Geophys. J . Int. (1994) 117, 749-762
Resonance prediction of deep sediment valleys through an eigenvalue
method
Tiao Zhou and Marijan Dravinski
Deparrment of Mechanical Engineering, University of Southern California, Los Angeles, California 913089-1453, USA
Accepted 1993 November 30. Received 1993 November 25; in original form 1992 October 20
SUMMARY
2-D resonance of different orders for a sediment valley embedded in a half-space is
studied for anti-plane and plane-strain models. It is found that the resonant
frequencies are properties of the valley while the incident wavefield only affects the
excitation of different modes. The dimensionless resonant frequencies are found to
be independent of impedance contrast between the half-space and the valley. Based
on this property, an eigenvalue method is proposed to determine resonant
frequencies of the valley. Resonant frequencies and mode shapes are obtained by
solving an eigenvalue problem. The method is valid for valleys of arbitrary shape.
The eigenvalue problem is solved numerically based on an indirect boundary
integral method. The corresponding eigenfrequencies are identified as the resonant
frequencies of the valley for all incident waves. The eigenvalue method results are
verified by a spectral-search method. The’ method is found to be numerically much
more efficient than the spectral-search method.
Key words: eigenvalue method, elastic wave, resonance, sediment valley.
INTRODUCTION
Knowledge of sediment-valley resonance is essential in
earthquake-resistance design. Research advances in elastic
waves diffraction by sediment type of scatters makes it
possible to investigate the resonance problem in considerable detail.
Bard & Bouchon (1985) made systematic study for
dynamic response of 2-D cosine-shaped valleys subjected to
vertical-plane SH-, SV- and P-incident waves. They found
that the dimensionless fundamental resonant frequencies are
invariant to change of impedance contrast and that they
only depend on the valley shape. Empirical relations were
proposed to calculate the fundamental resonant frequencies.
Jiang & Kuribayashi (1988) examined the resonance of 3-D
axisymmetric valleys. They showed that the fundamental
frequencies depend only on 1-D resonant frequency
(Haskell 1960) at the valley centre and valley geometry,
regardless of the impedance contrast. Resonance patterns
are classified into two categories: shear mode associated
with a vertical S wave and bulk mode excited by a vertical P
wave. Mossessian & Dravinski (1990) investigated the 3-D
resonance behaviour of semi-prolate sediment-filled valleys
subjected to vertical P- and S-wave incidence. They found
that the excitation of 3-D resonance is sensitive to the
change of valley geometry, and that 3-D fundamental
resonant frequencies are higher than the 2-D ones. Ohori,
Koketsu & Minami (1990) studied 3-D resonance and
compared it to the corresponding 1-D and 2-D resonance.
They concluded that both the amplification factor and the
resonant frequency for 3-D models are larger than those for
1-D and 2-D models.
It should be noticed that previous studies of valley
resonance were limited to symmetric valleys, vertical
incidence, and fundamental mode.
The method most often used in resonance investigations is
the spectral-search method (Bard & Bouchon 1985; Jiang &
Kuribayashi 1988; Mossessian & Dravinski 1990; Ohori et
al. 1990). In this approach, resonant frequencies are
identified from the valley surface-displacement-amplitude
spectral peaks. Another approach for evaluating resonant
frequencies approximately is through use of a ray method
with an acoustic assumption and the so-called ‘in-phase
condition’ under rigid bedrock (Rial 1989; Rial, Saltzman &
Ling 1991). This method is valid only at the valley centre
and for limited geometry.
In this paper, valley resonance of different orders is
investigated. The general model involves arbitrary valley
geometry, incidence of general direction and a wide range of
impedance contrasts. Based on the resonance properties, an
eigenvalue method is developed to predict resonance. The
results are verified by a spectral-search method.
749
T. Zhou and M. Dravinski
750
-1
2
Figure 1. Problem model. D and Do are domains of the valley and
the half-space. S, is interface between the valley and the half-space
and S, is the auxiliary surface (Dravinski 1982a. b).
PROBLEM STATEMENT
The geometry of the problem is depicted in Fig. 1. A valley
of domain D is perfectly embedded in a half-space domain
Do at the interface S,. Both the valley and the half-space are
assumed to be homogeneous, isotropic and linearly elastic.
The wave equations are satisfied in both domains D and D(,.
The traction-free condition at the flat surface S, and the
continuity conditions in displacement and tractions at S, are
assumed. In addition, appropriate radiation conditions
should be satisfied by the scattered wavefield at infinity.
When an incident wave from the half-space strikes the
valley, it is partially transmitted into the valley and partially
reflected back into the half-space. At certain frequencies,
the valley surface-displacement amplification (at S,) reaches
maximum and the valley is said to be in resonance. In this
paper, the valley resonant frequencies are defined as the
frequencies at which the valley surface-displacement peak
amplitude reaches maximum. Peak amplitude is the gravest
displacement amplitude across the valley surface S,. The
objective of this study is to predict resonance of different
orders and to explain its basic characteristics.
The parameters of the problem are defined as following:
the P- and S-wave velocities and the density of the valley
are denoted by a,/3 and p , respectively and those of the
Po and pa. The impedance contrast Is and
half-space by a(,,
dimensionless frequency SZ are defined as
POBO
4 =Bp
where h is the maximum depth of the valley, k is the shear
wavenumber and f; is the fundamental frequency of a flat
layer with thickness h, corresponding to a vertical S-wave
incidence (Haskell 1960).
RESONANCE PROPERTIES OF SEDIMENT
VALLEYS
Resonance properties are illustrated first through a spectral
search method for an anti-plane strain model. Amplitude
spectra for semi-circular valleys are obtained by using the
close-form solution (Trifunac 1971). For valleys of arbitrary
shape an indirect boundary integral method (Dravinski
1982a, b; Wong 1982; Dravinski & Mossessian 1987) is
employed to construct the corresponding spectra. Four
plane-harmonic SH-incident waves, with unit displacement
amplitude and different angles of incidence, are considered.
Figure 2 shows the surface-displacement amplitude
spectra for a semi-circular valley. The first mode is symmetric
with displacement amplitude largest at the valley centre
and decaying towards the edges. The second mode is
anti-symmetric with zero displacement at the centre. The
two displacement amplitude peaks appear at x e 0 . 5 . The
third and the fourth modes are symmetric and have two and
three peaks, respectively. A similar result for a nonsymmetric valley is shown by Fig. 3. The first mode has the
maximum displacement amplitude at the valley centre and
the second mode has two peaks between the centre and the
valley edges. It can be seen that for each valley the
displacement amplitude peaks appear at the same
frequencies for different angles of incidence.
To demonstrate the sensitivity of resonant frequencies to
impedance contrast change, three semi-circular valleys with
different impedance contrasts are considered. The peakamplitude spectra are utilized to better visualize the
amplitude-frequency relation. The peak amplitude is defined
as the gravest displacement amplitude across the valley free
surface. The peak-amplitude spectra for these valleys are
shown in Fig. 4. The first, third and fourth modes are
symmetric and the second one is anti-symmetric. The
symmetric modes are excited by all the incident waves
considered, while the antisymmetric modes are excited only
by an off-vertical incidence. It can be seen from Fig. 4 that
the resonance takes place at the same frequencies in each
model for different incident waves. This suggests that
resonant frequencies are properties of the valley geometry
and its boundary conditions, while the nature of incidence
only affects the excitation of resonant modes. The resonant
frequencies of different orders apparently are invariant to
change in the impedance contrast.
RESONANCE PREDICTION A N D
EIGENVALUE PROBLEMS
Since the resonant frequencies are invariant to change in
impedance contrast Ip, they can be calculated from the
extreme case of impedance contrast, corresponding to rigid
half-space (I, = 00). The resonant frequencies of this model
can be determined by solving an eigenvalue problem. This
method is referred to as the eigenvalue method.
The eigenvalue problem is associated with seeking
non-trivial solution of the equations of motion satisfying the
traction-free condition at S, and zero displacement
conditions at S,
(A + p)VV.u
+ ~ V ’ U+ PW’U
= 0; x E D .
(3)
tZ=O;
X€S,
(4)
u=o;
XES,
(5)
where A and p are the LamC’s constants, p is the density and
o is the circular frequency. Here x = ( x , y, z ) is the position
Resonance prediction
-2
-1
0
75 1
1
Figure 2. Surface-amplitude spectra for a semi-circular valley and incident plane harmonic S H waves. p = 1, p
the angle of incidence.
= 1, p,, = 3
and
= 5. Oinc is
752
T. Zhou and M . Dravinski
oinc
oinc
= 0"
oinc
= 30"
= 60"
-2
-1
0
1
2
Figure 3. Surface amplitude spectra for a non-symmetric valley and incident plane harmonic SH waves. p
the angle of incidence.
_.
=
1,
fi = 1,
pII = 2 and
fill = 4. o,,,
is
Resonance prediction
753
discussed first. Subsequently, the eigenvalue problems for
valleys of arbitrary shape are solved numerically.
SEMI-CIRCULAR VALLEY: THE ANTIPLANE STRAIN MODEL
For this valley, the general solution of eq. (3) satisfying
condition (4) is given by
c a,,(k)J,(kr) cos n 0
,,
cc
v(x) =
(6)
=o
0'
1
I
I
I
2
3
4
Frequency
where ( r , 0) are polar coordinates and a, are unknown
coefficients.
For a semi-circular valley of radius L I , condition (5) can be
expressed as
c a,(k)J,(ka) cos n o
m
= 0.
(7)
n=O
The condition for (7) to have a non-trivial solution is given
by
n = 0 , 1 , 2 , 3) . . .
J,,(ka)=0;
(8)
Therefore, the surface motion of the valley at free
oscillation can be written as
m
v(x, Q) =
1
3
2
4
Frequency
m
C C a,,TJ,t(jn.,x/a);
,,=o ., =o
-1 < x / a
<1
(9)
where the constants uns are determined by the initial
conditions and j,,,,7 is the sth zero of J,,(x) (Abramowitz &
Stegun 1972). J,a(j,z.7x/u)are the mode shapes. In terms of
dimensionless frequency, the eigenfrequencies are given by
Qi= { 1.531, 2.439, 3.269, 3.514, 4.062, . . .}.
(10)
VALLEYS OF ARBITRARY S H A P E
The eigenvalue problems for valleys of arbitrary shape in
general d o not have close-form solutions and various
numerical techniques must be used. In this work the
displacement field is expressed based on an indirect
boundary integral method (Dravinski 1982a, b).
Anti-plane strain model
Frequency
Figure 4. Peak-amplitude spectra for semi-circular valleys with
different impedance contrasts for incident plane harmonic SH
waves. Displacement amplitude is evaluated at 21 points equally
spaced along the valley free surface. -: O,nc = 0". --: O,nc = 30",
. . . : Olnc = 60" and -.-: O,,, = 90".
vector, u = (u, u, w ) is the displacement vector, and t' is a
traction vector on a z = const plane. For the anti-plane
strain model u = (0, u, 0) and for the plane-strain model
u = ( u , 0, w).
The eigenvalue problem for a semi-circular valley of the
anti-plane strain model, which has close-form solutions, is
If the displacement field u is approximated by a linear
combination of wavefields generated by L line sources along
an auxiliary surface S,, it follows then that (Dravinski
1982a)
L.
u ( x , 52) =
2 blV(x,x I , 52);
x, E S,
x E D,
(11)
1=1
where 6, are the unknown source densities. The Green's
function V , satisfies both the Helmholtz scalar equation and
the stress-free boundary condition (4).The auxiliary surface
S, is defined outside the domain D (see Fig. 1).
If condition (5) is imposed at N observation points along
the surface S,, the following system of equations is obtained
G,b = 0,
where G, is an N x L matrix and b is an L
(12)
X
1 coefficient
754
T. Zhou and M. Dravinski
vector with elements defined by
Gj(i, 1 ) = v ( X , ,
XI,
Q);
i = l , 2, . . . , N ;
X,
E s,;
XI
l = l , 2, . . . , L ;
E SA
N>L.
b = ( b , , b,, . . . , bL).
(13)
(14)
(15)
The eigenvalue problem therefore involves seeking the
frequencies for which there are non-trivial solutions for b of
system (12). This condition can be expressed as (Noble &
Daniel 1977)
det (G!'G,) = 0
(16)
components caused by these potentials. The wavefield given
by (20) and (21) satisfies the equations of motion (3) and the
traction-free condition (4) at S, (Dravinski 1982b). The
integral representation and evaluation of the Green's
functions are discussed in detail by Dravinski (1980) and
Dravinski & Mossessian (1988).
If conditions (5) are imposed at N points along S,, the
following system of equations is formed
G,c = 0
(22)
where 6, is known, 2N x 2L matrix, while c is a 2L
unknown coefficient vector with elements given by
X
1
where det( ) is the determinant of a matrix and G H is the
complex conjugate transpose of matrix G . By QR
decomposition (Ciariet & Lions 1987) it follows then
G I = Q[
]:
where Q is an N X N unitary matrix and R, is an L x L
upper triangular matrix.
Therefore, condition (16) can be expressed as
U"(i, I ) = U + ( x i , X I , Q);
(26)
uqi,I) = U*(Xi, x,,
9);
(27)
%(a)= ldet (R,)(
=0
(18)
w q i , 1 ) = W@(x;;x,,Q);
(28)
where R = Q, iRj is the complex frequency variable.
The true eigenfrequencies Q* of the system represented
by (3), (4) and (5) are real (Eringen & Suhubi 1975).
However, due to the numerical approximation, complex
roots R('=52:' + iQ:) are expected for (18) with IQyl<< 1
(Zhou 1993). Since R(R) is a continuous function, the
with the plane Q i = O must
intersection curve q of %!(a)
have a local minimum at Q,=Q),. Therefore, eigenfrequencies can be obtained at the local minimums of curve q .
In this study however, the eigenfrequencies are determined
equivalently from the local maximums of function g , ( Q )
defined by
W * ( i , I ) = W w ( x j x,,
, 52);
(29)
+
where s is a chosen constant and loyQis the scaling function
introduced for convenience. Function g , ( Q ) is called the
characteristic function and 52 is assumed to be real.
Plane-strain model
For this model the valley displacement field can be
expressed in terms of L pairs of potential sources along an
auxiliary surface S, according to Dravinski (1982b)
L
c [c,U%,
c [clW"(x,
x,, 9 )+ d/U*(x, X I ,
u ( x , Q) =
I=,
a)];
(20)
Q)l;
(21)
L
w ( x , Q) =
XI,
Q) + dlW'(x,
XI,
I=1
XED,
XIESA,
where c, and d, are the unknown coefficients. U " ( x , x , , Q)
and U*(x, x,, S2) are the Green's functions corresponding to
horizontal displacements induced by the dilatational and
shear potentials located at xI of the auxiliary surface S,
respectively, while W "(x, X I , 9 ) and W *(x, X I , Q) are the
Green's functions corresponding to vertical displacement
xi E
s,;
XI E
SA.
i = l , 2, . . . , N;
l = l , 2 , . . . , L.
Similarly to the anti-plane strain model, the frequencies at
which the characteristic function g 2 ( Q ) attains local
maximum are identified as the eigenfrequencies of the
problem. Function g,(Q) is defined as
where R, is the triangular matrix obtained by QR
decomposition of G, and l W n is a scaling function
introduced for convenience. The parameter s is chosen
constant and Q is assumed to be real.
NUMERICAL RESULTS
Resonance properties are studied by using both the
eigenvalue and spectral-search methods. The spectral-search
method requires more than one incident wave to be
considered in order to excite all the modes in the frequency
range of interest. For the anti-plane strain model, plane
harmonic incident SH waves are considered. The
off-vertical angles of incidence are chosen to be Oinc=
0", 30", 60" and 90", respectively. For the plane-strain model,
nine incident waves are considered: plane harmonic P-, SV(O,,, = 0", 30", 60" and 85") and the Rayleigh waves. The
material properties of the half-space are normalized as the
following: aO= 2, &, = 1 and p0 = 1. The total displacement
amplitude for P- and S-incident wavefields are normalized
to unity. For the Rayleigh incident waves, the amplitude of
the horizontal displacement component is chosen to be one.
For the plane-strain model, the displacement field has
both horizontal and vertical components, and the motion is
in general out of phase across the top valley S,. Therefore,
the total surface displacement amplitude of the valley is
Resonance prediction
used to represent the motion along surface S, (Zhou 1993).
The maximums of the peak-amplitude spectra correspond to
the resonant frequencies. Here the peak amplitude is
defined as the gravest total-displacement amplitude along
surface S,.
At this point, it is important to discuss the convergence
criteria for the spectral search and eigenvalue methods
based on the indirect boundary integral equation method.
For the spectral-search method the initial set of sources and
observation points is decided by the transparency test.
Namely, for a given geometry of the problem, the material
of the valley and the half-space are first assumed the same.
Deviation of the calculated response from the free-field
response is taken as a measure of the calculation error. If
the error is being judged to be unacceptable, the number of
sources and the number of collocation points are increased
until the error at each point, where the surface displacement
is calculated, is below the maximum error allowed. The
convergence test is performed next. The actual material
properties are taken and the response is calculated using the
set of sources and observation points determined by the
transparency test. The numbers of sources and observation
points are then increased and the surface response is
evaluated again. If the difference between the results
associated with the two sets of sources and observation
points is judged unacceptable, the numbers of sources and
observation points are further increased and the corresponding response is compared with the previous case.
This process is repeated until the difference between the
numerical results for the two succeeding cases is within the
tolerance required, and the latest set of sources and
observation points is taken as the final one.
For the eigenvalue method, the following convergence
criteria is used: the characteristic function is calculated using
the initial set of sources and observation points. The
numbers of sources and observation points are then
increased and the characteristic function is re-evaluated.
The location difference between the two sets of characteristic function peaks is considered as calculation error. If the
error is judged to be unacceptable, the numbers of sources
and observation points are increased until the error is below
the acceptable value. The latest set of sources and
observation points is taken as the final one.
Anti-plane strain model
A semi-circular valley is considered first in order to verify
the accuracy of the eigenfrequencies obtained through the
eigenvalue method. Fig. 5 depicts the characteristic function
g , for a semi-circular valley. In addition, the true
eigenfrequencies are plotted as well. It can be observed that
the eigenfrequencies obtained from the peaks of characteristic function are almost identical to the true eigenfrequencies (the difference is less than 1 per cent). In addition,
these values agree with the peak locations of the spectra
shown in Fig. 2. Therefore, the numerical approach appears
to be reliable in evaluation of eigenfrequencies.
To further illustrate the eigenvalue method, five types of
valley models are examined. The models are chosen in such
a way that the valley shape covers the range from perfectly
symmetric models to non-symmetric ones as shown in Fig. 6.
Type 1 valley is semi-circular, type 2 is semi-elliptical, type 3
755
'
? 6
1.5
1
2
2.5
3
3.5
Frequency
Figure 5. Characteristic function g , for a semi-circular valley and an
anti-plane strain model. 0:true eigenfrequencies.
is cosine shaped, while types 4 and 5 are non-symmetric.
The total width of the valley is normalized to 2. In this
study the shape factor r] is defined as the ratio of the
maximum depth h and average width A / h of the valley.
Namely
h
Alh
r]=-=-
h2
A
where A is cross-section area of the valley. The shape ratio
R is defined as the ratio of the maximum thickness of the
valley to the total width over which the sediment thickness
is more than half of its maximum value (Bard & Bouchon
1985). The parameters of the 22 models considered in this
study are listed in Table 1.
Figure 7 shows the characteristic function g, and the
peak-amplitude spectra for model 4(a). It can be seen that
the resonant frequencies predicted by the eigenvalue
method agree very well with the ones obtained by the
spectral-search method. The results for other models
considered have similar features and the details are omitted
(Zhou 1993). Instead, the resonant frequencies, for all the
models listed in Table 1, obtained by the eigenvalue method
and the spectral-search method are summarized by Table 2.
The results show that resonant frequencies are invariant
upon the change in impedance contrast (models l a , b, c; 2a,
b, c; and 3a, b, c). As the depth of a semi-elliptical valley
increases (0.5 for model 2a, 0.866 for 2d and 1 for model
la), the resonant frequencies of different orders also
increase. Model 4(a) (the geometry of which is a
combination of models 2a and 3a) produces the resonant
frequencies which are approximately the average of those
corresponding to models 2(a) and 3(a). Table 2 also shows
that the results obtained by the two methods agree very well
for all the models considered. To further illustrate this
agreement, the lowest two resonant frequencies of the first
kind for various valleys (see Table 1) obtained by both
methods are plotted in Fig. 8. Very good agreement for
resonant frequencies obtained by the eigenvalue method
and the spectral-search method can be observed. In
addition, the figure displays the fundamental frequencies
756
T. Zhou and M . Dravinski
0
X
X
0
(a)
type 1, semi-circle
(b) type
I"
(c)
2, semi-ellipse
I"
type 3, cosine
(d)
type 4, nonsymmetric
---a1 5
x50
X
2
= bl
+ + I1- cos
Figure 6. Geometry of the valleys for anti-plane strain model.
calculated by the Bard-Bouchon empirical relation proposed by Mossessian & Dravinski (1990). The eigenvalue
method better predicts the resonant frequencies than the
Bard-Bouchon empirical relation.
Therefore, the results presented demonstrate that the
eigenvalue method accurately predicts the resonant
frequencies for valleys of arbitrary shape.
Plane-strain model
Five cases are examined for this model. Since resonant
frequencies depend upon geometry and Poisson's ratio of
the valley, the five models incorporate different values of
these parameters. Model 1 is a cosine-shaped valley,
therefore Bard-Bouchon relations can be used to estimate
Resonance prediction
Table 1. Valley parameters for anti-plane strain model.
2 d ) ) 2
3a 11 1.5
2
1
5
20
1.5
2
4
3c
11
4a
4b112
4c
2
5a [I 2
1
5
5e
5f
1i
2
0.5
0.5
0.5
0.87
0.5
8 1 1
3 I 1
4
I
100 1
0.5
3 I 1
1
0.5
8 1 1
1
0.68
8 0.5 1.5
0.59
8 I 1 0.4 0.6 0.5 0.5
1 0.4 0.6 0.6 0.4
0.4 0.4 1.2 0.5 0.5
0.4 0.4 1.2 0.6 0.4
8 0.3 0.3 1.4 0.5 0.5
8 0.3 0.3 1.4 0.6 0.4
I
I
4
4
4
4
R ' I
0.58 0.64
0.58 0.64
0.58 0.64
0.29 0.32
0.29 0.32
0.29 0.32
0.5 0.55
0.5 0.5
0.5 0.5
0.5 0.5
0.37 0.39
0.5 0.53
0.5 0.52
0.71 0.85
0.58 0.81
1.25 1.07
0.68 0.96
1.67 1.11
0.73 0.98
R: Shape ratio according to Bard & Bouchon (1985).
v: Shape factor defined in this study.
Frequency
40
35
30
g
.B
z
da"
25
20
15
10
5
0
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
Frequency
Figure 7. Characteristic function g, (top) and the peak-amplitude
spectra (bottom) for non-symmetric valley 4(a) (see Table 1) and
anti-plane strain model. Displacement amplitude is evaluated at 15
points equally spaced along the valley free surface.
757
the resonant frequencies. Models 2-4 involve semi-circular
valley and model 5 is a non-symmetric one (see Fig. 11). In
order to assess the effect of Poisson's ratio, models 2, 3 and
4 are chosen to have different Poisson's ratio. The model
parameters are listed in Table 3.
The characteristic functions for models 4 and 5 are shown
in Fig. 9. Clear peaks can be observed for the functions. The
characteristic functions for the rest of the models have
similar features and are therefore omitted. The first four
eigenfrequencies for models 1-5, obtained by the eigenvalue
method, are summarized in Table 4. It can be observed that
the resonant frequencies vary not only with valley geometry
(models 1, 2 and 5) but also with Poisson's ratio (models 2,
3 and 4).
To gain further understanding of the valley resonance and
to verify the results obtained by the eigenvalue method, a
spectral-search method is used next.
The peak-amplitude spectra for models 2 and 4 are shown
in Fig. 10. It can be observed that for each model,
resonance takes place at the same frequencies for all the
incident waves considered. Some modes are not excited by
the vertical P of SV incidence. In addition, as the Poisson's
ratio increases, the resonant frequencies associated with
vertical P- and SV-incident waves also increase. They are
further apart as the Poisson's ratio increases. It is interesting
to see that for model 2 (Poisson's ratio v = 0.25), the lowest
resonant frequencies corresponding to the vertical P- and
SV-wave incidence are almost identical. The spectral-search
method is not able to distinguish them. Consequently, only
one resonant frequency is determined at Q i= 2. However,
for this model, the eigenvalue method produces two
resonant frequencies at Q = 2 (see Table 4) and it can be
shown that they are two very different modes. For that
purpose the eigenvalue problem for all the models was
reformulated using the finite-element approach (Zhou
1993). These calculations produced the resonant frequencies
which are very close to the one listed in Table 4. In
particular, for model 2 the resonant frequencies based on
the finite-element eigenvalue formulation are calculated to
be: S2:=2.04, S2:=2.07, Q;=2.49, S2,*=3.19. These
additional calculations confirmed the results of Table 4, and
they clearly showed that the first two resonant frequencies
for model 2 indeed correspond to two distinct modes.
Model 5 is a non-symmetric one. The valley geometry and
peak-amplitude spectra are given in Fig. 11. It can be
observed that the first mode is not excited by the vertical
P-wave incidence and the second one is not excited by the
vertical SV-wave incidence.
It should be noticed that for each of the valley models
considered in this paper, the resonant frequencies are the
same for different incident waves, although at these
frequencies the amplification varies with change of incident
wavefield. Therefore, the resonant frequencies are properties of the valley, while the incident wavefield only affects
the excitation of resonant modes. Since some modes may
not be excited by a particular incident wave, the regular
spectral-search method requires surface-response calculations for many incident waves. This is one of the main
difficulties of the spectral-search method in comparison to
the eigenvalue method.
The resonant frequencies for the four models obtained
through the spectral-search method are listed in Table 4. In
758
T. Zhou and M . Dravinski
Table 2. Resonant frequencies for anti-plane strain models
valley shape
I model 11
I
la
11
R1
R2
R3
1.51 2.39
3.28
11
11
R;
Q;
1.53 2.44
Table 3. Valley parameters for plane-strain models.
]I RSH I
3.29 11 1.53 I
[
0;
I
model
1
11
geometry
I
II
cosine
I
h I P l ” I P I b
0.5 10.5 I 1 / 3 I 1/16 I 8
~
]
I
semi-elliptical
1.19 1.59
1.38 2.14
cosine shaDe
1.41 2.07
1.43 2.11
3c
1
1.42
1.41
1.67
1.67
2.14
1.99
2.31
2.0
2.12
2.12
2.79
2.55
2.86
2.63
2.95
2.75
I[ 1.43
2.74
3.50
3.37
3.74
3.58
3.82
3.66
2.12
2.75
(1 1.41
The SV and P fundamental frequencies for model 1
calculated by Bard-Bouchon relations verify the two lowest
resonant frequencies predicted by the eigenvalue method. It
is evident that for models 2-5 the eigenvalue method in
general agrees with the spectral-search method better than
Bard-Bouchon relations. This is expected since the
Bard-Bouchon estimates are based on cosine-shaped
valleys.
The selected examples in this work show that the
eigenvalue method predicts resonant frequencies accurately.
i
1.44
1.45
1.75
1.69
2.20
1.97
2.33
2.05
2.16
2.17
2.77
2.58
2.86
2.69
2.96
2.77
2.85
2.84
3.53
3.41
3.76
3.60
3.82
3.66
1.41
1.41
1.74
1.53
2.69
1.69
3.48
1.77
Q,: the ith resonant frequency obtained through the spectral-search
method.
the ith resonant frequency obtained through the eigenvalue
method.
:,Q
,
the fundamental resonant frequency calculated by the
Bard-Bouchon relation.
Q:
EFFICIENCY O F T H E M E T H O D
Both spectral-search and eigenvalue methods used for
evaluation of resonant frequencies can be based on different
numerical approaches. The spectral-search method solves an
infinite problem involving the valley and the half-space for
different incident waves, while eigenvalue method solves a
finite problem involving only the valley without any
incidence. Hence the eigenvalue method is numerically
more efficient than the spectral-search method based on the
same numerical approach. To illustrate this point, the
efficiency of the eigenvalue methods and that of the
addition, the fundamental frequencies for models 2-5,
calculated by Bard-Bouchon relations, are also listed in
the table since these relations are based o n the
spectral-search results for cosine-shaped valleys. Table 4
clearly shows that the resonant frequencies for models 2-5,
obtained through the eigenvalue method and the spectralsearch method, agree very well.
3.5
3
a
second mode
2.5
8 ?!:
I
X
II
:
2
X
0
.
1.5
1
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
Shape factor
FQure 8. Resonant frequencies as a function of the shape factor calculated by different methods for the valleys of Fig. 6 and anti-plane strain
model. o: eigenvalue method; x: spectral search method; *: Bard-Bouchon empirical relation for the fundamental mode.
Resonance prediction
759
spectral-search method and the uppercase ones are
associated with the eigenvalue method.
If n, incident waves are considered and response is
evaluated by the spectral-search method at p locations on
the valley surface S, (see Fig. 1 ) with m inner and f outer
sources, together with n observation points along S,, the
number of flops to evaluate the surface response at each
frequency are given by (Zhou 1993):
For anti-plane strain model
nG = 2n(m
n,,=8
I
0‘
1.5
2.5
2
3
3.5
4
+ I) + lp
[( r n + f ) ’ (2 n - -m3+1)
+ ( m + f ) ( m+ I + 2n + 1 ) + f p
(33)
DimnsiMllws fraluary
(a)
and for the plane-strain model
x10‘
1
nG = 8n(m
+ 1) + 4Ip
(34)
+
x (2m
‘)+ 64(m + I )
+ 21 + 4n + 1 ) + 2Ip
1
n,.
(35)
The proposed eigenvalue method only requires N
observation points along S, and L sources along S,. At each
frequency the total numbers of Green’s function evaluations
NG and flops N F L required are then given by:
for the anti-plane strain model
2
-01.5
3
2.5
I
NG = N L
(36)
3.5
(37)
Dilmuionlusmquary
(b)
Figure 9. Characteristic functions g, for a plane-strain model. (a)
Semi-circular model (model 2 ) , N = 20, L = 10 and s = 3.5; and (b)
non-symmetric model (model 5 ) , N = 2 2 , L = 14 and s=6. For
descriptions of the models see Table 3 and Fig. 1 1 .
Table 4. Resonant frequencies for plane-strain models.
model
,,
. ,
,
- ,
I
I
1
. ,,
. ,
. ,
.
2
3
8,:
the ith resonant frequency obtained through the spectral-search
method.
Q,? the ith resonant frequency obtained through the eigenvalue
method.
8,: S V - and P-wave fundamental frequencies calculated by
the Bard-Bouchon relation.
spectral-search method, both based on the indirect
boundary integral approach (Dravinski 1982a, b), are
compared in terms of a number of Green’s function
evaluations and number of additional floating-point
operations (flops) required for solving the problem. In the
following discussion, the lower case symbols refer to the
and for plane-strain model
NG = 4NL
(38)
(
4) +
N F L =64L2 N - -
16L.
(39)
In general, the eigenvalue method has better convergence
since it requires less number of observation points N and
Green’s functions L than those needed for the spectralsearch method. For example, for model l(a) of anti-plane
strain model, the number of observation points and sources
are taken to be: n = 30, m = I = 12 and p = 16. Four plane
incident waves are considered. To calculate the surface
response at one frequency, it requires 1632 Green’s function
evaluations and approximately lo6 additional flops. Whereas
for the eigenvalue method, with N = 1 1 and L = 9 , only 99
Green’s function evaluations and 5256 additional flops are
required to evaluate the characteristic function at one
frequency.
Since evaluation of the Green’s functions for plane-strain
model requires a considerable amount of computation
(Dravinski & Mossessian 1988), reducing the number of
Green’s function evaluations is essential to improve the
numerical efficiency of the method. The eigenvalue method
requires much less Green’s function evaluations and flops,
hence it is more efficient. For example, for a semi-circular
valley (model 3), the spectral-search method requires 8792
760
T. Zhou and M . Dravinski
P wave incidence
20 I
1
Frequency
30,
20 I
I
,
SV wave inqdence
,
U
1.5
P wave incidence
I
I
1
2
2.5
Frequency
3
3.5
SV wave incidence
c
30
Frequency
Frequency
Rayleigh
wave incidence
,
Rayleigh wave incidence
I
1
I
1
2.5
Frequency
3
1
2
.
3
Frequency
4
(4
1.5
2
1
1
3.5
(b)
Figure 10. Peak-amplitude spectra for semi-circular plane-strain models 4 (a) and 2 (b) (see Table 3 ) . Displacement amplitude is evaluated at
21 points equally spaced along the valley free surface. N = 30, M = L = 12. For P- and SV-wave incidence, -: t,,,, = 0";--: t,tnc = 30"; . . . :
t,I"' = ( j o o " ; -.-: t, = g y .
I"C
Green's function evaluations and approximately 5.7 x lo7
additional flops to obtain the steady-state response at one
frequency. The eigenvalue method only requires 800
Green's function evaluations and approximately 1.1 x 10'
additional flops.
Therefore, the eigenvalue method is numerically much
more efficient than the spectral-search method for the
indirect boundary integral equation method. For other
numerical approaches, the essence of the eigenvalue method
remains the same. It solves a problem of a smaller size than
the spectral-search method based on the same numerical
approach. Hence the eigenvalue method is computationally
more efficient than the spectral-search method in determining the resonant frequencies. Use of different numerical
,
Pwavtincidence
15,
W
2
1.5
2.5
Resonance prediction
761
3
Frequency
wave incidence
SV
c
.--,
25
20 -
#
.-...
\
15-
,
;
,
I
I
V
1.5
#'
,
,
,,
I
,
'.
-
8,-
-
*- - _ _ _ _ - * *
2
2.5
3
Frequency
15 r
0'
1.5
Rayleigh wave incidence
I
1
1
I
2
2.5
3
Frequency
Figure 11. Peak-amplitude spectra for plane-strain model 5 (see Table 3). Displacement amplitude is evaluated at 21 points equally spaced
along the valley free surface. N = 34, M = L = 14. For P- and SV-wave incidence, -: Oino = 0";--: 0. = 30"; . . . . 0.In= = 60";-.-: 0. = 85".
8°C
approaches in analysing 3-D resonance are demonstrated in
the subsequent study of Zhou (1993).
CONCLUSIONS
Resonance properties for a valley of arbitrary shape
subjected to different incident SH; SV, P and Rayleigh
1°C
waves were investigated. A new eigenvalue method is
proposed to predict resonance.
In this paper the eigenvalue method is proposed to predict
the resonant frequencies of a valley for anti-plane strain and
plane-strain models. Resonant frequencies for selected
models are determined by the eigenvalue method and are
later verified by a spectral-search method or by the
762
T. Zhou and M.Dravinski
Bard-Bouchon relations. Presented results can be summarized as follows:
(1) resonant frequencies are properties of the valley,
while the incident wavefield only affects the excitation of the
resonant modes. Resonance of a valley takes place at the
same frequencies for different incident waves. Some modes
may not be excited by a n incident wave.
(2) Resonance prediction can be made through solving
appropriate eigenvalue problems. The resonant frequencies
are identical t o the corresponding eigenfrequencies. For a
symmetric valley, the symmetric and antisymmetric mode
shapes can b e used t o estimate the displacement amplitude
profile for vertical P- and SV-incident waves, respectively.
(3) Through the eigenvalue method, a general problem
involving the half-space is converted to a finite problem.
The eigenvalue method is computationally much more
efficient than the spectral-search method.
ACKNOWLEDGMENTS
The computations in this work were made possible through
a grant N00014-91-5-4163 by the San Diego Super Computer
Center at UC, San Diego and a grant INT-9021623 from the
National Science Foundation. The authors would like to
thank Keiiti Aki, Ta-Liang Teng, and Hossein Eshraghi for
their comments.
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