement amplitudes exhibit the same trends as the x-component displacement
amplitudes. For the graphs with incident angle θ = 30°, the x-component displacement
amplitudes tend to slightly oscillate about the free-field amplitudes on the front side of
the canyon, but exhibit a more shadowy behavior along the back side of the canyon.
Within the canyon surface, the displacement amplitudes are slightly oscillatory. The y-
component displacement amplitudes exhibit the same trends as the x-component
displacement amplitudes with slight oscillations appearing along the back side of the
canyon. The graphs with incident angle θ = 60° show the same trend as those for incident
angle θ = 30°. For the graphs with incident angle θ = 90°, the x-component displacement
amplitudes are symmetric about the origin and exhibit a shadowy behavior along the
about the free-field amplitudes on the front side and back side of the canyon. Within the
canyon surface, the displacement amplitudes are slightly oscillatory. The y-component
displacement amplitudes are symmetric about the origin, and are oscillatory about the
free-field amplitudes on the front and back side of the canyon. Within the canyon surface,
the displacement amplitudes are slightly oscillatory.
86
x-component displacement y-component displacement
x/a x/a
Fig. 3.11. Shallow trapezoidal canyon h/a = 0.25, η = 6, θ = 5°, 30°, 60°, 90°, N
max
=
48 Slope 60°.
0"
3"
6"
9"
12"
15"
18"
*4" *3" *2" *1" 0" 1" 2" 3" 4"
0"
3"
6"
9"
12"
15"
18"
*4" *3" *2" *1" 0" 1" 2" 3" 4"
0"
3"
6"
9"
12"
15"
18"
*4" *3" *2" *1" 0" 1" 2" 3" 4"
0"
3"
6"
9"
12"
15"
18"
*4" *3" *2" *1" 0" 1" 2" 3" 4"
0"
3"
6"
9"
12"
15"
18"
*4" *3" *2" *1" 0" 1" 2" 3" 4"
0"
3"
6"
9"
12"
15"
18"
*4" *3" *2" *1" 0" 1" 2" 3" 4"
0"
3"
6"
9"
12"
15"
18"
*4" *3" *2" *1" 0" 1" 2" 3" 4"
0"
3"
6"
9"
12"
15"
18"
*4" *3" *2" *1" 0" 1" 2" 3" 4"
Angle
of
Incidence=5°
Angle
of
Incidence=30°
Angle
of
Incidence=60°
Angle
of
Incidence=90°
87
Figure 3.11 shows the results for a trapezoidal canyon of depth h/a = 0.25, dimensionless
frequency η = 6, slope of 60°, and angles of incidence of θ = 5°, 30°, 60°, and 90°. For
the graphs with the incident angle θ = 5°, the x-component displacement amplitudes tend
to rapidly oscillate about the free-field amplitudes on the front side of the canyon, but
gradually oscillate on the back side of the canyon. Within the canyon surface, the
displacement amplitudes are slightly oscillatory with large spikes at the rims of the
canyon at x/a = -1 and x/a = +1. The y-component displacement amplitudes exhibit the
same trends as the x-component displacement amplitudes. Within the canyon surface, the
displacement amplitudes are slightly oscillatory. For the graphs with the incident angle θ
= 30°, the x-component displacement amplitudes tend to oscillate about the free-field
amplitudes on the front side of the canyon, but gradually oscillate on the back side of the
canyon. Within the canyon surface, the displacement amplitudes are slightly oscillatory
with large spikes at the rims of the canyon at x/a = -1 and x/a = +1. The y-component
displacement amplitudes exhibit the same trends as the x-component displacement
amplitudes with larger spikes at the rims of the canyon. Within the canyon surface, the
displacement amplitudes are slightly oscillatory. The graphs with incident angle θ = 60°
show the same trend as those for incident angle θ = 30°, with only a spike at the right rim
of the canyon at x/a = +1 for the x-component displacement amplitude and a smaller
spike for the y-component displacement amplitude. For the graphs with incident angle θ
= 90°, the x-component displacement amplitudes are symmetric about the origin and
exhibit a shadowy behavior along the about the free-field amplitudes on the front side and
back side of the canyon. Within the canyon surface, the displacement amplitudes are
slightly oscillatory with smaller spikes forming at the rims of the canyon surface. The y-
component displacement amplitudes are symmetric about the origin, and are oscillatory
about the free-field amplitudes on the front and back side of the canyon. Within the
canyon surface, the displacement amplitudes are slightly oscillatory with smaller spikes
forming at the rims of the canyon surface.
88
III.5.4 Nurek Dam
Fig. 3.12. Nurek Dam Model.
The results for the Nurek Dam, shown in Figure 3.12, are presented in this section. The
Nurek Dam, located in Tajikistan, is the largest earth-filled embankment dam in the
world, standing tall at 300 meters with a width of 700 meters supporting nine
hydroelectric generating units. The dam provides 98% of the region’s power. The results
are for incident angles of θ = 5°, 30°, 60°, and 90° for dimensionless frequency η = 2 and
8 (shown in Figs. 3.13 and 3.14). The incident P-waves are assumed to arrive from the
front side of the canyon for all cases. Each figure shows the x- and y-displacement
amplitudes for angles of incidence θ = 5°, 30°, 60°, and 90°.
The Nurek Dam was presented in the previous research of Lee and Wu (Lee and Wu
5,6
)
for incident P-waves. The results in this section compare favorably with Lee and Wu’s
results for η = 2, and an incident angle 90°. The Nurek Dam is not symmetrical, thus the
results at 90° are not symmetric (see in Fig. 3.13).
89
x-component displacement y-component displacement
x/a x/a
Fig. 3.13. Nurek Dam η = 2, θ = 5°, 30°, 60°, 90°, N
max
= 30.
Figure 3.13 shows the results for the Nurek Canyon for dimensionless frequency η = 2
and angles of incidence θ = 5°, 30°, 60°, and 90°. For the graphs with incident angle θ =
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
Angle
of
Incidence=5°
Angle
of
Incidence=30°
Angle
of
Incidence=60°
Angle
of
Incidence=90°
90
30°, the x-component displacement amplitudes show a shadowy behavior about the free-
field amplitudes on the front and back sides of the canyon. Within the canyon surface, the
displacement amplitudes are slightly oscillatory. The y-component displacement
amplitudes show a shadowy behavior about the free-field amplitudes on the front and
back sides of the canyon. Within the canyon surface, the displacement amplitudes are
slightly oscillatory. The graphs with incident angle θ = 60° show the same trend as those
for incident angle θ = 30°. The graphs with incident angle θ = 90° show the same trend.
The x-component displacement amplitudes tend to exhibit a shadowy behavior along the
about the free-field amplitudes on the front and back sides of the canyon, almost
becoming a constant value. Within the canyon surface, the displacement amplitudes are
slightly oscillatory. The y-component displacement amplitudes exhibit a shadowy
behavior about the free-field amplitudes on the front and back sides of the canyon.
Within the canyon surface, the displacement amplitudes are slightly oscillatory.
91
x-component displacement y-component displacement
x/a x/a
Fig. 3.14. Nurek Dam η = 8, θ = 5°, 30°, 60°, 90°, N
max
= 56.
Figure 3.14 shows the results for the Nurek Canyon for dimensionless frequency η = 8
and angles of incidence θ=5°, 30°, 60°, and 90°. For incident angle θ= 30°, the x-
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
Angle
of
Incidence=5°
Angle
of
Incidence=30°
Angle
of
Incidence=60°
Angle
of
Incidence=90°
92
component displacement amplitudes rapidly oscillate about the free field amplitudes on
the front side of the canyon and exhibit a more shadowy behavior along the back side of
the canyon. Within the canyon surface, large spikes appear between x/a = 0 and x/a = 1.
The y-component displacement amplitudes exhibit similar trends, except the free-field
amplitudes, which gradually oscillate along the back side of the canyon. The incident
angles θ = 60° and 90° exhibit the same trends as those observed for the graphs for
incident angle θ = 30°.
III.6 Observations
The results of this research are presented for various shaped shallow canyons: circle
segment, elliptical, and trapezoidal. These results have been presented in Figs. 3.4–3.14
and the following general observations are made:
(1) The method used in this paper is an expansion of Lee and Liu (Lee and Liu’s
7
)
redefined cylindrical-wave function with the coordinate system moved above the
half-space surface at a distance d/a, and with the application of the weighted
residual method to solve arbitrary-shaped shallow canyons.
(2) The weighted residual method is applied to a segment circle, a shallow elliptical
canyon, a shallow trapezoidal canyon, and a “real” canyon.
(3) This approach was applied to a circle segment of depth h/a = 0.5 with coordinates
located above the half-space surface at a distance d/a. The resulting plots for the
displacement amplitudes are for higher frequencies than η = 6, and greater than
those seen in previous studies.
(4) The circle segment canyon produced total displacement amplitudes on both the x-
and y-components for η = 6, which are highly oscillatory around the free-field
surface and along the backside of the canyon. Within the canyon, the oscillations
were highly oscillatory and produced spikes at the rims of the canyons at x/a = -1
and x/a = +1 for η = 6.
(5) This approach was then applied to shallow elliptical canyons with varying b/a
(depth/width) ratios = 0.25, 0.5, and 0.75, with the coordinate system located
above the half-space surface at a respective distance of d/a = 0.75, 0.5 and 0.25
93
for frequencies η = 6 and 8, which are much greater than those previously studied.
The total displacement amplitudes on both the x- and y-components for η = 6 and
8 were oscillatory around the free-field surface on the front side of the canyon and
less oscillatory on the backside of the canyon. Within the canyon surface, the
amplitudes were more oscillatory. As η increased, the oscillations increased and
spikes were observed at either the rims of the canyon at x/a = -1 and x/a = +1, the
center of the canyon at x/a = 0, or both.
(6) This approach was then applied to shallow trapezoidal canyons defined by a base
of a = 1 and a depth of h/a = 0.25, 0.5 and 0.75 for a coordinate system located
above the half-space surface at a respective distance d/a = 0.75, 0.5 and 0.25 for
sloping sides of 60° and 45°. The total displacement amplitudes on both the x-
and y-components for the shallow trapezoid slope of 60° showed a trend of being
oscillatory around the free-field surface on the front side of the canyon and
exhibiting a more shadowy behavior along the back side of the canyon. The
oscillations increased with higher frequencies and deeper depths. At higher
frequencies, spikes formed at the rims of the canyon. These same trends were
observed for the shallow trapezoids of slope 45°.
(7) This approach was then applied to the Nurek Dam. The dam produced total
displacement amplitudes on both the x- and y-components for η = 2 and 8. The
lowest frequency showed a trend of producing a shadowy behavior around the
free-field surface along the front and back sides of the canyon. Within the canyon
surface, it produced gradual oscillations. As the frequency increased, the
oscillations increased, and spikes formed on the back side of the canyon at x/a =
+1.
III.7 Summary
The two-dimensional diffraction of incident P-waves around a shallow arbitrary-shaped
canyon in an elastic half-space is presented in this paper. The scattered wave potentials
are defined by an infinite series of terms with Hankel functions and only sine terms.
Using the zero-stress boundary conditions, a solution is made using the weighted residual
method to create a set of simultaneous equations for the unknown coefficients on the
infinite series.
94
The solution is applied to shallow arbitrary-shaped canyons to verify the results and show
convergence with a finite number of terms. The method is applied to canyons of other
shapes to demonstrate the versatility of the methodology. This paper demonstrates that
good results can be achieved in the canyons that are shallow (b/a<1) for frequencies η =
4, 6, and 8. Good results were achieved for the trapezoidal canyons of slopes 60 and 45.
95
IV WEIGHTED RESIDUAL METHOD FOR DIFFRACTION OF PLANE P-
WAVES IN A 2-D ELASTIC HALF-SPACE ON AN IRREGULAR-SHAPED
ALLUVIAL VALLEY
IV.1 Introduction
The application of the weighted residual method for the diffraction of plane P-waves in a
2-D elastic half-space for irregular-shaped canyons was developed in Chapter II and
expanded in Chapter III for shallow irregular-shaped canyons. This chapter expands on
the previous work by applying the same methodology to irregular-shaped alluvial valleys.
Researchers have observed ground motion amplification in valleys, especially at the
boundaries, and this chapter develops tools that can enhance this research.
Expanding on Lee and Liu’s approach (Lee and Liu
10
) in Chapter I, this chapter applies
the weighted residual method for a solution of their wave function for an irregular-shaped
alluvial valley. The coordinate system is located at the half-space surface, which allows
for irregular-shaped alluvial valleys with h > a to be solved. Using this improved
weighted residual method, the results for Lee and Liu’s semi-circular canyon were
verified and new results for an ellipse and trapezoid are presented.
IV.2 Model
Fig. 4.1. Irregular-shaped alluvial valley with coordinates at the half space.
In Fig 4.1, the 2-D model of the irregular-shaped alluvial valley in an elastic half-space (y
> 0) is defined. Each point will have an (x,y) rectangular coordinate and, once
transformed into polar coordinates, each point will have an (r, θ) location. The geometry
96
of the canyon is transformed from the rectangular coordinate system into a cylindrical
coordinate system with the same origin at O, as shown:
x= rsinθ
y= rcosθ
r= x
2
+ y
2
(66)
θ = tan
−1
x
y
⎛
⎝
⎜
⎞
⎠
⎟
.
The figure encompasses the following: an incident plane P-wave defined by the potential
φ
i
, with an incidence angle θ
α
. The incident angle is measured with respect to the
horizontal x-axis.
The harmonic motion induced by an incident P- wave without the presence of an irregular
alluvial valley is defined by Lee and Liu (Lee and Liu
3
)in Chapter I. Resulting in the free
field P-wave potential
φ
ff
and the free field SV-wave potential
ψ
ff
which both satisfy the
zero-stress boundary conditions along the half-space surface.
φ
ff
=φ
i
+φ
r
= e
ik
α
xcosθ
α
−ysinθ
α
( )
+K
1
e
ik
α
xcosθ
α
+ysinθ
α
( )
(67)
ψ
ff
=ψ
r
= K
2
e
ik
β
xcosθ
β
+ysinθ
β ( )
.
As seen in Chapter II, the scattered wave potentials take a form that only utilized the
following sine terms (sin nθ):
φ
s
= A
n
H
n
(1)
(k
α
r)sinnθ
n=0
∞
∑
(68)
ψ
s
= C
n
H
n
(1)
(k
β
r)sinnθ
n=0
∞
∑ .
IV.3 The Presence of an Irregular Alluvial Valley
The irregular alluvial valley is defined by the following material properties: lame
constants λ
v
and µ
v
, mass density ρ
v
, and circular frequency ω. The longitudinal wave
velocity for the alluvial valley α
v
is
97
α
v
=
λ
v
+2µ
v
ρ
v
, (69)
and transverse wave velocity for the alluvial valley is
, (70)
therefore, k
αv
= ω/α
v
(P-wave number) and k
βv
= ω/β
v
(SV-wave number) .
In the irregular alluvial valley there exists a scattering P-wave and a scattering SV-wave
due to the reflection from the valley boundaries. These scattering waves take the
following form:
(71)
.
IV.4 Boundary Conditions for the Canyon Surface
Boundary conditions for the resulting combination of waves must satisfy displacement
continuity and traction continuity at the interface between the half space and the valley.
Therefore, boundary conditions for the displacement continuity on the half-space valley
interface are a combination of the scattered, free field and valley displacements
(Todorovska and Lee
6
and Liang and Yan
5
):
u
r
v
= u
r
s
+u
r
ff
u
θ
v
= u
θ
s
+u
θ
ff
.
(72)
In order to satisfy the boundary conditions of stress continuity on the canyon surface,
tractions are computed from the equations shown in Chapter II.
Therefore the tractions due to the scattered, incident, and reflected waves become
T
r
s
=σ
r
s
cosα +τ
rθ
s
sinα = 0
(73)
β
v
=
µ
v
ρ
v
φ
s
v
= B
n
J
n
(k
α
v
r)sinnθ
∑
ψ
s
v
= D
n
J
n
(k
β
v
r)sinnθ
∑
98
T
θ
s
=σ
θ
s
sinα +τ
rθ
s
cosα = 0
T
r
ff
=σ
r
ff
cosα +τ
rθ
ff
sinα = 0
T
θ
ff
=σ
θ
ff
sinα +τ
rθ
ff
cosα = 0
.
The traction due to the scattered valley waves then become
T
r
v
=σ
r
sv
cosα +τ
rθ
sv
sinα = 0
T
θ
v
=σ
θ
sv
sinα +τ
rθ
sv
cosα = 0
. (74)
On the canyon, the boundary condition of stress compatibility that must be satisfied on
the surface of the arbitrary shape are (Lee and Wu
9,10
and Todorovska and Lee
6
and
Liang and Yan
5
) the following:
T
r
s
+T
r
ff
=T
r
s
v
T
θ
s
+T
θ
ff
=T
θ
s
v
.
(75)
VI.5 The Application of the Weighted Residual Method
The functions that define the stresses are an infinite summation and thus the traction
equations are an infinite summation. An approximate solution will use a finite summation
with N terms and thus 4N unknowns: A
n
, B
n
, C
n,
D
n
, and n = 1 to N. The procedure for
solving these four equations, shown in Eqs. 25 and 26, is a special case of the method of
moments defined in Roger Harrington’s paper “Matrix Methods for Field Problems.”
(Harrington
4
) A set of weighting functions, w
1
, w
2
, w
3
, …w
N
, in the range of 0 to π, is
defined and applied to the traction terms, resulting in 4N equations that require
integration from 0 to π. Applying the method of weighted residuals, the weighting
function is:
w
m
= sin mθ
.
(76)
For m = 1,2….N, the function is:
99
T
r
s
( )
N
w
m
dθ
θ
∫
− T
r
sv
( )
N
w
m
dθ
θ
∫
= − T
r
ff
( )
N
w
m
dθ
θ
∫
T
θ
s
( )
N
w
m
dθ
θ
∫
− T
θ
sv
( )
N
w
m
dθ
θ
∫
= − T
θ
ff
( )
N
w
m
dθ
θ
∫
(77)
u
r
s
( )
N
w
m
dθ −
θ
∫
u
r
sv
( )
N
θ
∫
w
m
dθ = − u
r
ff
( )
θ
∫
w
m
dθ
u
θ
s
( )
N
w
m
dθ −
θ
∫
u
θ
sv
( )
N
θ
∫
w
m
dθ = − u
θ
ff
( )
θ
∫
w
m
dθ
. (78)
Since sin(mθ) equals zero when m = 0, Eqs. 77 and 78 represent 4N pairs of equations
with 4N unknowns, A
n
, B
n,
C
n
,, D
n
for n = 1, 2,….N.
IV.6 Numerical Solutions
The equations for traction, Eq. 78, and the equations for displacement, Eq. 78, form a set
of complex simultaneous equations with unknowns A
n
, C
n,
B
n
, D
n
, and a coefficient
matrix with terms a
ij
, c
ij
, b
ij
, d
ij
, and constant terms r
i
shown in Eq. 79. The upper half of
the 4N set of equations represents Eq. 77 and the lower half represents Eq. 78. The
coefficients and constant terms are determined by integration along the boundary of the
valley for θ from 0 to π:
a
1,1
... c
1,1
... b
1,1
... d
1,1
...
... ...
... ...
a
2N,1
... c
2N,1
... b
2N,1
... d
2N,1
...
a
2N+1,1
... c
2N+1,1
... b
2N+1,1
... d
2N+1,1
...
... ...
... ...
a
4N,1
... c
4N,1
... b
4N,1
... d
4N,1
...
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
=
A
1
.
A
N
C
1
.
C
N
B
1
.
B
N
D
1
.
D
N
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
=
r
1
.
.
r
N
r
2N+1
.
.
r
4N
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
.
(79)
The set of 4N complex equations is solved for the unknowns A
1
- A
N
, C
1
- C
N
, B
1
– B
N
,
and D
1
-D
N
.
100
IV.7 Results
From the above analysis that determined the coefficients A
n
,, B
n,
C
n
, and D
n
of the P- and
SV-wave potentials, the displacement amplitudes can now be determined. The
displacement amplitudes that occur on the surface of the half space and within the canyon
surface are important in studying the variability of ground motions in the canyon vicinity.
The total displacement amplitudes are a linear combination of the scattered, free-field,
and valley waves.
Fig. 4.2. Semi-circular-shaped alluvial valley.
Figure 4.2 shows a semi-circular shaped alluvial valley. This figure has a depth-to-half-
width ratio h/a = 1, and their displacement amplitudes on the surface of the canyon (r
1
=
a) are plotted along the horizontal x-axis in the interval -1 ≤ x/a ≤ 1. Their point x/a = -1
corresponds to the left rim of the canyon, x/a = 0 to the bottom, and x/a = 1 to the right
rim. The incident P-waves are assumed to arrive from the left (x/a < 0) in all cases except
of course for that of vertical incidence (θ
α
=0).
101
x-component displacement y-component displacement
x/a x/a
Fig. 4.3. Irregular valley and arbitrary semi-circle canyon η = 2, θ = 60°, and N
max
=
80.
In Fig. 4.3, results are shown for the semi-circular alluvial valley with a coordinate
system at the half space for angle of incidence θ
α
= 60°, dimensionless frequency η = 2,
ratio of alluvial valley, half- space shear velocity β
1/
β = 0.1, and mass density ρ
1
/ρ = 0.1.
The shear velocity ratio and mass density for the alluvial valley represents an almost
empty valley, thus becoming a canyon. Using the weighted residual method, the almost
empty alluvial valley recreating a semi-circular canyon results were verified against Lee
and Liu’s (Lee and Liu
10
) analytic solution for their semi-circular canyon. The results
correlated with Lee and Liu’s results.
IV.7.1 Application of Other Arbitrary-Shaped Irregular Alluvial Valleys
This improved methodology to solve the arbitrary-shaped semi-circular canyon from
Chapter II can be applied to any arbitrary-shaped irregular alluvial valley. This section of
the report will look at an elliptical-shaped alluvial valley and a trapezoidal-shaped
alluvial valley. For each shape, the number of equations, N
max
, is increased until
0"
1"
2"
3"
4"
5"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
5"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
5"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
5"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
Irregular
Valley
Semi-‐Circular
Canyon
102
convergence is reached or the problem becomes numerically unstable. The procedure that
has been applied in this paper, method of moments using weighting functions, is an
approximate technique and its accuracy is a function of convergence as the infinite series
are truncated to a finite number of unknown terms A
1
….A
n
, B
1
….B
n
, C
1
….C
n
, D
1
….D
n
.
Three factors that affect the results are the number of terms, stability of the Bessel
functions, and integration.
Fig. 4.4. Elliptical-shaped alluvial valley.
IV.7.2 Elliptical-Shaped Alluvial Valley—SOFT SOILS β
1
/β = ½, ρ
1
/ρ = 2/3, and
µ
1
/µ = 0.1667
The results for an elliptical-alluvial valley of depths b =1, 1.25, and 1.5, with a ratio of
alluvial valley, half-space shear velocity β
1/
β=1/2, and mass density ρ
1
/ρ=2/3 as shown in
Fig. 4.4, are presented in this section. In all cases, the incident P-waves are assumed to
arrive from the front side of the valley. The results are presented in figures that show the
x- and y-displacement amplitudes for an angle of incidence of θ= 5°, 30°, 60°, and 90°.
The case for θ = 30° is for an oblique incident P-wave and the case of θ= 90° is for a
vertical incident P-wave. The x- and y-displacement amplitudes are plotted in the figures
on the horizontal axis from x/a = -4 to x/a = +4. For convenience, each graph has three
areas of interest: the front side (where the waves arrive first x/a=-4 to x/a=-1), the back
side (from x/a=+1 to x/a=+4), and the surface of the valley (from x/a=-1 to x/a=+1). The
results for b = 1 are shown in Fig. 4.5a,b, the results for b = 1.25 are shown in Fig. 4.6
and the results for b = 1.5 are shown in Figs. 4.7 and 4.8:
103
x-component displacement y-component displacement
x/a x/a
Fig. 4.5a. Ellipse alluvial valley b/a = 1, η = 2, θ = 5°, 30°, and N
max
= 20.
0"
2"
4"
6"
&4" &3" &2" &1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
&4" &3" &2" &1" 0" 1" 2" 3" 4"
!6#
!5#
!4#
!3#
!2#
!1#
0#
1#
2#
3#
4#
5#
6#
!4# !3# !2# !1# 0# 1# 2# 3# 4#
!6#
!5#
!4#
!3#
!2#
!1#
0#
1#
2#
3#
4#
5#
6#
!4# !3# !2# !1# 0# 1# 2# 3# 4#
0"
1"
2"
3"
4"
5"
6"
)4" )3" )2" )1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
5"
6"
)4" )3" )2" )1" 0" 1" 2" 3" 4"
!6#
!5#
!4#
!3#
!2#
!1#
0#
1#
2#
3#
4#
5#
6#
!4# !3# !2# !1# 0# 1# 2# 3# 4#
!6#
!5#
!4#
!3#
!2#
!1#
0#
1#
2#
3#
4#
5#
6#
!4# !3# !2# !1# 0# 1# 2# 3# 4#
Angle
of
Incidence=5°
Angle
of
Incidence=30°
Phase
5°
Phase
30°
104
x-component displacement y-component displacement
x/a x/a
Fig. 4.5b. Ellipse alluvial valley b/a = 1, η = 2, θ = 60°, 90°, and N
max
=20.
0"
1"
2"
3"
4"
5"
6"
)4" )3" )2" )1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
5"
6"
)4" )3" )2" )1" 0" 1" 2" 3" 4"
!6#
!5#
!4#
!3#
!2#
!1#
0#
1#
2#
3#
4#
5#
6#
!4# !3# !2# !1# 0# 1# 2# 3# 4#
!6#
!5#
!4#
!3#
!2#
!1#
0#
1#
2#
3#
4#
5#
6#
!4# !3# !2# !1# 0# 1# 2# 3# 4#
0"
1"
2"
3"
4"
5"
6"
)4" )3" )2" )1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
5"
6"
)4" )3" )2" )1" 0" 1" 2" 3" 4"
!6#
!5#
!4#
!3#
!2#
!1#
0#
1#
2#
3#
4#
5#
6#
!4# !3# !2# !1# 0# 1# 2# 3# 4#
!6#
!5#
!4#
!3#
!2#
!1#
0#
1#
2#
3#
4#
5#
6#
!4# !3# !2# !1# 0# 1# 2# 3# 4#
Angle
of
Incidence=60°
Angle
of
Incidence=90°
Phase
60°
Phase
90°
105
Figure 4.5a,b shows the results for an elliptical alluvial valley of depth b/a = 1,
dimensionless frequency η = 2, and an angle of incidence of θ= 5°, 30°, 60°, and 90°. For
the graphs with incident angle θ = 5°, the x-component displacement amplitudes slightly
oscillate about the free-field amplitude on the front side of the valley and produce a
shadowy behavior along the back side of the valley. Along the surface of the alluvial
valley, the x-component displacement amplitudes are oscillatory, producing a spike on
the front side rim of the valley at x/a = -1 and a smaller spike on the back side rim at x/a
=+1. The y-component displacement amplitudes slightly oscillate about the free-field
amplitude on the front side of the valley and produce a shadowy behavior along the back
side. Along the surface of the alluvial valley, the displacement amplitudes are oscillatory
with spikes at the rims of the valley at x/a = -1 and x/a =+1. For the graphs with incident
angle θ= 30°, the x-component displacement amplitudes tend to oscillate about the free-
field amplitude on the front side of the valley, but gradually oscillate and produce a
shadowy behavior along the back side. Along the surface of the alluvial valley, both the
x-component displacement amplitudes are highly oscillatory and produce a spike on the
front side and back side rim of the valley. The y-component displacement amplitudes
tend to oscillate about the free-field amplitude on the front side of the valley and produce
a shadowy behavior along the back side. Along the surface of the alluvial valley, the
displacement amplitudes are highly oscillatory with spikes at the rims of the valley at x/a
=+1 and x/a =-1. For the graphs with the incident angle θ = 60°, the x-component
displacement amplitudes tend to slightly oscillate about the free-field amplitude on the
front side of the valley, but exhibit decaying oscillations along the back side. Along the
surface of the alluvial valley, both the x-component displacement amplitudes are
oscillatory and produce a spike between the center of the valley and the back side rim.
The y-component displacement amplitudes tend to produce a shadowy behavior about the
free-field amplitude on the front side of the valley and along the back side. Along the
surface of the alluvial valley, the displacement amplitudes are oscillatory with small
spikes at the rims of the valley at x/a =+1 and x/a =-1. For the graphs with incident angle
θ= 90°, the x- and y-component displacement amplitudes are symmetric about the origin
0. The x-component displacement amplitudes produce no motion on the canyon front side
or back side. Along the surface of the alluvial valley, the displacement amplitudes are
106
oscillatory with spikes at both rims for displacement x/a =+1 and x/a =-1. The y-
component free-field amplitudes exhibit a decaying oscillatory motion on the front side
of the valley and along the back side. Along the surface of the alluvial valley, the
displacement amplitudes are very slightly oscillatory with downward spikes at both rims
of the valley for displacement x/a =+1 and x/a =-1.
Figure 4.5a,b also shows the phase diagrams alongside the corresponding displacement
amplitudes. All phase diagrams have been scaled by π and have been shifted arbitrarily to
have a zero phase angle at x/a = 0.
107
x-component displacement y-component displacement
x/a x/a
Fig. 4.6. Ellipse alluvial valley b/a = 1.25, η = 2, θ = 60°, 90°, and N
max
= 24.
Figure 4.6 shows the results for an elliptical alluvial valley of depth b =1.25,
dimensionless frequency η = 2, and an angle of incidence of θ = 5°, 30°, 60°, and 90°,
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
Angle
of
Incidence=5°
Angle
of
Incidence=30°
Angle
of
Incidence=60°
Angle
of
Incidence=90°
108
with a ratio of alluvial valley, half-space shear velocity β
1/
β = 1/2, and mass density ρ
1
/ρ
= 2/3. The displacement amplitudes tend to look similar along the free-field surface of the
front and back sides of the valley as those seen in Fig. 4.5 for the b =1 ellipse. Figure 4.6
shows smaller spikes at the rims of the valleys and more oscillatory behavior within the
valley surface, with the exception of the y-component displacement amplitude for angle
of incidence θ = 90° that produces a large spike at the alluvial valley center.
109
x-component displacement y-component displacement
x/a x/a
Fig. 4.7. Ellipse alluvial valley b/a = 1.5, η = 2, θ = 5°, 30°, 60°, 90°, and N
max
= 36.
0"
2"
4"
6"
8"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
Angle
of
Incidence=5°
Angle
of
Incidence=30°
Angle
of
Incidence=60°
Angle
of
Incidence=90°
110
Figure 4.7 shows the results for an elliptical alluvial valley of depth b =1.25,
dimensionless frequency η = 2, and an angle of incidence of θ = 5°, 30°, 60°, and 90°,
with a ratio of alluvial valley and half- space shear velocity β
1/
β=1/2, and mass density
ρ
1
/ρ=2/3. The displacement amplitudes tend to look similar along the free-field surface of
the front and back sides of the valley as those seen in Fig. 4.6 for the ellipse of b = 1.25.
Figure 4.7 shows x- and y-component displacement amplitudes that are slightly larger
and more oscillatory about the free-field amplitudes on the front side of the valley than
those in Fig. 4.6, but they still produce a shadowy behavior along the back side of the
valley. Very small spikes are seen at the rims of the alluvial valley.
111
x-component displacement y-component displacement
x/a x/a
Fig. 4.8. Ellipse alluvial valley b/a = 1.25, η = 8, θ = 5°, 30°, 60°, 90°, and N
max
= 64.
Figure 4.8 shows the results for an elliptical alluvial valley of depth b = 1.5,
dimensionless frequency η = 8, and an angle of incidence of θ= 5°, 30°, 60°, and 90°,
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
Angle
of
Incidence=5°
Angle
of
Incidence=30°
Angle
of
Incidence=60°
Angle
of
Incidence=90°
112
with a ratio of alluvial valley, half-space shear velocity β
1/
β = ½, and mass density ρ
1
/ρ =
2/3. For the graphs with an incident angle of θ= 5°, the x-component displacement
amplitudes are oscillatory about the free-field amplitude on the front side of the valley,
and exhibit decaying oscillatory amplification along the back side of the valley. Along
the surface of the alluvial valley, the x-component displacement amplitudes are highly
oscillatory, producing large spikes on the rims of the valley at x/a = -1 and x/a =+1. The
y-component displacement amplitudes are oscillatory about the free-field amplitude on
the front side of the valley and produce a shadowy behavior along the back side of the
valley. Along the surface of the alluvial valley the displacement amplitudes are highly
oscillatory with large spikes at the rims of the valley at x/a = -1 and x/a =+1. The graphs
with incident angles of θ = 30° and 60° produce similar trends as those seen in the θ = 5°
incidence angle. For the graphs with the incident angle of θ = 90°, the x- and y-
component displacement amplitudes are symmetric about the origin 0. The x-component
displacement amplitudes produce no motion on the front or back side of the canyon.
Along the surface of the alluvial valley, the displacement amplitudes are highly
oscillatory with large spikes at both rims of the valley for displacement at x/a =+1 and x/a
=-1. The y-component free-field amplitudes are oscillatory along the front and back sides
of the valley. Along the surface of the alluvial valley, the displacement amplitudes are
highly slightly oscillatory with large spikes at both rims of the valley for displacement at
x/a = +1 and x/a = -1.
IV.7.3 Trapezoidal Alluvial Valley—Soft Soils β
1
/β = 1/2, ρ
1
/ρ = 2/3, and µ
1
/µ =
0.1667
Results for the trapezoidal alluvial valley, as shown in Fig. 4.9, are presented in this
section. The trapezoidal valley is defined by a base of a = 1 and a depth of h = 1, with
sloping sides of 60° and 45° measured from the horizontal axis with a ratio of alluvial
valley, half-space shear velocity β
1/
β = 1/2, and mass density ρ
1
/ρ = 2/3. In all cases, the
incident P-waves are assumed to arrive from the front side of the valley. Each figure
shows the x- and y-displacement amplitudes for an angle of incidence of θ = 30°, 60°,
and 90°. The case for the θ= 30° angle is for an oblique incident P-wave, and the case of
113
θ = 90° is for a vertical incident P-wave. The x- and y-displacement amplitudes are
plotted in the figures on the horizontal axis from x/a = -4 to x/a = +4. For convenience,
each graph has three areas of interest: the front side (from x/a=-4 to x/a-1), the back side
(from x/a+1 to x/a=+4), and the surface of the valley (from x/a=-1 to x/a+1).
Fig. 4.9. Trapezoidal alluvial valley.
114
x-component displacement y-component displacement
x/a x/a
Fig. 4.10. Trapezoidal alluvial valley h/a = 1, η = 2, θ = 5°, 30°, 60°, 90°; N
max
=16,
and slope 60°.
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
Angle
of
Incidence=5°
Angle
of
Incidence=30°
Angle
of
Incidence=60°
Angle
of
Incidence=90°
115
Figure 4.10 shows the results for a trapezoidal alluvial valley of a depth to width ratio
h/a=1 for dimensionless frequency η = 2, a slope of 60° for an angle of incidence of θ=
5°, 30°, 60°, and 90°, with a ratio of alluvial valley, half-space shear velocity β
1/
β = 1/2,
and mass density ρ
1
/ρ = 2/3. The x- and y-displacement amplitudes are plotted in the
figures on the horizontal axis from x/a = -4 to x/a = +4. For the graphs with incident
angle θ = 5°, the x-component displacement amplitudes oscillate about the free-field
amplitude on the front side of the valley, and produce a shadowy behavior along the back
side of the valley. Along the surface of the alluvial valley, the amplitudes are oscillatory,
producing a large spike at the back side canyon rim at x/a = -1. The y-component
displacement amplitudes produce a steady shadowy behavior along the front side and
back side of the valley. Along the surface of the alluvial valley, the displacement
amplitudes are slightly oscillatory with a large spike at both rims of the valley. The
graphs with incident angle θ= 30° produce a similar trend as those seen for the incident
angle θ = 5° except with much smaller spikes at the rims of the canyon. The graphs with
the incident angle θ = 60° display the same trends as the ones for incidence angle θ = 30°
except that the oscillations along the valley surface are slightly larger with slightly larger
spikes at the canyon rims at x/a = +1 and x/a = -1. For the graphs with the incident angle
θ = 90°, the x-component displacement amplitudes are symmetric about the origin and
exhibit a shadowy behavior about the free-field amplitudes on the front and back sides of
the valley. Along the surface of the alluvial valley, the displacement amplitudes are
oscillatory with small spikes on the sloping sides. The y-component displacement
amplitudes are symmetric about the origin and produce a shadowy behavior about the
free-field amplitudes on the front and back sides of the valley. Along the surface of the
alluvial valley, displacement amplitudes are slightly oscillatory with a large spike at the
rims of the valley.
116
x-component displacement y-component displacement
x/a x/a
Fig. 4.11. Trapezoidal alluvial valley h/a = 1, η = 8, θ = 5°, 30°, 60°, 90°; N
max
= 48,
and slope 45°.
0"
2"
4"
6"
8"
10"
12"
14"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
12"
14"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
12"
14"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
12"
14"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
12"
14"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
12"
14"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
12"
14"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
12"
14"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
Angle
of
Incidence=5°
Angle
of
Incidence=30°
Angle
of
Incidence=60°
Angle
of
Incidence=90°
117
Figure 4.11 shows the results for a trapezoidal alluvial valley for a depth to width ratio
h/a=1, for dimensionless frequency η = 8, slope 45° for an angle of incidence of θ= 5°,
30°, 60°, and 90°, with a ratio of alluvial valley, half-space shear velocity β
1/
β = 1/2, and
mass density ρ
1
/ρ = 2/3. For the graphs with incident angle θ = 5°, the x-component
displacement amplitudes oscillate about the free-field amplitude on the front side of the
valley and produce a shadowy behavior along the back side. Along the surface of the
alluvial valley, the amplitudes are highly oscillatory. The y-component displacement
amplitudes produce a slight oscillatory behavior along the front side of the valley, and a
shadowy behavior along the back side. Along the surface of the alluvial valley, the
displacement amplitudes are slightly oscillatory with a large spike at both valley rims.
The graphs with incident angle θ = 30° produce a similar trend as those seen for incident
angle θ = 5° except for the x-component-displacements clustered spikes that now form at
the canyon rim. The graphs with incident angle θ = 60° display the same trends as the
graphs for angle of incidence θ = 30° except that the x-component displacements exhibit
decaying oscillatory amplification along the back side of the valley and the y-component
displacements show a slightly smaller spike forming at the back side rim of the canyon at
x/a = -1. For the graphs with incident angle θ = 90°, the x-component displacement
amplitudes are symmetric about the origin and exhibit a shadowy behavior about the free-
field amplitudes on the front and back side of the valley. Along the surface of the alluvial
valley, the displacement amplitudes are highly oscillatory. The y-component
displacement amplitudes are symmetric about the origin and produce a shadowy behavior
about the free-field amplitudes on the front side and back side of the valley. Along the
surface of the alluvial valley, displacement amplitudes are slightly oscillatory with a large
spike at the valley rims.
118
IV.7.4 Trapezoidal Alluvial Valley—Hard Soils β
1
/β = 2, ρ
1
/ρ = 3/2, and µ
1
/µ = 6
x-component displacement y-component displacement
x/a x/a
Fig. 4.12. Trapezoidal alluvial valley h/a = 1, η=2, θ = 5°, 30°, 60°, 90°, N
max
=20,
and slope 45°.
0"
1"
2"
3"
4"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
Angle
of
Incidence=5°
Angle
of
Incidence=30°
Angle
of
Incidence=60°
Angle
of
Incidence=90°
119
Figure 4.12 shows the results for a trapezoidal alluvial valley for a depth to width ratio
h/a=1, for dimensionless frequency η = 2 and slope of 45° for an angle of incidence of θ=
5°, 30°, 60°, and 90°, with a ratio of alluvial valley, half-space shear velocity β
1/
β = 2,
and mass density ρ
1
/ρ=3/2, which is a much harder soil. For the graphs with incident
angles of θ = 5°, 30°, and 60°, the x-component displacement amplitudes gradually
oscillate about the free-field amplitude on the front and back side of the valley. Along the
alluvial valley surface, both the x- and y-displacement amplitudes are gradually
oscillatory. The graphs with the incident angle of 90° produce the same trend as the
others except that they are symmetric. The magnitudes of the amplitudes are
approximately 20% of the size of those in softer soils.
IV.8 Observations
The results of this research has been presented for various shaped alluvial valleys: semi-
circular, elliptical, and trapezoidal. These results have been presented in Figs. 4.3 – 4.12
and the following general observations can be made:
(1) Lee and Liu’s (Lee and Liu
10
) analytical approach was applied to a semi-circular
canyon with η = 2 and θ = 60° and compared with an alluvial valley using the
weighted residual method. The results were shown to match. The same amplification
of surface displacements, as seen in previous studies, is also shown in these results.
(2) This approach using the weighted residual method was applied to an elliptical-
shaped soft alluvial valley with varying b/a (depth/width) ratios of 1, 1.25, and 1.5.
The resulting plots for the displacement amplitudes are for elliptical canyons with a
higher frequency range than for those alluvial valleys in previous studies.
(3) The elliptical alluvial valley with b/a = 1 and 1.25 produced total displacement
amplitudes on both the x- and y-components for η = 8, which were highly
oscillatory around the free-field surface on the front side of the valley and produced
a more shadowy gradual oscillation along the backside of the valley. Within the
valley, there were oscillations with spikes at the rims of the valley at x/a =- 1 and x/a
= +1. The oscillations within the valley became more rapid as η increased.
120
Increasing the elliptical depth to b = 1.5 decreased the oscillations within the valley
surface and produced larger spikes with greater magnitude at the valley rims.
(4) This approach was then applied to the trapezoidal soft alluvial valleys. The total
displacement amplitudes of both the x- and y-components on the half space showed
a trend of being oscillatory on the front side of the valley around the free-field
amplitudes and produced a shadowy behavior about the free field around the back
side. Within the valley, both the x- and y-components were highly oscillatory.
(5) At the trapezoidal-alluvial valley rims, spikes in the displacement amplitudes were
observed for slope 45° for the y-displacement amplitude.
(6) Changing the slope of the trapezoidal alluvial valley walls from 60° to 45°
demonstrated that the amplitudes of both the x- component and y-component appear
sensitive to the angle of incidence and the slope of the valley sides. The
amplification of the x- component amplitudes and y-component amplitudes were
seen to change from the valley sides to the canyon bottom. The degree of the spikes
of the valley rims also changed.
(7) This approach was then applied to the trapezoidal hard alluvial valleys. The total
displacement amplitudes of both the x- and y-components on the half-space were
20% of the amplitude magnitudes of the soft alluvial valleys. The x- and y-
component amplitudes gradually oscillated about the front side, back side, and
within the valley.
(8) This method provided good results for an arbitrary-shaped valley in which the radius
to the canyon surface was approximately equal to a. The solutions became
numerically unstable as the radius to the canyon boundary decreased because the
Bessel functions values exponentially increased.
IV.9 Summary
The two-dimensional diffraction of incident P-waves around an arbitrary-shaped irregular
alluvial valley in an elastic half space is presented in this paper. The scattered wave
potentials are defined by an infinite series of terms with Hankel functions and only sine
terms. Using the zero stress boundary conditions, a solution is made using the weighted
121
residual method to create a set of simultaneous equations for the unknown coefficients
for a finite number of terms for the series.
The solution is applied to a traditional semi-circular canyon to verify the results of an
almost-empty alluvial valley. The method is then applied to other shaped alluvial valleys
to demonstrate the versatility of the methodology. This paper demonstrates that good
results can be achieved in deep (b >= a) valleys with frequencies as high as η = 8.
However, the results become unstable for shallow alluvial valleys (b < a). Good results
were achieved for the trapezoidal alluvial valleys of slopes of 60° and 45°.
The method in this paper has versatility and may be applied to other incident waves as
well as alluvial valleys of different shapes.
122
V CONCLUSIONS
V.1 Purpose
The purpose of this thesis was to study the scattering and diffraction of P-waves on an
arbitrary shaped canyon or valley because nothing is regular in nature. Prior research
focused on using the analytical methods to solve a variety of regularly shaped canyons
and valleys. The goal of this thesis is to generate a method that would have the flexibility
to define any surface that could potentially represent the complexity of a real life canyon
or valley. This would provide researchers a model to analyze the observed variations in
ground motions caused by local topography. This thesis is an initial step in achieving
that goal.
V.2 The Numerical Method
This thesis builds upon Lee and Liu’s analytical method which redefined the cylindrical-
wave function to satisfy the zero stress-boundary condition along the half-space for semi-
circular canyons by defining the wave potentials in terms of a single sine or cosine
function only, instead of a combination of both. The analytical method could only solve
simple geometrically shaped canyons because the wave equation could only be defined
by one of the six coordinate systems-rectangular, cylindrical, elliptical, parabolical,
spherical and spheroidal. Combining Lee and Liu’s formulation with the use of the
weighted residual method for the solution of the wave function, allows for a solution to
an arbitrary shaped canyon and valley. In the weighted residual method the two traction
equations on the boundaries contain an infinite number of unknowns due to the function
being an infinite series. Applying the weighted residual method with the weighting
functions and truncating the series to N terms, 2N independent equations can be used to
solve for the 2N unknowns. With the use of the weighted residual method, which is
simple to formulate, the geometric shape restrictions can be removed and the arbitrary
shape can be solved because the weighted residual method only requires integration along
the original boundaries.
123
V.3 The Physics of the problem
In reality the application of the weighted residual method addresses the physics of the
problem since the waves travelling through a medium take on the simplest wave form to
satisfy the physics of nature. The physics is the diffraction and scattering of the waves at
the surface of the canyon or at the interface between the surface and the sediment. For
the canyon, the waves need to satisfy the zero stress at the surface. For the valley, the
waves need to satisfy the continuity of stress and displacement at the interface in addition
to the zero stress at the surface. The weighted residual method satisfies these boundary
conditions by integrating the traction equations along the arbitrary shaped surfaces and
interfaces.
V.4 Achieving Good Results
Success or failure of the application of the weighted residual method is dependent upon
managing the Hankel function so that the numerical problem doesn’t become unstable.
Hankel functions are used because they represent an outgoing cylindrical wave that
originates from a source point and propagates radially outward. The Hankel function
goes to negative infinity if the surface edge gets too close to the origin. Therefore, for
best results the shape of the canyon or valley must oscillate about a shape closely related
to an arc of a circle. For example good results are achieved with a trapezoid as long as
the sides follow the circumference of a circle. When the canyon was a circular shape it
matched exactly Lee and Liu’s analytic solution. Other shapes such as deep ellipses,
deep trapezoids, shallow ellipses, shallow trapezoids, nurek dams, hard and medium
stiffness valleys all produced results that converged. When other shapes such as a
rectangle were tried the solution didn’t converge because the rectangle deviated too much
from the general shape of a circle.
V.5 Arbitrary Shaped Canyons
In Chapter II canyons: semi-circular, elliptical, and trapezoidal were studied with the
coordinate system located at the origin. This approach was applied to elliptical-shaped
canyons with varying b/a (depth/width) ratios of 1.25 and 1.5. This approach was then
applied to the trapezoidal canyon. Changing the slope of the trapezoidal canyon walls
124
from 60° to 45° demonstrated that the amplitudes of both the x- component and y-
component appeared sensitive to the angle of incidence and the slope of the canyon sides.
For the trapezoidal canyon, larger values of η (greater than 6) caused convergence
problems. This method provides good results for arbitrary-shaped canyons in which the
radius to the canyon surface is approximately equal to r = a. The solutions became
numerically unstable as the radius to the canyon boundary became small, because the
Bessel functions caused numerical problems.
V.6 Shallow Arbitrary Shaped Canyons
In Chapter III various shaped shallow canyons: circle segment, elliptical, and trapezoidal
were studied. For the shallow canyon to be studied the coordinate system was moved
above the half-space surface at a distance d, and the application of the weighted residual
method was applied to solve for the arbitrary-shaped shallow canyons. The coordinate
system could not be defined at the half-space because the edge of the canyon and the
origin of the canyon are too close causing the Hankel function to lead to convergence
problems. This approach was first applied to a circle segment of depth h/a = 0.5 with
coordinates located above the half-space surface at a distance d. The results for the
displacement amplitudes are for higher frequencies than η = 6, and greater than those
seen in previous studies. This approach was then applied to shallow elliptical canyons
with varying b/a (depth/width) ratios = 0.25, 0.5, and 0.75, with the coordinate system
located above the half-space surface at a respective distance of d/a = 0.75, 0.5 and 0.25
for frequencies η = 6 and 8, which are much greater than those previously studied.
Convergence problems occurred for distances d/a < 0.25. This analytical approach was
then applied to shallow trapezoidal canyons defined by a base of a = 1 and a depth of h/a
= 0.25, 0.5 and 0.75 for a coordinate system located above the half-space surface at a
respective distance d/a = 0.75, 0.5 and 0.25 for sloping sides of 60° and 45°.
Convergence problems also occurred for distances d/a < 0.25. This analytical approach
was then applied to a Nurek Dam model. The dam produced total displacement
amplitudes on both the x- and y-components for η = 2 and 8.
125
V.7 Arbitrary Shaped Valleys
In Chapter IV, various shaped alluvial valleys: semi-circular, elliptical, and trapezoidal.
This approach was first applied to an almost empty alluvial valley with β
1
/β
= 0.1 and
mass density ρ
1
/ρ
= 0.1 and compared to Lee and Liu’s analytical approach of a semi-
circular canyon with η = 2 and θ = 60°. The results were shown to match. This approach
was then applied to an elliptical-shaped soft alluvial valley, meaning that the alluvial
valley is softer than the surrounding half-space with a wave speed ratio β
1
/β < 1. For this
case β
1
/β
= 1/2 and mass density ρ
1
/ρ
= 2/3with varying b/a (depth/width) ratios of 1,
1.25, and 1.5. The resulting plots for the displacement amplitudes are for elliptical
canyons with a higher frequency range than for those alluvial valleys in previous studies.
Depths less than 1 and greater than 1.5 were shown to have convergence problems. This
approach was then applied to the trapezoidal soft alluvial valleys. Changing the slope of
the trapezoidal alluvial valley walls from 60° to 45° demonstrated that the amplitudes of
both the x- component and y-component appear sensitive to the angle of incidence and
the slope of the valley sides. This analytical approach was then applied to the trapezoidal
hard alluvial valleys meaning the alluvial valley is harder than the surrounding half speed
with a wave speed ratio β
1
/β > 1. For this case β
1
/β
= 3/2 and mass density ρ
1
/ρ
= 6 with
a b/a (depth/width) ratios of 1. This method provided good results for an arbitrary-
shaped valley in which the radius to the valley surface was approximately equal to r = a.
The solutions became numerically unstable as the radius to the canyon boundary
decreased because the Bessel functions values exponentially increased.
V.8 Programming Challenges
The numerical results were all achieved using visual basic and excel. This was a
challenge because visual basic doesn’t recognize imaginary numbers which required
programming all the complex operations. On the other hand programming using macros
with visual basic allowed for an easy interface between the results and the graphical
presentations. Visual basic is a disadvantage for large systems of equations when the
terms require integration because it is very slow. Some of the computer trials would take
up to ten hours to run. Future work might consider the use of Fortran.
126
VI Future Work
The application of the weighted residual method for the diffraction of plane P-waves in a
2-D elastic half-space for irregular shaped canyons was developed in Chapter II and
expands it in Chapter III for shallow irregular shaped canyons and expands it even further
by applying the same methodology to irregular shaped alluvial valleys. Researchers have
observed that when earthquake waves travel through these basins filled with soft, deep
sedimentary layers and alluvial deposits the waves slow down, and the amplitude
increases in order to carry the same amount of energy as the waves in the rock. This
amplification of ground motions in valleys especially at the boundaries may have
significant impact on structures located at these boundaries. This section on proposed
future work develops the tools that can enhance this research by expanding on Lee and
Liu’s approach (Lee and Liu
11
), and by applying the weighted residual method for the
solution of their wave function for a shallow moon shaped irregular alluvial valley.
where the bottom and top of the alluvial filled valley is defined by circle segments that
meet at the valley boundary. Some preliminary results are shown in the following figures.
VI.1 MODEL
Fig. 6.1. Irregular Moon Shaped Alluvial Valley with Coordinates at the Half Space
In Figure 6.1, the two-dimensional model of the irregular shaped alluvial valley in an
elastic half-space (y > 0) is defined. Each point will have an (x,y) rectangular coordinate
and once transformed into polar coordinates each point will have an (r, θ) location. The
127
geometry of the canyon is transformed from the rectangular coordinate system into a
cylindrical coordinate system with the same origin at O.
x= rsinθ
y= rcosθ
r= x
2
+ y
2
(80)
θ = tan
−1
x
y
⎛
⎝
⎜
⎞
⎠
⎟
The figure encompasses the following: an incident Plane P-wave defined by the Potential
i
φ
i
, with an incidence angle θ
α
. The incident angle is measured with respect to the
horizontal x-axis. The waves have a circular frequency ω=2πf, a longitudinal wave
velocity α and a transverse wave velocity β.
Using Lee and Liu’s (Lee and Liu
11
)methodology which was summarized in Chapter I,
the harmonic motion induced by incident P-wave without the presence of the irregular
valley results in
the same free field P-wave Potential, SV-wave potential, and scattered P-
and SV- wave potentials, Equations 67 and 68.
VI.2 Presence of Irregular Alluvial Valley
The Irregular Alluvial Valley properties are defined in Chapter IV. In the Irregular
Alluvial Valley there exist an additional scattering P-wave and a scattering SV-wave due
to the reflection from the valley boundaries. (Lee
5
, Moeen-Vaziri and Trifunac
6,7
)
These
scattering waves take the following form:
φ
v
= B
n
H
n
(1)
(k
α1
r)sinnθ
n=0
∞
∑
+B
2n
H
n
(2)
(k
α1
r)sinnθ
(81)
ψ
v
= D
n
H
n
(1)
(k
β1
r)sinnθ
n=0
∞
∑
+D
2n
H
n
(2)
(k
β1
r)sinnθ
The use of the Hankel function of the 2
nd
order, H
n
(2)
, represents incoming waves
converging on the origin.
128
VI.3 Boundary Conditions for the Interface
The Boundary Conditions for the resulting combination of waves must satisfy
displacement continuity and traction continuity at the interface between the half-space
and the valley and the traction continuity along the top of the valley.
Therefore, boundary conditions for the displacement continuity on the half-space valley
interface are:
u
r
v
= u
r
s
+u
r
ff
u
θ
v
= u
θ
s
+u
θ
ff
(82)
In order to satisfy the boundary conditions of stress continuity on the surface of the
canyon, tractions are computed. On the canyon, the Boundary Condition of stress
compatibility must be satisfied on the surface of the arbitrary shape are: (Lee and Wu
9,10
)
T
r
s
+T
r
ff
=T
r
s
v
T
θ
s
+T
θ
ff
=T
θ
s
v
(83)
Boundary Conditions along the top of the canyon, the traction on the free surface must be
zero:
T
r
v
=0
T
θ
v
=0
(84)
VI.4 Application of Weighted Residual Method
The functions that define the stresses are an infinite summation and thus the traction
equations, Equations 73 and 74, are an infinite summation. Applying the method of
weighted residuals, the weighting function is:
w
m
= sin mθ
(85)
and the three sets of boundary condition equations become:
T
r
s
( )
N
w
m
dθ
θ
∫
− T
r
sv
( )
N
w
m
dθ
θ
∫
= − T
r
ff
( )
N
w
m
dθ
θ
∫
T
θ
s
( )
N
w
m
dθ
θ
∫
− T
θ
sv
( )
N
w
m
dθ
θ
∫
= − T
θ
ff
( )
N
w
m
dθ
θ
∫
(86)
129
u
r
s
( )
N
w
m
dθ −
θ
∫
u
r
sv
( )
N
θ
∫
w
m
dθ = − u
r
ff
( )
θ
∫
w
m
dθ
u
θ
s
( )
N
w
m
dθ −
θ
∫
u
θ
sv
( )
N
θ
∫
w
m
dθ = − u
θ
ff
( )
θ
∫
w
m
dθ
(87)
(T
r
v
)
N
w
m
dθ = 0
θ
∫
(T
θ
v
)
N
w
m
dθ = 0
θ
∫
(88)
Since sin(mθ) equals zero when m=0, Equations 86 and 87 and 88 represent 3N pairs of
equations with 6N unknowns, A
n
, B
n,
C
n
,, D
n
for n=1, 2,….N.
VI.5 Results for a Semi-Circular Moon Shaped Valley- HARD SOIL β
1
/β= 2,
ρ
1
/ρ=3/2, µ
1
/µ=6
130
x-component displacement y-component displacement
x/a x/a
Fig. 6.2a. Moon Shaped Valley η=2, H1/a=1, H2/a=0.8, N
max
=12,
θ=5°, 30°
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
!8#
!6#
!4#
!2#
0#
2#
4#
6#
8#
10#
!4# !3# !2# !1# 0# 1# 2# 3# 4#
!8#
!6#
!4#
!2#
0#
2#
4#
6#
8#
10#
!4# !3# !2# !1# 0# 1# 2# 3# 4#
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
!8#
!6#
!4#
!2#
0#
2#
4#
6#
8#
10#
!4# !3# !2# !1# 0# 1# 2# 3# 4#
!8#
!6#
!4#
!2#
0#
2#
4#
6#
8#
10#
!4# !3# !2# !1# 0# 1# 2# 3# 4#
Amplitude
θ=5°
Amplitude
θ=30°
Phase/π,
θ=5°
Phase/π,
θ=30°
131
x-component displacment y-component displacement
x/a x/a
Fig. 6.2b. Moon Shaped Valley η=2, H1/a=1, H2/a=0.8, N
max
=12,
θ=60°, 90°
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
!8#
!6#
!4#
!2#
0#
2#
4#
6#
8#
10#
!4# !3# !2# !1# 0# 1# 2# 3# 4#
!8#
!6#
!4#
!2#
0#
2#
4#
6#
8#
10#
!4# !3# !2# !1# 0# 1# 2# 3# 4#
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
!8#
!6#
!4#
!2#
0#
2#
4#
6#
8#
10#
!4# !3# !2# !1# 0# 1# 2# 3# 4#
!8#
!6#
!4#
!2#
0#
2#
4#
6#
8#
10#
!4# !3# !2# !1# 0# 1# 2# 3# 4#
Amplitude
θ=60°
Amplitude
θ=90°
Phase/π,
θ=90°
Phase/π,
θ=60°
132
Figure 6.2a,b shows the results for a semi-circular moon shaped alluvial valley of depth
H1/a=1 and H2/a=0.8, with dimensionless frequency η=2, and an angle of incidence θ=
5°, 30°, 60°, and 90°, hard soils β
1
/β= 2, ρ
1
/ρ=3/2, µ
1
/µ=6. For the graphs with incident
angle θ= 5°, the x-component displacement amplitudes slightly oscillate about the free-
field amplitude on the front side of the valley, and produce a shadowy behavior along the
backside of the valley. Along the top surface of the moon shaped alluvial valley, the x-
component displacement amplitudes are highly oscillatory, producing a spikes on the
front side rim of the valley x/a = -1 and on the backside rim of the valley at x/a =+1. The
y-component displacement amplitudes are slightly oscillate about the free-field amplitude
on the front side of the valley and produce a shadowy behavior along the backside of the
valley. Along the surface of the moon shaped alluvial valley the displacement amplitudes
are highly oscillatory with spikes at the rims of the valley at x/a = -1 and x/a =+1. For the
graphs with the incident angle θ= 30°, the x-component displacement amplitudes tend to
oscillate about the free-field amplitude on the front side of the valley, but gradually
oscillate, and produce a shadowy behavior along the backside of the valley. Along the
surface of the moon shaped alluvial valley both the x- component displacement
amplitudes are highly oscillatory and produce a spike on the front side and backside rim
of the valley. The y-component displacement amplitudes tend to slightly oscillate about
the free-field amplitude on the front side of the valley, and produce a shadowy behavior
along the backside of the valley. Along the surface of the moon shaped alluvial valley the
displacement amplitudes are highly oscillatory with large spikes at the rims of the valley
x/a =+1 and x/a =-1. For the graphs with the incident angle θ= 60°, the x-component
displacement amplitudes tend to slightly oscillate about the free-field amplitude on the
front side of the valley, but produce a shadowy behavior along the backside of the valley.
Along the surface of the moon shaped alluvial valley both the x- component displacement
amplitudes are highly oscillatory producing a spike at both rims of the valley. The y-
component displacement amplitudes tend to produce a shadowy behavior about the free-
field amplitude on the front side of the valley and along the backside of the valley. Along
the surface of the moon shaped alluvial valley the displacement amplitudes are highly
oscillatory with a large spike at the rims of the valley x/a =+1 and x/a =-1. For the graphs
with the incident angle θ= 90°, the x- and y-component displacement amplitudes are
133
symmetric about the origin 0. The x-component displacement amplitudes produce no
motion on the front side and backside of the canyon. Along the surface of the moon
shaped alluvial valley, the displacement amplitudes are highly oscillatory with spikes at
both rims of the valley at x/a =+1 and x/a =-1. The y- component free-field amplitudes
produce a shadowy behavior on the front side of the valley and along the backside of the
valley. Along the surface of the moon shaped alluvial valley, the displacement
amplitudes are very highly oscillatory with spikes at both rims of the valley at x/a =+1
and x/a =-1.
Figure 6.2a,b also shows the phase diagrams alongside their corresponding displacement
amplitudes. All phase diagrams have been scaled by π and have been shifted arbitrarily
to have a zero phase angle at x/a=0
134
Fig. 6.3a. 3-Dimensional Moon Shaped Valley- x/a vs. U
x
for η=2, H1=1 H2=0.8,
N
max
=12,
θ=5°, 10°, 15°, 20°, 25°, 30°, 35°, 40°, 45°, 50°, 55°, 60°65°, 70°, 75°, 80°, 85°, 90°
135
Fig. 6.3b. 3-Dimensional Moon Shaped Valley- x/a vs. U
y
for η=2, H1=1 H2=0.8,
N
max
=12,
θ=5°, 10°, 15°, 20°, 25°, 30°, 35°, 40°, 45°, 50°, 55°, 60°65°, 70°, 75°, 80°, 85°, 90°
136
Figure 6.3a, b, show the 3-Dimensional results for a semi-circular moon shaped alluvial
valley with hard soil at a depth of H1 = 1 and H2 = 0.8, with dimensionless frequency
η=2, and an angle of incidence θ=5°, 10°, 15°, 20°, 25°, 30°, 35°, 40°, 45°, 50°, 55°, 60°,
65°, 70°, 75°, 80°, 85°, 90°, hard soils β
1
/β= 2, ρ
1
/ρ=3/2, µ
1
/µ=6.
137
VI.6 Results for a Semi-Circular Moon Shaped Valley- Medium SOILS β
1
/β= 0.95,
ρ
1
/ρ=0.90, µ
1
/µ=1
x-component displacement y-component displacement
x/a x/a
Fig. 6.4a. Moon Shaped Valley η=2, H1/a=1, H2/a=0.8, N
max
=15, θ=5°, 30°
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
!8#
!6#
!4#
!2#
0#
2#
4#
6#
8#
!4# !3# !2# !1# 0# 1# 2# 3# 4#
!8#
!6#
!4#
!2#
0#
2#
4#
6#
8#
!4# !3# !2# !1# 0# 1# 2# 3# 4#
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
!8#
!6#
!4#
!2#
0#
2#
4#
6#
8#
!4# !3# !2# !1# 0# 1# 2# 3# 4#
!8#
!6#
!4#
!2#
0#
2#
4#
6#
8#
!4# !3# !2# !1# 0# 1# 2# 3# 4#
Amplitude
θ=5°
Amplitude
θ=30°
Phase/π,
θ=5°
Phase/π,
θ=30°
138
x-component displacement y-component displacement
x/a x/a
Fig. 6.4b. Moon Shaped Valley η=2, H1/a=1, H2/a=0.8, N
max
=15, θ=60°, 90°
Figure 6.4a, b shows the results for a semi-circular moon shaped alluvial valley of depth
H1/a=1 and H2/a=0.8, with dimensionless frequency η=2, and an angle of incidence θ=
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
!8#
!6#
!4#
!2#
0#
2#
4#
6#
8#
!4# !3# !2# !1# 0# 1# 2# 3# 4#
!8#
!6#
!4#
!2#
0#
2#
4#
6#
8#
!4# !3# !2# !1# 0# 1# 2# 3# 4#
0"
1"
2"
3"
4"
5"
6"
7"
8"
9"
10"
,4" ,3" ,2" ,1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
5"
6"
7"
8"
9"
10"
,4" ,3" ,2" ,1" 0" 1" 2" 3" 4"
!8#
!6#
!4#
!2#
0#
2#
4#
6#
8#
!4# !3# !2# !1# 0# 1# 2# 3# 4#
!8#
!6#
!4#
!2#
0#
2#
4#
6#
8#
!4# !3# !2# !1# 0# 1# 2# 3# 4#
Amplitude
θ=60°
Amplitude
θ=90°
Phase/π,
θ=90°
Phase/π,
θ=60°
139
5°, 30°, 60°, and 90°, medium soils β
1
/β= 0.95, ρ
1
/ρ=0.9, µ
1
/µ=1. For the graphs with
incident angle θ= 5°, the x-component displacement amplitudes slightly oscillate about
the free-field amplitude on the front side of the valley, and produce a shadowy behavior
along the backside of the valley. Along the top surface of the moon shaped alluvial
valley, the x- component displacement amplitudes are highly oscillatory, producing a
spikes on the front side rim of the valley x/a = -1. The y-component displacement
amplitudes are slightly oscillate about the free-field amplitude on the front side of the
valley and produce a shadowy behavior along the backside of the valley. Along the
surface of the moon shaped alluvial valley the displacement amplitudes are highly
oscillatory with a small spike at the rims of the valley at x/a = -1 and x/a =+1. For the
graphs with the incident angle θ= 30°, the x-component displacement amplitudes tend to
oscillate about the free-field amplitude on the front side of the valley, but produce a
shadowy behavior along the backside of the valley. Along the surface of the moon shaped
alluvial valley both the x- component displacement amplitudes are highly oscillatory and
produce a spike on the front side rim of the valley. The y-component displacement
amplitudes produce a shadowy behavior about the free-field amplitude on the front side
of the valley, and along the backside of the valley. Along the surface of the moon shaped
alluvial valley the displacement amplitudes are highly oscillatory. For the graphs with
the incident angle θ= 60°, the x-component displacement amplitudes tend to slightly
oscillate about the free-field amplitude on the front side of the valley and along the
backside of the valley. Along the surface of the moon shaped alluvial valley both the x-
component displacement amplitudes are highly oscillatory producing a spike at the front
side rim of the valley. The y-component displacement amplitudes tend to produce a
shadowy behavior about the free-field amplitude on the front side of the valley and along
the backside of the valley. Along the surface of the moon shaped alluvial valley the
displacement amplitudes are highly oscillatory. For the graphs with the incident angle θ=
90°, the x- and y-component displacement amplitudes are symmetric about the origin 0.
The x-component displacement amplitudes produce no motion on the front side and
backside of the canyon. Along the surface of the moon shaped alluvial valley, the
displacement amplitudes are highly oscillatory. The y- component free-field amplitudes
produce a shadowy behavior on the front side of the valley and along the backside of the
140
valley. Along the surface of the moon shaped alluvial valley, the displacement
amplitudes are very highly oscillatory.
Figure 6.4a,b also shows the phase diagrams alongside their corresponding displacement
amplitudes. All phase diagrams have been scaled by π and have been shifted arbitrarily
to have a zero phase angle at x/a=0
Fig. 6.5a. Fig. 4a. 3-Dimensional Moon Shaped Valley- x/a vs. U
x
for η=2, H1=1
H2=0.8, N
max
=15, θ=5°, 30°, 60°, 90°
Fig. 6.5b. 3-Dimensional Moon Shaped Valley- x/a vs. U
y
for η=2, H1=1 H2=0.8,
N
max
=15,θ=5°, 30°, 60°, 90°
Figure 6.3a, b, show the 3-Dimensional results for a semi-circular moon shaped alluvial
valley with medium stiff soil β
1
/β= 0.95, ρ
1
/ρ=0.9, µ
1
/µ=1 at a depth of H1=1 and
H2=0.8, with dimensionless frequency η=2, and an angle of incidence θ=5°, 10°, 15°,
20°, 25°, 30°, 35°, 40°, 45°, 50°, 55°, 60°, 65°, 70°, 75°, 80°, 85°, 90°.
141
The figures show preliminary results for two moon shaped valleys with depth H1/a=1 and
H2/a=0.8 for a hard soil and a medium stiff soil. The preliminary results were favorable
for a circle segment moon shaped valley in which the radius to top canyon surface was
approximately equal to H1/a=1 and the radius to the bottom canyon surface was
approximately equal to H2/a=0.8. The solutions became numerically unstable as the
radius to the top canyon boundary H2/a decreased <0.8, or the radius to the bottom
canyon boundary H1/a increased >1. Large amplifications due result from constructive
interference of direct waves with the basin-edge-generated scattered waves due to the
variable thickness of soft sediments. These results are compatible with observed ground
motions in alluvial valleys, especially at the boundaries of the valleys due to
reflection/formation at basin edges and other discontinuities in seismic properties.
The methodology used in this potential future work could potentially be applied to actual
alluvial valleys where recorded ground motions have been obtained.
142
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145
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Chapter II
1 Achenbach, J.D. Wave Propagation in Elastic Solids. 1973 North Holland,
Elsevier Science
2 Cao, H., and Lee, V.W. Scattering and Diffraction of Plane SH waves by
Circular Cylindrical Canyons with Variable Depth-to-Width Ratio. European J.
of Earthquake Engineering (in Press)1989, 2:29-37
3 Cao, H., and Lee, V.W. Scattering and Diffraction of Plane P-waves by
Circular Cylindrical Canyons with Variable Depth-to-Width Ratio. Soil
Dynamics and Earthquake Engineering, 1990, Vol. 9, No.3 pp 141–150.
4 Harrington, R.F. Matrix Methods for Field Problems, Proceedings of the IEEE,
Vol., 55, No.2, February, 1967.
5 Manoogian, M.E., and Lee, V.W. Scattering of SH-Waves by Arbitrary Surface
Topograhpy, 3
rd
Int. Conference on Recent Advances in Geotechnical
Earthquake Engineering & Soil Dynamics, Univ. of Missouri-Rolla, Conf.
Report Vol. II; 665-670, Apr 2-9, St. Louis, Missouri, 95
6 Manoogian, M.E., and Lee, V.W. Diffraction of SH-Waves by Subsurface
Inclusions of Arbitrary Shape, J. Eng. Mech., A.S.C.E., 122(2), 123-129, 7
Pages, 15 Refs., Feb, 96
7 Manoogian, M.E., and Lee, V.W. Review of Methods to Calculate
Displacements near Surface and Sub-Surface Irregular Topography in an
Elastic Half-Space, 11
th
World Conference on Earthquake Engineering,
Acapulco, MEXICO, Jun 23-28, 96.
8 Manoogian, M.E., and Lee, V.W. Antiplane Deformations Near Arbitrary-
Shape Alluvial Valleys, ISET J. Earthquake Technology, 36(2), 107-120, 14
Parges, 63 Refs., June, 99.
9 Moeen-Vaziri and Trifunac, M.D. Scattering and diffraction of plane SH waves
146
by two-dimensional inhomogeneities, Int. J. Soil Dynamics and Earthquake
Engineering. Vol. 7, No.4, pp 179–188, 1987.
10 Moeen-Vaziri, N. and Trifunac, M.D. Scattering and Diffraction of plane P and
SV waves by two-dimensional inhomogeneities., Int. J. Soil Dynamics and
Earthquake Eng. Vol. 7, No..4 pp 189–200, 1988.
11 Pao, Y.H. and Mow. C.C. Diffraction of elastic waves and dynamic stress
concentrations. Crane, Russak and Company, Inc. New York, 1973
12 Trifuanc, M.D. A note on scattering of plane SH waves by a semi-cylindrical
canyon, Int. J. Earthquake Eng. And Struct. Dynamics, 1973, 1, 267–281.
13 V.W. Lee and J. Karl Diffraction of SV Waves by underground, Circular,
Cylindrical Cavities, Int. J. Soil Dynamics & Earthquake Eng., 11(8), (1992),
445-456, 12 Pages, 18 Refs., Jun, 93
14 V.W. Lee and J. Karl Diffraction of Elastic Plane P Waves by Circular
Underground Unlined Tunnels, European J. Earthquake Eng., VI(1), 29-36, 8
Pages, 15 Refs., Aug, 93
15 V.W. Lee and M.E. Manoogian Surface Motion above an Arbitrary shape
Underground Cavity for Incident SH Waves, European J. Earthquake Eng.,
VII(1), 3-11, 9 Pages, 13 Refs., Aug 95
16 V.W. Lee, S. Chen and M.E. Manoogian Deformations Near Surface and Sub-
Surface (Foundations Above Subway) Topographies, First China-USA-Japan
Workshop on Civil Infrastructure Systems, Shanghai, China, Nov 4-6, 98.
17 V.W. Lee & X.Y. Wu Application of the Weighted Residual Method to
Diffraction by 2-D Canyons of Arbitrary Shape, I: Incident SH Waves, Int. J.
Soil Dynamics & Earthquake Engineering. 13(5), 1994, 355–364, 10 Pages, 17
Refs, Oct. 94 (a).
18 V.W. Lee & X.Y. Wu Application of the Weighted Residual Method to
Diffraction by 2-D Canyons of Arbitrary Shape, II: Incident P, SV, & Rayleigh
Waves, Int. J. Soil Dynamics & Earthquake Engineering. 13(5), 1994, 365–373,
9 Pages, 24 Refs, Oct. 94 (b).
19 V.W. Lee & W-Y Liu. Two Dimensional Diffraction of P- and SV- Waves
Around a Semi-Circular Canyon in an Elastic Half-Space: An Analytic Solution
via a Stress Free Wave Function, Soil Dynamics and Earthquake Engineering,
(accepted for publication) 2014.
20 William H. Press, S.A.T., William T. Vetterling, Brian P. Flannery (1986).
Numerical Recipes in Fortran 77 The Art of Scientific Computing, Cambridge
University Press
147
21 Wong, H.L. and Trifunac, M.D. Scattering of plane SH waves by a semi-
cylindrical canyon, Int.J. Earthquake Eng. and Struct. Dynamics, 1974, 3, 159–
169.
Chapter III
1 Achenbach, J.D. Wave Propagation in Elastic Solids. 1973 North Holland,
Elsevier Science.
2 Cao, H., and Lee, V.W. Scattering and Diffraction of Plane P waves by Circular
Cylindrical Canyons with Variable Depth-to-Width Ratio. Soil Dynamics and
Earthquake Engineering, 1990, Vol. 9, No.3 pp 141–150.
3 Harrington, R.F. Matrix Methods for Field Problems, Proceedings of the IEEE,
Vol., 55, No.2, February, 1967.
4 Pao, Y.H. and Mow. C.C. Diffraction of elastic waves and dynamic stress
concentrations. Crane, Russak and Company, Inc. New York, 1973.
5 V.W. Lee & X.Y. Wu Application of the Weighted Residual Method to
Diffraction by 2-D Canyons of Arbitrary Shape, I: Incident SH Waves, Int. J.
Soil Dynamics & Earthquake Engineering. 13(5), 1994, 355–364, 10 Pages, 17
Refs, Oct. 94 (a).
6 V.W. Lee & X.Y. Wu Application of the Weighted Residual Method to
Diffraction by 2-D Canyons of Arbitrary Shape, II: Incident P, SV, & Rayleigh
Waves, Int. J. Soil Dynamics & Earthquake Engineering. 13(5), 1994, 365–373,
9 Pages, 24 Refs, Oct. 94 (b).
7 V.W. Lee & W-Y Liu. Two Dimensional Diffraction of P- and SV- Waves
Around a Semi-Circular Canyon in an Elastic Half-Space: An Analytic Solution
via a Stress Free Wave Function, Soil Dynamics and Earthquake Engineering,
(accepted for publication) 2014.
Chapter IV
1 Achenbach, J.D. Wave Propagation in Elastic Solids. 1973 North Holland,
Elsevier Science
2 Cao, H., and Lee, V.W. Scattering and Diffraction of Plane SH waves by
Circular Cylindrical Canyons with Variable Depth-to-Width Ratio. European J.
of Earthquake Engineering (in Press)1989, 2:29-37
148
3 Cao, H., and Lee, V.W. Scattering and Diffraction of Plane P waves by Circular
Cylindrical Canyons with Variable Depth-to-Width Ratio. Soil Dynamics and
Earthquake Engineering, 1990, Vol. 9, No.3 pp 141-150
4 Harrington, R.F. Matrix Methods for Field Problem, Proceedings of the IEEE,
Vol., 55, No.2, February, 1967.
5 Jian-wen Liang. Lin-jun Yan. Scatting of plane P waves by circular-arc layered
alluvial valleys: An analytical . ACTA SEISMOLOGICA SINICA, 2001, Vol.14,
No.2 (176~195)
6 M.I. Todorovska & V.W.Lee. Surface motion of shallow circular alluvial
valleys for incident plane SH waves-analytical solution. Soil Dynamics and
Earthquake Engineering, 1991, Vol.10, No.4 pp.192-200
7 Pao, Y.H. and Mow. C.C. Diffraction of elastic waves and dynamic stress
concentrations. Crane, Russak and Company, Inc. New York, 1973
8 V.W. Lee & X.Y. Wu Application of the Weighted Residual Method to
Diffraction by 2-D Canyons of Arbitrary Shape, I: Incident SH Waves, Int. J.
Soil Dynamics & Earthquake Engineering. 13(5), 1994, 355-364, 10 Pages, 17
Refs, Oct. 94
9 V.W. Lee & X.Y. Wu Application of the Weighted Residual Method to
Diffraction by 2-D Canyons of Arbitrary Shape, II: Incident P, SV, & Rayleigh
Waves, Int. J. Soil Dynamics & Earthquake Engineering. 13(5), 1994, 365-373,
9 Pages, 24 Refs, Oct. 94
10 V.W. Lee & W-Y Liu. Two Dimensional Diffraction of P- and SV- Waves
Around a Semi-Circular Canyon in an Elastic Half-Space: An Analytic Solution
via a Stress Free Wave Function, Soil Dynamics and Earthquake Engineering,
(accepted for publication) 2014
Chapter VI
1 Achenbach, J.D. Wave Propagation in Elastic Solids. 1973 North Holland,
Elsevier Science
2 Cao, H., and Lee, V.W. Scattering and Diffraction of Plane SH waves by
Circular Cylindrical Canyons with Variable Depth-to-Width Ratio. European J.
of Earthquake Engineering (in Press)1989, 2:29-37
3 Cao, H., and Lee, V.W. Scattering and Diffraction of Plane P waves by
Circular Cylindrical Canyons with Variable Depth-to-Width Ratio. Soil
Dynamics and Earthquake Engineering, 1990, Vol. 9, No.3 pp 141-150
149
4 Harrington, R.F. Matrix Methods for Field Problems, Proceedings of the IEEE,
Vol., 55, No.2, February, 1967
5 Lee, V.W. On deformation near circular underground cavity subjected to
incident plane SH-waves, Proc. of Symposium on Applications on Computer
Methods in Engineering, Vol II, USC 951-962
6 Moeen-Vaziri and Trifunac, M.D. Scattering and diffraction of plane SH waves
by two-dimensional inhomogeneities, Int. J. Soil Dynamics and Earthquake
Engineering. Vol. 7, No.4, pp 179-188, 1987
7 Moeen-Vaziri, N. and Trifunac, M.D. Scattering and Diffraction of plane P and
SV waves by two-dimensional inhomogeneities., Int. J. Soil Dynamics and
Earthquake Eng. Vol. 7, No..4 pp 189-200, 1988
8 Pao, Y.H. and Mow. C.C. Diffraction of elastic waves and dynamic stress
concentrations. Crane, Russak and Company, Inc. New York, 1973
9 V.W. Lee & X.Y. Wu Application of the Weighted Residual Method to
Diffraction by 2-D Canyons of Arbitrary Shape, I: Incident SH Waves, Int. J.
Soil Dynamics & Earthquake Engineering. 13(5), 1994, 355-364, 10 Pages, 17
Refs, Oct. 94
10 V.W. Lee & X.Y. Wu Application of the Weighted Residual Method to
Diffraction by 2-D Canyons of Arbitrary Shape, II: Incident P, SV, & Rayleigh
Waves, Int. J. Soil Dynamics & Earthquake Engineering. 13(5), 1994, 365-373,
9 Pages, 24 Refs, Oct. 94
11 V.W. Lee & W-Y Liu. Two Dimensional Diffraction of P- and SV- Waves
Around a Semi-Circular Canyon in an Elastic Half-Space: An Analytic Solution
via a Stress Free Wave Function, Soil Dynamics and Earthquake Engineering,
(accepted for publication) 2014
150
APPENDIX
The Figures in the Appendix are a continuation of trials run in Chapter II, III and IV.
x-component displacement y-component displacement
x/a x/a
Fig. A.1. Shallow Elliptical Shaped Canyon η=2, θ=5°, 30°,60°,90°, b/a=0.5
N
max
=20
0"
1"
2"
3"
4"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
Angle
of
Incidence=90°
Angle
of
Incidence=60°
Angle
of
Incidence=30°
Angle
of
Incidence=5°
151
x-component displacement y-component displacement
x/a x/a
Fig. A.2. Shallow Elliptical Shaped Canyon η=8, θ=5°, 30°,60°,90°, b/a=0.5
N
max
=72
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
Angle
of
Incidence=5°
Angle
of
Incidence=30°
Angle
of
Incidence=60°
Angle
of
Incidence=90°
152
x-component displacement y-component displacement
x/a x/a
Fig. A.3. Shallow Elliptical Shaped Canyon η=2, θ=5°, 30°,60°,90°, b/a=0.75
N
max
=24
0"
1"
2"
3"
4"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
Angle
of
Incidence=90°
Angle
of
Incidence=60°
Angle
of
Incidence=30°
Angle
of
Incidence=5°
153
x-component displacement y-component displacement
x/a x/a
Fig. A.4. Shallow Elliptical Shaped Canyon η=8, θ=5°, 30°,60°,90°, b/a=0.75
N
max
=80
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
Angle
of
Incidence=5°
Angle
of
Incidence=30°
Angle
of
Incidence=60°
Angle
of
Incidence=90°
154
x-component displacement y-component displacement
x/a x/a
Fig. A.5. Shallow Trapezoidal Shaped Canyon η=2, θ=5°, 30°, 60°, 90°, h/a=0.25,
N
max
=14 Slope 45°
0"
1"
2"
3"
4"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
5"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
5"
(4" (3" (2" (1" 0" 1" 2" 3" 4" Angle
of
Incidence=90°
Angle
of
Incidence=60°
Angle
of
Incidence=30°
Angle
of
Incidence=5°
155
x-component displacement y-component displacement
x/a x/a
Fig. A.6. Shallow Trapezoidal Shaped Canyon η=6, θ=5°, 30°, 60°, 90°, h/a=0.25,
N
max
=52 Slope 45°
0"
3"
6"
9"
12"
15"
18"
*4" *3" *2" *1" 0" 1" 2" 3" 4"
0"
3"
6"
9"
12"
15"
18"
*4" *3" *2" *1" 0" 1" 2" 3" 4"
0"
3"
6"
9"
12"
15"
18"
*4" *3" *2" *1" 0" 1" 2" 3" 4"
0"
3"
6"
9"
12"
15"
18"
*4" *3" *2" *1" 0" 1" 2" 3" 4"
0"
3"
6"
9"
12"
15"
18"
*4" *3" *2" *1" 0" 1" 2" 3" 4"
0"
3"
6"
9"
12"
15"
18"
*4" *3" *2" *1" 0" 1" 2" 3" 4"
0"
3"
6"
9"
12"
15"
18"
*4" *3" *2" *1" 0" 1" 2" 3" 4"
0"
3"
6"
9"
12"
15"
18"
*4" *3" *2" *1" 0" 1" 2" 3" 4"
Angle
of
Incidence=5°
Angle
of
Incidence=30°
Angle
of
Incidence=60°
Angle
of
Incidence=90°
156
x-component displacement y-component displacement
x/a x/a
Fig. A.7. Shallow Trapezoidal Shaped Canyon η=2, θ=5°, 30°, 60°, 90°, h/a=0.5,
N
max
=16 Slope 45°
0"
1"
2"
3"
4"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
Angle
of
Incidence=90°
Angle
of
Incidence=60°
Angle
of
Incidence=30°
Angle
of
Incidence=5°
157
x-component displacement y-component displacement
x/a x/a
Fig. A.8. Shallow Trapezoidal Shaped Canyon η=6, θ=5°, 30°, 60°, 90°, h/a=0.5,
N
max
=60 Slope 45°
0"
3"
6"
9"
12"
15"
18"
*4" *3" *2" *1" 0" 1" 2" 3" 4"
0"
3"
6"
9"
12"
15"
18"
*4" *3" *2" *1" 0" 1" 2" 3" 4"
0"
3"
6"
9"
12"
15"
18"
*4" *3" *2" *1" 0" 1" 2" 3" 4"
0"
3"
6"
9"
12"
15"
18"
*4" *3" *2" *1" 0" 1" 2" 3" 4"
0"
3"
6"
9"
12"
15"
18"
*4" *3" *2" *1" 0" 1" 2" 3" 4"
0"
3"
6"
9"
12"
15"
18"
*4" *3" *2" *1" 0" 1" 2" 3" 4"
0"
3"
6"
9"
12"
15"
18"
*4" *3" *2" *1" 0" 1" 2" 3" 4"
0"
3"
6"
9"
12"
15"
18"
*4" *3" *2" *1" 0" 1" 2" 3" 4"
Angle
of
Incidence=5°
Angle
of
Incidence=30°
Angle
of
Incidence=60°
Angle
of
Incidence=90°
158
x-component displacement y-component displacement
x/a x/a
Fig. A.9. Shallow Trapezoidal Shaped Canyon η=2, θ=5°, 30°, 60°, 90°, h/a=0.5,
N
max
=14 Slope 60°
0"
1"
2"
3"
4"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
Angle
of
Incidence=90°
Angle
of
Incidence=60°
Angle
of
Incidence=30°
Angle
of
Incidence=5°
159
x-component displacement y-component displacement
x/a x/a
Fig. A.10. Shallow Trapezoidal Shaped Canyon η=6, θ=30°, 60°, 90°, h/a=0.5,
N
max
=52 Slope 60°
0"
3"
6"
9"
12"
15"
18"
*4" *3" *2" *1" 0" 1" 2" 3" 4"
0"
3"
6"
9"
12"
15"
18"
*4" *3" *2" *1" 0" 1" 2" 3" 4"
0"
3"
6"
9"
12"
15"
18"
*4" *3" *2" *1" 0" 1" 2" 3" 4"
0"
3"
6"
9"
12"
15"
18"
*4" *3" *2" *1" 0" 1" 2" 3" 4"
0"
3"
6"
9"
12"
15"
18"
*4" *3" *2" *1" 0" 1" 2" 3" 4"
0"
3"
6"
9"
12"
15"
18"
*4" *3" *2" *1" 0" 1" 2" 3" 4"
0"
3"
6"
9"
12"
15"
18"
*4" *3" *2" *1" 0" 1" 2" 3" 4"
0"
3"
6"
9"
12"
15"
18"
*4" *3" *2" *1" 0" 1" 2" 3" 4"
Angle
of
Incidence=5°
Angle
of
Incidence=30°
Angle
of
Incidence=60°
Angle
of
Incidence=90°
160
x-component displacement y-component displacement
x/a x/a
Fig. A.11. Shallow Trapezoidal Shaped Canyon η=2, θ=5°, 30°, 60°, 90°, h/a=0.75,
N
max
=14 Slope 60°
0"
1"
2"
3"
4"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
Angle
of
Incidence=90°
Angle
of
Incidence=60°
Angle
of
Incidence=30°
Angle
of
Incidence=5°
161
x-component displacement y-component displacement
x/a x/a
Fig. A.12. Shallow Trapezoidal Shaped Canyon η=6, θ=5°, 30°, 60°, 90°, h/a=0.75,
N
max
=58 Slope 60°
0"
3"
6"
9"
12"
15"
18"
*4" *3" *2" *1" 0" 1" 2" 3" 4"
0"
3"
6"
9"
12"
15"
18"
*4" *3" *2" *1" 0" 1" 2" 3" 4"
0"
3"
6"
9"
12"
15"
18"
*4" *3" *2" *1" 0" 1" 2" 3" 4"
0"
3"
6"
9"
12"
15"
18"
*4" *3" *2" *1" 0" 1" 2" 3" 4"
0"
3"
6"
9"
12"
15"
18"
*4" *3" *2" *1" 0" 1" 2" 3" 4"
0"
3"
6"
9"
12"
15"
18"
*4" *3" *2" *1" 0" 1" 2" 3" 4"
0"
3"
6"
9"
12"
15"
18"
*4" *3" *2" *1" 0" 1" 2" 3" 4"
0"
3"
6"
9"
12"
15"
18"
*4" *3" *2" *1" 0" 1" 2" 3" 4"
Angle
of
Incidence=5°
Angle
of
Incidence=30°
Angle
of
Incidence=60°
Angle
of
Incidence=90°
162
x-component displacement y-component displacement
x/a x/a
Fig. A.13. Shallow Trapezoidal Shaped Canyon η=2, θ=5°, 30°, 60°, 90°, h/a=0.75,
N
max
=12 Slope 45°
0"
1"
2"
3"
4"
'4" '2" 0" 2" 4"
0"
1"
2"
3"
4"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
'4" '2" 0" 2" 4"
0"
1"
2"
3"
4"
'4" '2" 0" 2" 4"
0"
1"
2"
3"
4"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
Angle
of
Incidence=90°
Angle
of
Incidence=60°
Angle
of
Incidence=30°
Angle
of
Incidence=5°
163
x-component displacement y-component displacement
x/a x/a
Fig. A.14. Shallow Trapezoidal Shaped Canyon η=6, θ=5°, 30°, 60°, 90°, h=0.75,
N
max
=62 Slope 45°
0"
3"
6"
9"
12"
15"
18"
*4" *3" *2" *1" 0" 1" 2" 3" 4"
0"
3"
6"
9"
12"
15"
18"
*4" *3" *2" *1" 0" 1" 2" 3" 4"
0"
3"
6"
9"
12"
15"
18"
*4" *3" *2" *1" 0" 1" 2" 3" 4"
0"
3"
6"
9"
12"
15"
18"
*4" *3" *2" *1" 0" 1" 2" 3" 4"
0"
3"
6"
9"
12"
15"
18"
*4" *3" *2" *1" 0" 1" 2" 3" 4"
0"
3"
6"
9"
12"
15"
18"
*4" *3" *2" *1" 0" 1" 2" 3" 4"
0"
3"
6"
9"
12"
15"
18"
*4" *3" *2" *1" 0" 1" 2" 3" 4"
0"
3"
6"
9"
12"
15"
18"
*4" *3" *2" *1" 0" 1" 2" 3" 4"
Angle
of
Incidence=5°
Angle
of
Incidence=30°
Angle
of
Incidence=60°
Angle
of
Incidence=90°
164
x-component displacement y-component displacement
x/a x/a
Fig. A.15. Ellipse Alluvial Valley x/a vs. U
x
and U
y
for η=8, b=1 N
max
=60, β
1
/β= ½,
ρ
1
/ρ=2/3, µ
1
/µ=0.1667
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
Angle
of
Incidence=5°
Angle
of
Incidence=30°
Angle
of
Incidence=60°
Angle
of
Incidence=90°
165
x-component displacement y-component displacement
x/a x/a
Fig. A.16. Ellipse Alluvial Valley x/a vs. U
x
and U
y
for η=8, b=1.25 N
max
=64
β
1
/β= ½, ρ
1
/ρ=2/3, µ
1
/µ=0.1667
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
Angle
of
Incidence=5°
Angle
of
Incidence=30°
Angle
of
Incidence=60°
Angle
of
Incidence=90°
166
x-component displacement y-component displacement
x/a x/a
Fig. A.17. Trapazoid Alluvial Valley x/a vs. U
x
and U
y
for η=8, N
max
=34, Slope 60°
β
1
/β= ½, ρ
1
/ρ=2/3, µ
1
/µ=0.1667
0"
2"
4"
6"
8"
10"
12"
14"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
12"
14"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
12"
14"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
12"
14"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
12"
14"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
12"
14"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
12"
14"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
12"
14"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
Angle
of
Incidence=5°
Angle
of
Incidence=30°
Angle
of
Incidence=60°
Angle
of
Incidence=90°
167
x-component displacement y-component displacement
x/a x/a
Fig. A.18. Trapazoid Alluvial Valley x/a vs. U
x
and U
y
for η=2, N
max
=20, Slope 45°
β
1
/β= ½, ρ
1
/ρ=2/3, µ
1
/µ=0.1667
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
Angle
of
Incidence=5°
Angle
of
Incidence=30°
Angle
of
Incidence=60°
Angle
of
Incidence=90°
168
x-component displacement y-component displacement
x/a x/a
Fig. A.19. Trapazoid Alluvial Valley x/a vs. U
x
and U
y
for η=2, N
max
=30, Slope 60°
β
1
/β= 2, ρ
1
/ρ=3/2, µ
1
/µ=6
0"
1"
2"
3"
4"
'4" '2" 0" 2" 4"
0"
1"
2"
3"
4"
'4" '2" 0" 2" 4"
0"
1"
2"
3"
4"
'4" '2" 0" 2" 4"
0"
1"
2"
3"
4"
'4" '2" 0" 2" 4"
0"
1"
2"
3"
4"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
'4" '3" '2" '1" 0" 1" 2" 3" 4"
0"
1"
2"
3"
4"
'4" '2" 0" 2" 4"
0"
1"
2"
3"
4"
'4" '2" 0" 2" 4"
Angle
of
Incidence=5°
Angle
of
Incidence=30°
Angle
of
Incidence=60°
Angle
of
Incidence=90°
169
x-component displacement y-component displacement
x/a x/a
Fig. A.20. Trapazoid Alluvial Valley x/a vs. U
x
and U
y
for η=8, N
max
=54, Slope 60°
β
1
/β= 2, ρ
1
/ρ=3/2, µ
1
/µ=6
0"
2"
4"
6"
8"
10"
12"
14"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
12"
14"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
12"
14"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
12"
14"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
12"
14"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
12"
14"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
12"
14"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
12"
14"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
Angle
of
Incidence=5°
Angle
of
Incidence=30°
Angle
of
Incidence=60°
Angle
of
Incidence=90°
170
x-component displacement y-component displacement
x/a x/a
Fig. A.21. Trapazoid Alluvial Valley x/a vs. U
x
and U
y
for η=8, N
max
=36, Slope 45°
β
1
/β= 2, ρ
1
/ρ=3/2, µ
1
/µ=6
0"
2"
4"
6"
8"
10"
12"
14"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
12"
14"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
12"
14"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
12"
14"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
12"
14"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
12"
14"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
12"
14"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
0"
2"
4"
6"
8"
10"
12"
14"
(4" (3" (2" (1" 0" 1" 2" 3" 4"
Angle
of
Incidence=5°
Angle
of
Incidence=30°
Angle
of
Incidence=60°
Angle
of
Incidence=90°
Abstract (if available)
Abstract
This thesis studied the scattering and diffraction of P-waves on an arbitrary shaped canyon or valley. The analytical method defined the cylindrical-wave function to satisfy the zero stress-boundary condition along the half-space for semi-circular canyons by defining the wave potentials in terms of a single sine or cosine function only, instead of a combination of both. Combining this formulation with the use of the weighted residual method for the solution of the wave function, allows for a solution to an arbitrary shaped canyon and valley. The goal of this thesis is to generate a method that would have the flexibility to define any surface that could potentially represent the complexity of a real life canyon or valley.
Linked assets
University of Southern California Dissertations and Theses
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Asset Metadata
Creator
Brandow, Heather
(author)
Core Title
Two-dimensional weighted residual method for scattering and diffraction of elastic waves by arbitrary shaped surface topography
School
Viterbi School of Engineering
Degree
Doctor of Philosophy
Degree Program
Civil Engineering
Publication Date
07/06/2015
Defense Date
04/23/2015
Publisher
University of Southern California
(original),
University of Southern California. Libraries
(digital)
Tag
OAI-PMH Harvest,weighted residual method
Format
application/pdf
(imt)
Language
English
Contributor
Electronically uploaded by the author
(provenance)
Advisor
Lee, Vincent W. (
committee chair
), Carlson, Anders (
committee member
), Soibelman, Lucio (
committee member
), Trifunac, Mihailo D. (
committee member
), Wellford, L. Carter (
committee member
)
Permanent Link (DOI)
https://doi.org/10.25549/usctheses-c3-586826
Unique identifier
UC11301274
Identifier
etd-BrandowHea-3549.pdf (filename),usctheses-c3-586826 (legacy record id)
Legacy Identifier
etd-BrandowHea-3549.pdf
Dmrecord
586826
Document Type
Dissertation
Format
application/pdf (imt)
Rights
Brandow, Heather
Type
texts
Source
University of Southern California
(contributing entity),
University of Southern California Dissertations and Theses
(collection)
Access Conditions
The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...
Repository Name
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Repository Location
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Tags
weighted residual method