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Scattering of plane SH-waves by a cylindrical canyon of arbitrary shape in anisotropic media
ht. 3. Engng Sci. Vol. 30, No. 12, pp. 1773-1787, Printed in Great Britain. At1 rights reserved
$5.00 + 0.00 Km-722.5192 Copyright @ 1992 Pergamon Press Ltd
1992
SCATTERING OF PLANE Z?H-WAVES BY A CYLINDRICAL CANYON OF ARBITRARY SHAPE IN ANISOTROPIC MEDIA RAY P. S. HAN”, KAI-YUAN ‘Department
YEH’,
GUQLI
LIU2 and DIANKUI
LIU3
of Mechanical and Industrial Engineering, University of Manitoba, Winnipeg, Manitoba, Canada R3T 2N2
*Department of Mechanics, Lanzhou University, Lanzhou, Gansu 730000, People’s Republic of China ‘Institute of Engineering Mechanics, State Seismological Bureau, Harbin, People’s Republic of China Ah@raet--The scattering and diffraction of plane Sn-waves by an arbitrary-shaped cySmdricaI canyon in anisotropic media is foliated here. Analytical solutions are obtained via the complex function theory, using the orthogonal property of the Hermite functions to solve the resulting set of infinite algebraic equations. Expressions for scattered displacements and scattered stresses are given. Three cross-sectional profile types are used in the numerical simulation of the ~o~imensional canyon topography: (a) a semi-circular profile, (b) a semi-elliptical profile and (c) a triangular profile. The results obtained in (a) and (b) are consistent with known solutions computed by Trifunac and his co-workers [1,3] using a different method. As the exact solution for (c) is not known to exist, the result given here is beliived to be new and would therefore serve as a useful check for numerical analysts working in this area.
1. INTRODUCTION In earthquake
engineering, it is important to understand the role played by the surface topography on the ground ampli~~tion effects during strong ground motion. Thus, it is not surprising that the scattering and diffraction of the earthquake waves by surface and subsurface irregul&ies have been extensively investigated over the past two decades. Both analytical and numerical methods have been used by various researchers. The complexity of the problem implies that closed form analytical solutions are possible only for the simplest geometries and for linear, isotropic and homogeneous materials. Also, the incident waves are usually assumed to be the horizontally polarized shear waves @H-waves). Exact solutions for excitation by SH_waves were presented by Trifunac [l] for semi-cylindrical valleys, Trifunac [Z] for semi-circular canyons and for semi-elliptical canyons by Wong and Trifunac [3]. Quasianalytical procedures are also very popular as they can handle reasonably complex situations. The Fourier-Bessel series of Lee and his co-workers [4-‘71 and perturbation-typed techniques [&lo] are some examples of this type of approximate solutions. The most versatile methods of all, are the numerical-based procedures, such as the boundary element methods [ll-191, the finite element methods [20,21] and the finite difference techniques [22,23]. Numerical methods, however, have some serious disadvantages. They are considerably more expensive to use and they can suffer from numerical stability problems, particularly at high frequencies of excitation of the waves. Whenever feasible, analytical fo~ulations are desired over the more powerful semianalytical and numerical methods: they serve as a check over these approximate procedures. Using a method based on the complex function theory, Liu et al. [24] and Liu [25] analyzed the stress concentrations around a circular hole excited by $R-waves, and in a later paper, Liu and Han [26] extended the method to solve a semi-circular canyon in an anisotropic media. In this paper, an attempt is made to handle arbitrary-shaped canyon in an anisotropic media using an analytical formulation. The procedure involves transforming a cylindrical topography of any arbitrary shape with domain S and boundary L in the z-plane into a half space in the A-plane representation, by means of conformal mapping technique such as the Schwarz-Christoffel transformation technique. The general solution of scattering of the SH-waves in the k-plane is then expressed as the ~o~uin action. Employing the Hermite function and its orthogonality 1773
1774
R. P. S. HAN et ul.
condition leads to an infinite set of algebraic equations, from which solutions for the surface displacement amplitudes are calculated. Three types of canyon cross-sectional profiles are investigated: semi-circular, semi-ellpitical and triangular shapes.
2. FORMULATION Consider a cylindrical canyon of arbitrary shape in an anisotropic medium as depicted in Fig. 1. The propagation of the horizontally polarized shear waves is governed by the twodimensional wave equation:
where W(X, y, t) is the displacement normal to the xy-plane, t the time, p the mass density and t YZPt,,* are the shear stress components. Using Hooke’s law, the relationships between stress and strain in an anisotropic medium are given by,
qJz
=
c45
$T
+
CT44 cw
aY in which c~, cqg, cgs are elements of the 6 x 6 square matrix containing the elastic moduli of the anisotropic medium. Since the elastic matrix possesses only positive eigenvalues, the following relationship is valid: c44
>
0,
c44css
-
6s
>
0.
(41
In view of equations (2) and (3), the wave equation given by equation (1) becomes, d2W
c55
-+2c4.5&2
d2W ax ay
a2w
d2W
+C&w=P3fZ.
Introducing the following complex variables z=x+iy,
f=x-iy
(6)
-
(7)
and substituting into equation (5) yields
(--a f
Cszif
2&s)
$$+2(c44+ Css)g
To facilitate the study of the scattering of SN-waves by a cylindrical canyon of arbitrary shape, a conformal mapping function is introduced: 2 = w(A),
(h = E + io).
Fig. 1. An arbitrary-shaped cylindrical canyon in an anisotropic half space.
(8)
Scatteringof plane SH-waves
1775
Equation (7) then becomes (-CM + c55+ 2ic45) &j$
+2(c,+cs5&;
($$)
($3
- (CM- c55+ 2ic45) =$-$(&)~)=P$.
(9)
The wave equation can be expressed in a compact form by the use of these new variables: 5‘=; [(l - iy)o@)
+ (1+ iy)a@)]
f =;
+ (1+ iF)C@)]
[(l - ij++I)
(10)
where y=
2 [-c45 + ia1.
01)
Using equation (lo), the wave equation in equation (9) becomes 4
a2w _=_acal;
1 d2W
(12)
C$ at2
in which c”T= -
1
PC44
Assuming steady-state
(C&5 - cfs).
response, the displacement W(<,
5, t)
(13)
W( 5, c, t) can be represented
by
= %[w(l;, f)e-‘““I
(14)
where ~(5, 4) is a complex function of 5; and t, o* is the circular frequency of an incident wave and 9 implies the real part of the complex function. Substituting equation (14) into equation (12) leads to
where kT = co*/+. The displacement
(16)
of the scattered waves is a solution of equation
(15) which is of the form
given by [24,25]:
(17) Note that A,, is an unknown coefficient to be determined function of the first kind with argument X.
3. SCATTERED
and H:‘)(x) is the nth order Hankel
STRESSES
From equations (2) and (3), the stresses in the complex plane can be written as (18)
qz = (65 + b-d
5: +(65 -ih)z.
aw (19)
1776
R. P. S. HAN et al.
In the mapping plane (A, 1?) these stress components
More convenient That is,
expressions
become
for these stress components
can be derived using equation
+ [(-c+$ + cs5 + 2ic‘#.s)(l - ip) + (CM+ c55)(1 + iy)] $}
zin
=
f
([i(~~ +css)(i - iy)
+ (3.k - i(c, -
3
CSS))(~ + WI
(14).
(22)
5
dW w’(n) ~ + [i(cM + css)(l - iy) + (2ch5 - i(cd4 - cs5))(1 + iy)) 7
a< 1 l@‘(n)1
+ a [(2cd5 + i(cM - CSS))(~ - iy) - i(c44 + Ml r
+
ir)l~~
+ (f&s + i(c44- css))(l - iP) - i(c, + c55)(1+ iY)l~~] ~~
(23)
Introduce the following relationships,
Substitute equation (14) into equations (22) and (23), and the resulting equations are simplified using equation (24). Having obtained the scattered displacement field in equation (17), the following expressions for scattered stresses can be derived, namely:
1777
Scattering of plane SH-waves
(26) where
b = c’ = -(CACTI -
c&)
,=-d=y~.
4. REFLECTION
(27)
OF SH-WAVES
The incident plane SH-wave propagating
IN AN
ANISOTROPIC
HALF-SPACE
in the n-direction is given by
Wci)= W,exp[--i{o*t-k,(xcos
8-y
sin 8)}]
(28)
where W,, w* and 8 are the amplitude, frequency and angle of incidence respectively, and ke = d/co is the wave number. The incident wave propagates in an anisotropic medium with a velocity ce in the n-direction. This shear wave velocity can be computed by substituting equation (28) into equation (5). That is f (cU sin2 8 - 2cd5 cos 8 sin
CfJ = [
8 + c55cos2e$“.
The reflected steady-state plane H-wave by the surface of the anisotropic half-space shown in Fig. 2, propagates in the first quadrant with the displacement given by WC’) = WI exp[ -i{ w*t - ke,(x cos O1+ y sin e,)}].
(30)
Similar to the notation employed for the incident wave, WI denotes the amplitude of the reflected BY-wave, O1 is the reflected angle, and the wave number ke, = o*/c~,. Just as in equation (29), the shear wave velocity of the reflected wave can be calculated from ce, = i
(CM SirI2 &+
2C45 COS 81 Sin &+
C55 COS2 &)]
Fig. 2. Incident and reflected SH-waves.
ln.
(31)
1778
R. P. S. HAN
et al.
In the absence of the canyon, the incident wave Wci) and the reflected interfere and the total wave field in the half-space W@)would be
wave w@) would
w(t) = w”’ + w”’ This far-field motion must satisfy the traction-free half-space
(32)
boundary condition along the surface of the
( -cU sin 8 + c45cos 8)iWokoeikRx‘OS’ + (ca sin 8i + c45 cos O,)iWlk,,eik@l”‘OS‘1 = 0,
(33)
ke cos 0 = ke, cos 8,
(34)
that is and (-cM sin 8 + c45cos B)W,k, +
sin 8, +
(c44
81)Wlke, = 0.
c45 cos
(35)
From the definition of the wave numbers ke, ke,, together with the aid of equations (29) and (31), equations (34) and (35) reduce to C44 tan*
81 +
k45
and
tan
8,
-
(C44
tan*
6 -
2C45
tan
6)
=
W, ca tan 8 - c45 W,=c,tan 8, +c45’
0
(36)
(37)
Solving equations (36) and (37) produces tan&=
(
tan0-2%
c441
,
tan 822%
(38)
c44
w,/w, = 1.
(39)
Equation (39) shows that, as expected, the amplitude of the reflected wave is always equal to the amplitude of the incident wave. If 0 < tan-‘(2c,,/c,), the reflected wave will propagate in the second quadrant and can be expressed as WC’)= WI exp[-i{w*t
+ k,& cos 8, -y sin e,)}]
(40)
and the shear wave velocity can be computed from cl?, = f [
(C44 Sin*
&)]?
(41)
Proceeding in a similar fashion as given in the derivation of equations after invoking the traction-free boundary condition at the free-edge:
(38) and (39), we get
tan&=-
8,
( w,/w,=
-
2C45
tane-2%
COS
81 Sin
c44>
;
8,
+
C55 COS*
tan 8<2%
-c&tan 8 = 1. c45 c45 + cM tan 6,
c44
(43)
A graph of the reflected angle 8i versus the angle of incidence 8 for some selected anisotropic materials is illustrated in Fig. 3. This plot is obtained using equation (38) for tan 0 2 2c4JcM or Critical position is attained when the incident angle equation (42) for tan 8 < %45/c++. ecrit = tan-1(2c45/c,) and the reflected angle 8, = 0. As shown, for an isotropic material, namely cz = c45/c55 = 0 , /3 = c~/c~~ = 1.0, the reflected wave, for the prescribed range of incident angle 0“ I 8 5 90”, will always propagate in the first quadrant. This is not true in the case of an anisotropic medium. The reflected wave can propagate either in the first or second quadrant, depending on the magnitude of the incident angle. If the incident angle is smaller than Ocrit, which is material-dependent, the reflected wave will propagate in the second quadrant.
1779
Scattering of plane SH-waves
1st. Quadrant
2nd. Quadrant
Critical Angle of Incidence
C
60° 500 4o” 3o” 200
a =
c45/95
P = c,r/cs5
100 00
900
700
500
300
- 300
- 100
100
Reflected Angle
- 500
- 700
- 900
81
Fig. 3. Relationship between the incident angle 0 and the reflected angle 8, in anisotropic medium.
Otherwise, the propagation will be in the first quadrant. equation (28) can be expressed as W@ = W, exp ?
In the z-plane,
the incident wave
(z&e + ze+“)
C’
(W
I
and the reflected waves equations (30) and (40) are given respectively
by
ike, WC’)= W, exp 2 (fe’@ + ze -i&)1
(45)
and WC’) = W, exp ?
(zei% + ie-ielj].
(4)
[
In the A-plane, the incident wave equation (28) can be written as W@)= W, exp[ 2 and the reflected wave equations (30)-(40)
{o(3L)eie + oOe_“}I
(47)
are
WC’) = W, exp[ 2
{o(l)eiel
WC’) = W, exp[ 2
{o(L)e’“l + o(n)eTiel}].
+ W(A)eSiel}]
(48)
and
5. SCATTERING
OF SH-WAVES BY A CYLINDRICAL ANISOTROPIC MEDIUM
(4%
CANYON
IN AN
The scattering of plane W-wave by a cylindrical canyon of arbitrary shape in an anisotropic half-space is considered here. Close to the region around the canyon, the incident wave W”) and the reflected wave WC” are scattered and diffracted by the outer boundary of the canyon.
1780
R. P. S. HAN et 01.
Thus the total wave field WC*)is obtained from w(t) = w(i) + ~(0 + ~(9
(50) where WCs’denotes the scattered and diffracted wave by the two-dimensional arbitrary-shaped canyon, Imposing the stress-free boundary condition at y = 0, the scattered wave as given by equation (17) when expressed in the A-plane becomes
If r$i represents the scattered stress corresponding to W (‘), then the traction-free condition for the stresses, in the A-plane representation, is r$; = rz; + r’d! + r$ = 0, It is more convenient (52) as follows:
at
boundary
u =O.
(52)
and more compact to express the boundary condition given by equation
c &,A, = E PI=0
(53)
in which
(54) For tan 8 L ~c~=Jc~ [-(c,
+ c55)eiB+ i(2cd5 + i(c, - c55))e-io] ~w’(h) lo’(n)1 w’(A)
+ [i(2q5 + i(cez - c&)eiB + (cd4 + c&e-‘tt] ~
e’“@‘“[w(n)e’”+ o(A)e-‘e] b’(n)1 >
w’(A) - ke, W, [ -(cM + c55)e-iBI + i(2cd5 + i(c, - c55))eiel] { Iw’(A)I + [i(2cd5 + i(c, - c=J)e+
w’(k) eiksl’2[w(jl)e-ie1 + w(A)eiel] + (cd4 + c5s)eie1]Iw’(A)l 1
(55)
and for tan 8 < 2c4.JcU o’(A) E = -ke W, [ -cu + cs5)eie + i(2cd5 + i(ca - c=J)epi@] ~ 1 Iw’(h)l o’(A) + [i(2c45 + i(ca - cs5))eie + (cu + c&e-‘“1 ___ b’(n)1
eiko’2[w(3c)eie+ o(l2)e+‘]
w’(A) + ke, WI [-(c4 + c55)eiB1+ i(2ca5 + i(cM - css))e-iel] __ I~‘(~)1
+ [i(2cds + i(c4 - css))eieI + (cw + css)e+‘l] &%!_)e-ib,~2[w(~)ei% w(n)1
+ o(n)e-i%].
(56)
1781
Scattering of plane SH-waves
Using the fact that Hermite functions are convergent and orthogonal in the interval (-W, +w) on the O&axis, equation (53) can be expanded in terms of these functions. Note that the diplacement field satisfies the traction-free boundary condition on the surface of the two-dimensional canyon except at the convex. At this location, the convex in the A-plane representation, is mapped into the interval [-6,, a,] on the O&axis. That is,
c
k = 0, 1, 2, . . . , (n - 1)
G&, = Ek,
n=O
(57)
where 62
e,, Hek(&T)e+ de
Enk =
82
I
e Hek(QeT5’ dE (59) -61 where Hek(Q is the kth order Hermite function with ee5’ as the weighting function. Equation (57) together with equation (53) represent the set of infinite algebraic equations for the determination of the unknown coefficient A,,. &k =
6. NUMERICAL
EXAMPLES
As a numerical simulation of an arbitrary-shaped canyon, three cross-sectional profile types are analyzed: (a) a semi-circular profile, (b) a semi-elliptical profile and (c) a triangular profile. Both isotropic and anisotropic materials are employed in the model. For all three canyon types, n = 4, k = 3 are used to truncate equation (57) and to generate plots of surface displacement amplitudes versus the dimensionless distance in the cross-sectional direction x/A for incident angles 8 = 0, 30, 60, and 90“. Thus, x/A = f 1 locate the right and left edges of the canyon respectively and x/A = 0 represents the bottom of the canyon. Note that A is a characteristic dimension pertaining to the particular type of canyon. Comparison of the computed results for profile type (a) and (b) with the known exact solutions are made. No known exact result is believed to exist for profile type (c). 6.1. Scattering of plane SH-wave by a semicircular canyon Figure 4 shows a two-dimensional semi-circular canyon of unit radius (i.e. A = l), excited by a plane SH-wave of unit amplitude (i.e. W. = 1). The conformal mapping function used for the transformation is 2 = w(n) =
&-VTT, rZ+ ijKZF, rz + Vm,
AC-1 -11Asl 3c>l.
Fig. 4. Scattering of SH-wave by a semi-circular canyon.
(60)
1782
R. P. S. HAN et
41.
Cross-Sectional
Distance
z/A
Cross-Sectional
Distance
z/A
Cross-Sectional
Distance
z/A
Cross-Sectional
Distance
z/A
Fig. 5. Surface displacement amplitudes for a semi-circular canyon for an isotropic medium (cM/css = 1.0, c4s/c55 = 0).
Figure 5 shows the total displacement amplitudes 1WC*)] along the surface of the semi-circular canyon topography corresponding to the dimensionless frequency rl= k,A/n = 0.1, 0.25, 0.75 and 1.25 for an isotropic medium (cM/cS5 = 1.0, c4JcS5 = 0), and these results are in agreement with those given by Trifunac [l, 21. Figure 6 depicts similar results, but for an anisotropic model (cJcS5 = 0.8, c4JcS5 = 0.2). It can be seen from these plots that for the anisotropic model at high q, the maximum displacement amplitudes are larger than the corresponding isotropic model by lo-20%. 6.2. Scattering of plane SH-wave by a semi-elliptical canyon The model is given in Fig. 7. As before, the incident wave is a plane SH-wave of unit amplitude (i.e. W, = 1). The conformal mapping function used for the transformation is &-jlFTS+
2 = o(A)=
1-E’ 2
d-7
1
-
d-7
where A, B are the semi-major
A+iVm+
amplitudes
(61)
E’R
and semi-minor axis of the ellipse respectively E = BJA,
The total surface displacement
-Rs.$‘R
A+iVFZ+
E2
2
Zf<--R
R=A
and
l+E d 1-E’ -
]W(“( of the semi-elliptical
canyon
topography
Scattering
plane
-waves
Cross-Sectional Distance c/A
Cross-Se.ctiond Distance
z/A
Cross-Sectional Distance
z/A
Cross-Sectional Distance
z/A
6. Surface displacement amplitudes for a semi-circular canyon for an anisotropic medium (Q.& c,5/c*s = 0.2).
= 0.8,
corresponding to rl= 0.1, 0.5, 1.0 and 1.5 for an isotropic medium (c4/cs5 = 1.0, c&c55 = 0) are given in Fig. 8. These results are consistent with those obtained by Wong and Trifunac [3] using Mathieu function. Similar plots based on an anisotropic medium (ceq/cs5 = 0.8, cJc~~ =0.2) are illustrated in Fig. 9. It can be seen that the maximum anisotropic displacement amplitudes aretlarger than the corresponding isotropic quantities by approx. 10% in the high rl range. 6.3. Scattering of plane SH-wave by a triangular canyon A triangular canyon of dimension A = 1 and excited by a plane SH-wave of unit amplitude is considered here. The model is illustrated in Fig. 10. As far as we know, this problem has not been solved analytically. The conformal mapping function used for the transformation is 2A
\
w(i) L4
0 I
Y Fig. 7. Scattering of SH-wave by a semi-elliptical canyon.
z
gs
2.0
3.0
4.0
Displacement Amplitude
1.0
Displacement Amplitude
lW(‘fl
1
ELC
\W(*)j
WI
c:
1.0
2.0
3.0
4.0
Displacement Amplitude
Displacement Ampiitude
5.0
IW(‘)j
5.C
/IV(‘)] Amplitude
Displacement Amplitude
Displacement
iW(‘)l
II@‘)1
Displacement Amplitude
Displacement Amplitude
jW(f)/
1W(‘f 1
-‘r 1
1785
Scattering of plane SH-waves
_Z
60°
B
’
H
W(‘)
e
8
Fig. 10. Scattering of SH-wave by a triangular canyon.
obtained via the Schwarz-Christoffel
transformation,
namely
T,(a)
E>l,
w&=1’ z = o(n) =
l-iH T,(Q(,=,
T,()c),
EC-1
(62)
O
l+iH
-l
T&&z,
‘@)’
11 31
1.211 3.63h3
where 11 31
T,(a)=a+--+--+---+---
u4 = -3nl13+S.,2~413+~~~l~3+.
1.2.511 3.6.95AS+“’ . . .
and H is the depth of the canyon. Figure 11 shows the total surface displacement amplitudes IW(‘)l of this triangular canyon corresponding to rl= 0.1, 0.25, 0.75 and 1.25 for an isotropic medium (c,.,/c~~ = 1.0, c~/c~~ = 0), whereas in Fig. 12, the results pertain to an anisotropic half-space (c.,.Jcs~ = 0.8, c4Jcss = 0.2). As shown, the maximum anisotropic displacement amplitudes are larger than the corresponding isotropic quantities by -15% to 70% at the high 7 values. For all three canyon profiles, the displacement amplitudes have a similar pattern. In an isotropic medium, when the incident SH-waves are vertical, that is (6 = 90”), the displacement amplitudes are symmetric with respect to x/A = 0 axis. This symmetry is not observed for an anisotropic medium, which should be the case. As 8 approaches zero, due to interference between the incident and scattered waves, the motion becomes more complex for x/A < 0 side (2nd quadrant) than the x/A > 0 side (1st quadrant).
7.
SUMMARY
AND
CONCLUSIONS
In this paper, the method of complex function theory is employed to analyze the steady-state scattering of plane SH-waves by a two-dimensional cylindrical canyon of arbitrary shape in anisotropic media. Computational results of surface displacement amplitudes for a cylindrical canyon with three types of cross-sections: a semi-circular profile, a semi-elliptical profile and a triangular profile, are presented. Based on these solutions, the following comments apply.
1786
R. P. S. HAN
et al.
0
LA
------. dy
T)=oolo
0
YEI _g
---_.
4
sl tit
~=“__._”
-3
-2
-1
1
0
Cross-Sectional
Distance
i
2
3
z/A
Cross-Sectional
I
Fig. 11. Surface
Distance
displacement
for a triangular
canyon
for au isotropic
5
I -3
-2
-1
Cross-Sectional Fig.
12. Surface
displacement
1
Distance
,
-3
-2
1
2
3
I -3
-2
z/A
amplitudes
0
1
medium
(cJcss
-1
0
-1
Cross-Sectional for
a
triangular canyon c&55 = 0.2).
for
an
anisotropic
2
Distance
Cross-Sectional
z/A
Distance
0
-1
Cross-Sectional
% Cross-Sectional
-2
z/A
amplitudes
z/A
,
-3
Cross-Sectional
Distance
z/A
= 1.0, cd5/cs5 = 0).
1
Distance
0
medium
3
2
3
z/A
1
Distance
2
I
z/A (c++/c~~= 0.8,
1787
Scattering of plane SH-waves
(a) To use this procedure, it is necessary to obtain the conformal mapping function defined by equation (8). It is convenient to use the Schwarz-Christoffel transformation technique to arrive at the required conformal mapping function, as was demonstrated for the case of the triangular canyon. (b) As shown, results converged when only n = 4, k = 3 number of terms are used in the truncation of the infinite set of algebraic equations. (c) From the various plots, it appears that generally, anisotropy produces larger surface displacement amplitudes in the canyon than isotropic materials, particularly at high ‘I. (d) Not surprisingly, for the vertical incidence of H-waves, symmetry of the displacement amplitudes with respect to the x/A = 0 axis is destroyed in an anisotropic medium compared to an isotropic medium. Acknowledgements-Financial support from the University of Manitoba is gratefully acknowledged. like to convey our thanks to the reviewer for his comments.
Also we would
REFERENCES M. D. TRIFUNAC, Bull. Seirm. Sot. Am. 61,175s (1971). M. D. TRIFUNAC, ht. J. Earthquuke Engng Struct. Dyn. 1, 267 (1973). H. L. WONG and M. D. TRIFUNAC, Znt. J. Earthquake Engng Struct. Dyn. 3, 157 (1974). H. CA0 and V. W. LEE, Eur. J. Eurthquake Engng 3,29 (1989). H. CA0 and V. W. LEE, Int. J. Soil Dyn. Earthquuke Engng 9, 141 (1990). V. W. LEE and H. CAO, J. Engng Mech. 115,2035 (1989). M. I. TODOROVSKA and V. W. LEE, ht. J. Soil Dyn. Earthquake Engng 10, 192 (1991). I. McIVOR, Bull. Seism. Sot. Am. 59, 1359 (1%9). F. J. SABINA and J. R. WILLIS, Geophys. J. 42, 685 (1975). K. WATANABE, Wave Motion 6,477 (1984). H. S. WONG and P. C. JENNINGS, Bull. Seism. Sot. Am. 65, 1239 (1975). F. J. SANCHEZ-SESMA and J. A. ESQUIVEL, Bull. Seism. Sot. Am. 69, 1107 (1979). H. L. WONG, Bull. Seism. Sot. Am. 72, 1167 (1982). M. DRAVINSKI, J. Engng Mech. 108,1 (1982). F. J. SANCHEZ-SESMA, M. A. BRAVO and I. HERRERA, Bull. Seism. Sot. Am. 75,297 (1985). Y. NIWA and S. HIROSE, Proc. JSCE Struct. EngnglEarthquake Engng 2 (1985). N. MOEEN-VAZIRI and M. D. TRIFUNAC, Int. J. Soil Dyn. Eurthquoke Engng 4, 18 (1985). N. MOEEN-VAZIRI and M. D. TRIFUNAC, Int. J. Soil Dyn. Earthquake Engng 4,179 (1988). H. KAWASE, Bull. Seism. Sot. Am. 78, 1415 (1988). J. LYSMER and G. WAAS, J. Engng Mech. 98,85 (1972). M. N. A. EILOUGH and R. S. SANDHU, Int. J. Soil Dyn. Earthquake Engng 6,70 (1987). Z. ALTERMAN and F. C. KARAL, Bull. Seism. Sot. Am. 58,367 (1968). D. M. BOORE, 1. Geophys Rev. 75, 1512 (1970). D. LIU, B. GA1 and G. TAO, Wave Motion 4,293 (1982). D. LIU, Actu Mech. Sinica 4, 146 (1988). D. LIU and F. HAN, Earthquake Engng Engng Vibr. 10, 11 (1990) (in Chinese). (Received 23 December 1991; accepted 13 February 1992)
ES 30:12-H
$5.00 + 0.00 Km-722.5192 Copyright @ 1992 Pergamon Press Ltd
1992
SCATTERING OF PLANE Z?H-WAVES BY A CYLINDRICAL CANYON OF ARBITRARY SHAPE IN ANISOTROPIC MEDIA RAY P. S. HAN”, KAI-YUAN ‘Department
YEH’,
GUQLI
LIU2 and DIANKUI
LIU3
of Mechanical and Industrial Engineering, University of Manitoba, Winnipeg, Manitoba, Canada R3T 2N2
*Department of Mechanics, Lanzhou University, Lanzhou, Gansu 730000, People’s Republic of China ‘Institute of Engineering Mechanics, State Seismological Bureau, Harbin, People’s Republic of China Ah@raet--The scattering and diffraction of plane Sn-waves by an arbitrary-shaped cySmdricaI canyon in anisotropic media is foliated here. Analytical solutions are obtained via the complex function theory, using the orthogonal property of the Hermite functions to solve the resulting set of infinite algebraic equations. Expressions for scattered displacements and scattered stresses are given. Three cross-sectional profile types are used in the numerical simulation of the ~o~imensional canyon topography: (a) a semi-circular profile, (b) a semi-elliptical profile and (c) a triangular profile. The results obtained in (a) and (b) are consistent with known solutions computed by Trifunac and his co-workers [1,3] using a different method. As the exact solution for (c) is not known to exist, the result given here is beliived to be new and would therefore serve as a useful check for numerical analysts working in this area.
1. INTRODUCTION In earthquake
engineering, it is important to understand the role played by the surface topography on the ground ampli~~tion effects during strong ground motion. Thus, it is not surprising that the scattering and diffraction of the earthquake waves by surface and subsurface irregul&ies have been extensively investigated over the past two decades. Both analytical and numerical methods have been used by various researchers. The complexity of the problem implies that closed form analytical solutions are possible only for the simplest geometries and for linear, isotropic and homogeneous materials. Also, the incident waves are usually assumed to be the horizontally polarized shear waves @H-waves). Exact solutions for excitation by SH_waves were presented by Trifunac [l] for semi-cylindrical valleys, Trifunac [Z] for semi-circular canyons and for semi-elliptical canyons by Wong and Trifunac [3]. Quasianalytical procedures are also very popular as they can handle reasonably complex situations. The Fourier-Bessel series of Lee and his co-workers [4-‘71 and perturbation-typed techniques [&lo] are some examples of this type of approximate solutions. The most versatile methods of all, are the numerical-based procedures, such as the boundary element methods [ll-191, the finite element methods [20,21] and the finite difference techniques [22,23]. Numerical methods, however, have some serious disadvantages. They are considerably more expensive to use and they can suffer from numerical stability problems, particularly at high frequencies of excitation of the waves. Whenever feasible, analytical fo~ulations are desired over the more powerful semianalytical and numerical methods: they serve as a check over these approximate procedures. Using a method based on the complex function theory, Liu et al. [24] and Liu [25] analyzed the stress concentrations around a circular hole excited by $R-waves, and in a later paper, Liu and Han [26] extended the method to solve a semi-circular canyon in an anisotropic media. In this paper, an attempt is made to handle arbitrary-shaped canyon in an anisotropic media using an analytical formulation. The procedure involves transforming a cylindrical topography of any arbitrary shape with domain S and boundary L in the z-plane into a half space in the A-plane representation, by means of conformal mapping technique such as the Schwarz-Christoffel transformation technique. The general solution of scattering of the SH-waves in the k-plane is then expressed as the ~o~uin action. Employing the Hermite function and its orthogonality 1773
1774
R. P. S. HAN et ul.
condition leads to an infinite set of algebraic equations, from which solutions for the surface displacement amplitudes are calculated. Three types of canyon cross-sectional profiles are investigated: semi-circular, semi-ellpitical and triangular shapes.
2. FORMULATION Consider a cylindrical canyon of arbitrary shape in an anisotropic medium as depicted in Fig. 1. The propagation of the horizontally polarized shear waves is governed by the twodimensional wave equation:
where W(X, y, t) is the displacement normal to the xy-plane, t the time, p the mass density and t YZPt,,* are the shear stress components. Using Hooke’s law, the relationships between stress and strain in an anisotropic medium are given by,
qJz
=
c45
$T
+
CT44 cw
aY in which c~, cqg, cgs are elements of the 6 x 6 square matrix containing the elastic moduli of the anisotropic medium. Since the elastic matrix possesses only positive eigenvalues, the following relationship is valid: c44
>
0,
c44css
-
6s
>
0.
(41
In view of equations (2) and (3), the wave equation given by equation (1) becomes, d2W
c55
-+2c4.5&2
d2W ax ay
a2w
d2W
+C&w=P3fZ.
Introducing the following complex variables z=x+iy,
f=x-iy
(6)
-
(7)
and substituting into equation (5) yields
(--a f
Cszif
2&s)
$$+2(c44+ Css)g
To facilitate the study of the scattering of SN-waves by a cylindrical canyon of arbitrary shape, a conformal mapping function is introduced: 2 = w(A),
(h = E + io).
Fig. 1. An arbitrary-shaped cylindrical canyon in an anisotropic half space.
(8)
Scatteringof plane SH-waves
1775
Equation (7) then becomes (-CM + c55+ 2ic45) &j$
+2(c,+cs5&;
($$)
($3
- (CM- c55+ 2ic45) =$-$(&)~)=P$.
(9)
The wave equation can be expressed in a compact form by the use of these new variables: 5‘=; [(l - iy)o@)
+ (1+ iy)a@)]
f =;
+ (1+ iF)C@)]
[(l - ij++I)
(10)
where y=
2 [-c45 + ia1.
01)
Using equation (lo), the wave equation in equation (9) becomes 4
a2w _=_acal;
1 d2W
(12)
C$ at2
in which c”T= -
1
PC44
Assuming steady-state
(C&5 - cfs).
response, the displacement W(<,
5, t)
(13)
W( 5, c, t) can be represented
by
= %[w(l;, f)e-‘““I
(14)
where ~(5, 4) is a complex function of 5; and t, o* is the circular frequency of an incident wave and 9 implies the real part of the complex function. Substituting equation (14) into equation (12) leads to
where kT = co*/+. The displacement
(16)
of the scattered waves is a solution of equation
(15) which is of the form
given by [24,25]:
(17) Note that A,, is an unknown coefficient to be determined function of the first kind with argument X.
3. SCATTERED
and H:‘)(x) is the nth order Hankel
STRESSES
From equations (2) and (3), the stresses in the complex plane can be written as (18)
qz = (65 + b-d
5: +(65 -ih)z.
aw (19)
1776
R. P. S. HAN et al.
In the mapping plane (A, 1?) these stress components
More convenient That is,
expressions
become
for these stress components
can be derived using equation
+ [(-c+$ + cs5 + 2ic‘#.s)(l - ip) + (CM+ c55)(1 + iy)] $}
zin
=
f
([i(~~ +css)(i - iy)
+ (3.k - i(c, -
3
CSS))(~ + WI
(14).
(22)
5
dW w’(n) ~ + [i(cM + css)(l - iy) + (2ch5 - i(cd4 - cs5))(1 + iy)) 7
a< 1 l@‘(n)1
+ a [(2cd5 + i(cM - CSS))(~ - iy) - i(c44 + Ml r
+
ir)l~~
+ (f&s + i(c44- css))(l - iP) - i(c, + c55)(1+ iY)l~~] ~~
(23)
Introduce the following relationships,
Substitute equation (14) into equations (22) and (23), and the resulting equations are simplified using equation (24). Having obtained the scattered displacement field in equation (17), the following expressions for scattered stresses can be derived, namely:
1777
Scattering of plane SH-waves
(26) where
b = c’ = -(CACTI -
c&)
,=-d=y~.
4. REFLECTION
(27)
OF SH-WAVES
The incident plane SH-wave propagating
IN AN
ANISOTROPIC
HALF-SPACE
in the n-direction is given by
Wci)= W,exp[--i{o*t-k,(xcos
8-y
sin 8)}]
(28)
where W,, w* and 8 are the amplitude, frequency and angle of incidence respectively, and ke = d/co is the wave number. The incident wave propagates in an anisotropic medium with a velocity ce in the n-direction. This shear wave velocity can be computed by substituting equation (28) into equation (5). That is f (cU sin2 8 - 2cd5 cos 8 sin
CfJ = [
8 + c55cos2e$“.
The reflected steady-state plane H-wave by the surface of the anisotropic half-space shown in Fig. 2, propagates in the first quadrant with the displacement given by WC’) = WI exp[ -i{ w*t - ke,(x cos O1+ y sin e,)}].
(30)
Similar to the notation employed for the incident wave, WI denotes the amplitude of the reflected BY-wave, O1 is the reflected angle, and the wave number ke, = o*/c~,. Just as in equation (29), the shear wave velocity of the reflected wave can be calculated from ce, = i
(CM SirI2 &+
2C45 COS 81 Sin &+
C55 COS2 &)]
Fig. 2. Incident and reflected SH-waves.
ln.
(31)
1778
R. P. S. HAN
et al.
In the absence of the canyon, the incident wave Wci) and the reflected interfere and the total wave field in the half-space W@)would be
wave w@) would
w(t) = w”’ + w”’ This far-field motion must satisfy the traction-free half-space
(32)
boundary condition along the surface of the
( -cU sin 8 + c45cos 8)iWokoeikRx‘OS’ + (ca sin 8i + c45 cos O,)iWlk,,eik@l”‘OS‘1 = 0,
(33)
ke cos 0 = ke, cos 8,
(34)
that is and (-cM sin 8 + c45cos B)W,k, +
sin 8, +
(c44
81)Wlke, = 0.
c45 cos
(35)
From the definition of the wave numbers ke, ke,, together with the aid of equations (29) and (31), equations (34) and (35) reduce to C44 tan*
81 +
k45
and
tan
8,
-
(C44
tan*
6 -
2C45
tan
6)
=
W, ca tan 8 - c45 W,=c,tan 8, +c45’
0
(36)
(37)
Solving equations (36) and (37) produces tan&=
(
tan0-2%
c441
,
tan 822%
(38)
c44
w,/w, = 1.
(39)
Equation (39) shows that, as expected, the amplitude of the reflected wave is always equal to the amplitude of the incident wave. If 0 < tan-‘(2c,,/c,), the reflected wave will propagate in the second quadrant and can be expressed as WC’)= WI exp[-i{w*t
+ k,& cos 8, -y sin e,)}]
(40)
and the shear wave velocity can be computed from cl?, = f [
(C44 Sin*
&)]?
(41)
Proceeding in a similar fashion as given in the derivation of equations after invoking the traction-free boundary condition at the free-edge:
(38) and (39), we get
tan&=-
8,
( w,/w,=
-
2C45
tane-2%
COS
81 Sin
c44>
;
8,
+
C55 COS*
tan 8<2%
-c&tan 8 = 1. c45 c45 + cM tan 6,
c44
(43)
A graph of the reflected angle 8i versus the angle of incidence 8 for some selected anisotropic materials is illustrated in Fig. 3. This plot is obtained using equation (38) for tan 0 2 2c4JcM or Critical position is attained when the incident angle equation (42) for tan 8 < %45/c++. ecrit = tan-1(2c45/c,) and the reflected angle 8, = 0. As shown, for an isotropic material, namely cz = c45/c55 = 0 , /3 = c~/c~~ = 1.0, the reflected wave, for the prescribed range of incident angle 0“ I 8 5 90”, will always propagate in the first quadrant. This is not true in the case of an anisotropic medium. The reflected wave can propagate either in the first or second quadrant, depending on the magnitude of the incident angle. If the incident angle is smaller than Ocrit, which is material-dependent, the reflected wave will propagate in the second quadrant.
1779
Scattering of plane SH-waves
1st. Quadrant
2nd. Quadrant
Critical Angle of Incidence
C
60° 500 4o” 3o” 200
a =
c45/95
P = c,r/cs5
100 00
900
700
500
300
- 300
- 100
100
Reflected Angle
- 500
- 700
- 900
81
Fig. 3. Relationship between the incident angle 0 and the reflected angle 8, in anisotropic medium.
Otherwise, the propagation will be in the first quadrant. equation (28) can be expressed as W@ = W, exp ?
In the z-plane,
the incident wave
(z&e + ze+“)
C’
(W
I
and the reflected waves equations (30) and (40) are given respectively
by
ike, WC’)= W, exp 2 (fe’@ + ze -i&)1
(45)
and WC’) = W, exp ?
(zei% + ie-ielj].
(4)
[
In the A-plane, the incident wave equation (28) can be written as W@)= W, exp[ 2 and the reflected wave equations (30)-(40)
{o(3L)eie + oOe_“}I
(47)
are
WC’) = W, exp[ 2
{o(l)eiel
WC’) = W, exp[ 2
{o(L)e’“l + o(n)eTiel}].
+ W(A)eSiel}]
(48)
and
5. SCATTERING
OF SH-WAVES BY A CYLINDRICAL ANISOTROPIC MEDIUM
(4%
CANYON
IN AN
The scattering of plane W-wave by a cylindrical canyon of arbitrary shape in an anisotropic half-space is considered here. Close to the region around the canyon, the incident wave W”) and the reflected wave WC” are scattered and diffracted by the outer boundary of the canyon.
1780
R. P. S. HAN et 01.
Thus the total wave field WC*)is obtained from w(t) = w(i) + ~(0 + ~(9
(50) where WCs’denotes the scattered and diffracted wave by the two-dimensional arbitrary-shaped canyon, Imposing the stress-free boundary condition at y = 0, the scattered wave as given by equation (17) when expressed in the A-plane becomes
If r$i represents the scattered stress corresponding to W (‘), then the traction-free condition for the stresses, in the A-plane representation, is r$; = rz; + r’d! + r$ = 0, It is more convenient (52) as follows:
at
boundary
u =O.
(52)
and more compact to express the boundary condition given by equation
c &,A, = E PI=0
(53)
in which
(54) For tan 8 L ~c~=Jc~ [-(c,
+ c55)eiB+ i(2cd5 + i(c, - c55))e-io] ~w’(h) lo’(n)1 w’(A)
+ [i(2q5 + i(cez - c&)eiB + (cd4 + c&e-‘tt] ~
e’“@‘“[w(n)e’”+ o(A)e-‘e] b’(n)1 >
w’(A) - ke, W, [ -(cM + c55)e-iBI + i(2cd5 + i(c, - c55))eiel] { Iw’(A)I + [i(2cd5 + i(c, - c=J)e+
w’(k) eiksl’2[w(jl)e-ie1 + w(A)eiel] + (cd4 + c5s)eie1]Iw’(A)l 1
(55)
and for tan 8 < 2c4.JcU o’(A) E = -ke W, [ -cu + cs5)eie + i(2cd5 + i(ca - c=J)epi@] ~ 1 Iw’(h)l o’(A) + [i(2c45 + i(ca - cs5))eie + (cu + c&e-‘“1 ___ b’(n)1
eiko’2[w(3c)eie+ o(l2)e+‘]
w’(A) + ke, WI [-(c4 + c55)eiB1+ i(2ca5 + i(cM - css))e-iel] __ I~‘(~)1
+ [i(2cds + i(c4 - css))eieI + (cw + css)e+‘l] &%!_)e-ib,~2[w(~)ei% w(n)1
+ o(n)e-i%].
(56)
1781
Scattering of plane SH-waves
Using the fact that Hermite functions are convergent and orthogonal in the interval (-W, +w) on the O&axis, equation (53) can be expanded in terms of these functions. Note that the diplacement field satisfies the traction-free boundary condition on the surface of the two-dimensional canyon except at the convex. At this location, the convex in the A-plane representation, is mapped into the interval [-6,, a,] on the O&axis. That is,
c
k = 0, 1, 2, . . . , (n - 1)
G&, = Ek,
n=O
(57)
where 62
e,, Hek(&T)e+ de
Enk =
82
I
e Hek(QeT5’ dE (59) -61 where Hek(Q is the kth order Hermite function with ee5’ as the weighting function. Equation (57) together with equation (53) represent the set of infinite algebraic equations for the determination of the unknown coefficient A,,. &k =
6. NUMERICAL
EXAMPLES
As a numerical simulation of an arbitrary-shaped canyon, three cross-sectional profile types are analyzed: (a) a semi-circular profile, (b) a semi-elliptical profile and (c) a triangular profile. Both isotropic and anisotropic materials are employed in the model. For all three canyon types, n = 4, k = 3 are used to truncate equation (57) and to generate plots of surface displacement amplitudes versus the dimensionless distance in the cross-sectional direction x/A for incident angles 8 = 0, 30, 60, and 90“. Thus, x/A = f 1 locate the right and left edges of the canyon respectively and x/A = 0 represents the bottom of the canyon. Note that A is a characteristic dimension pertaining to the particular type of canyon. Comparison of the computed results for profile type (a) and (b) with the known exact solutions are made. No known exact result is believed to exist for profile type (c). 6.1. Scattering of plane SH-wave by a semicircular canyon Figure 4 shows a two-dimensional semi-circular canyon of unit radius (i.e. A = l), excited by a plane SH-wave of unit amplitude (i.e. W. = 1). The conformal mapping function used for the transformation is 2 = w(n) =
&-VTT, rZ+ ijKZF, rz + Vm,
AC-1 -11Asl 3c>l.
Fig. 4. Scattering of SH-wave by a semi-circular canyon.
(60)
1782
R. P. S. HAN et
41.
Cross-Sectional
Distance
z/A
Cross-Sectional
Distance
z/A
Cross-Sectional
Distance
z/A
Cross-Sectional
Distance
z/A
Fig. 5. Surface displacement amplitudes for a semi-circular canyon for an isotropic medium (cM/css = 1.0, c4s/c55 = 0).
Figure 5 shows the total displacement amplitudes 1WC*)] along the surface of the semi-circular canyon topography corresponding to the dimensionless frequency rl= k,A/n = 0.1, 0.25, 0.75 and 1.25 for an isotropic medium (cM/cS5 = 1.0, c4JcS5 = 0), and these results are in agreement with those given by Trifunac [l, 21. Figure 6 depicts similar results, but for an anisotropic model (cJcS5 = 0.8, c4JcS5 = 0.2). It can be seen from these plots that for the anisotropic model at high q, the maximum displacement amplitudes are larger than the corresponding isotropic model by lo-20%. 6.2. Scattering of plane SH-wave by a semi-elliptical canyon The model is given in Fig. 7. As before, the incident wave is a plane SH-wave of unit amplitude (i.e. W, = 1). The conformal mapping function used for the transformation is &-jlFTS+
2 = o(A)=
1-E’ 2
d-7
1
-
d-7
where A, B are the semi-major
A+iVm+
amplitudes
(61)
E’R
and semi-minor axis of the ellipse respectively E = BJA,
The total surface displacement
-Rs.$‘R
A+iVFZ+
E2
2
Zf<--R
R=A
and
l+E d 1-E’ -
]W(“( of the semi-elliptical
canyon
topography
Scattering
plane
-waves
Cross-Sectional Distance c/A
Cross-Se.ctiond Distance
z/A
Cross-Sectional Distance
z/A
Cross-Sectional Distance
z/A
6. Surface displacement amplitudes for a semi-circular canyon for an anisotropic medium (Q.& c,5/c*s = 0.2).
= 0.8,
corresponding to rl= 0.1, 0.5, 1.0 and 1.5 for an isotropic medium (c4/cs5 = 1.0, c&c55 = 0) are given in Fig. 8. These results are consistent with those obtained by Wong and Trifunac [3] using Mathieu function. Similar plots based on an anisotropic medium (ceq/cs5 = 0.8, cJc~~ =0.2) are illustrated in Fig. 9. It can be seen that the maximum anisotropic displacement amplitudes aretlarger than the corresponding isotropic quantities by approx. 10% in the high rl range. 6.3. Scattering of plane SH-wave by a triangular canyon A triangular canyon of dimension A = 1 and excited by a plane SH-wave of unit amplitude is considered here. The model is illustrated in Fig. 10. As far as we know, this problem has not been solved analytically. The conformal mapping function used for the transformation is 2A
\
w(i) L4
0 I
Y Fig. 7. Scattering of SH-wave by a semi-elliptical canyon.
z
gs
2.0
3.0
4.0
Displacement Amplitude
1.0
Displacement Amplitude
lW(‘fl
1
ELC
\W(*)j
WI
c:
1.0
2.0
3.0
4.0
Displacement Amplitude
Displacement Ampiitude
5.0
IW(‘)j
5.C
/IV(‘)] Amplitude
Displacement Amplitude
Displacement
iW(‘)l
II@‘)1
Displacement Amplitude
Displacement Amplitude
jW(f)/
1W(‘f 1
-‘r 1
1785
Scattering of plane SH-waves
_Z
60°
B
’
H
W(‘)
e
8
Fig. 10. Scattering of SH-wave by a triangular canyon.
obtained via the Schwarz-Christoffel
transformation,
namely
T,(a)
E>l,
w&=1’ z = o(n) =
l-iH T,(Q(,=,
T,()c),
EC-1
(62)
O
l+iH
-l
T&&z,
‘@)’
11 31
1.211 3.63h3
where 11 31
T,(a)=a+--+--+---+---
u4 = -3nl13+S.,2~413+~~~l~3+.
1.2.511 3.6.95AS+“’ . . .
and H is the depth of the canyon. Figure 11 shows the total surface displacement amplitudes IW(‘)l of this triangular canyon corresponding to rl= 0.1, 0.25, 0.75 and 1.25 for an isotropic medium (c,.,/c~~ = 1.0, c~/c~~ = 0), whereas in Fig. 12, the results pertain to an anisotropic half-space (c.,.Jcs~ = 0.8, c4Jcss = 0.2). As shown, the maximum anisotropic displacement amplitudes are larger than the corresponding isotropic quantities by -15% to 70% at the high 7 values. For all three canyon profiles, the displacement amplitudes have a similar pattern. In an isotropic medium, when the incident SH-waves are vertical, that is (6 = 90”), the displacement amplitudes are symmetric with respect to x/A = 0 axis. This symmetry is not observed for an anisotropic medium, which should be the case. As 8 approaches zero, due to interference between the incident and scattered waves, the motion becomes more complex for x/A < 0 side (2nd quadrant) than the x/A > 0 side (1st quadrant).
7.
SUMMARY
AND
CONCLUSIONS
In this paper, the method of complex function theory is employed to analyze the steady-state scattering of plane SH-waves by a two-dimensional cylindrical canyon of arbitrary shape in anisotropic media. Computational results of surface displacement amplitudes for a cylindrical canyon with three types of cross-sections: a semi-circular profile, a semi-elliptical profile and a triangular profile, are presented. Based on these solutions, the following comments apply.
1786
R. P. S. HAN
et al.
0
LA
------. dy
T)=oolo
0
YEI _g
---_.
4
sl tit
~=“__._”
-3
-2
-1
1
0
Cross-Sectional
Distance
i
2
3
z/A
Cross-Sectional
I
Fig. 11. Surface
Distance
displacement
for a triangular
canyon
for au isotropic
5
I -3
-2
-1
Cross-Sectional Fig.
12. Surface
displacement
1
Distance
,
-3
-2
1
2
3
I -3
-2
z/A
amplitudes
0
1
medium
(cJcss
-1
0
-1
Cross-Sectional for
a
triangular canyon c&55 = 0.2).
for
an
anisotropic
2
Distance
Cross-Sectional
z/A
Distance
0
-1
Cross-Sectional
% Cross-Sectional
-2
z/A
amplitudes
z/A
,
-3
Cross-Sectional
Distance
z/A
= 1.0, cd5/cs5 = 0).
1
Distance
0
medium
3
2
3
z/A
1
Distance
2
I
z/A (c++/c~~= 0.8,
1787
Scattering of plane SH-waves
(a) To use this procedure, it is necessary to obtain the conformal mapping function defined by equation (8). It is convenient to use the Schwarz-Christoffel transformation technique to arrive at the required conformal mapping function, as was demonstrated for the case of the triangular canyon. (b) As shown, results converged when only n = 4, k = 3 number of terms are used in the truncation of the infinite set of algebraic equations. (c) From the various plots, it appears that generally, anisotropy produces larger surface displacement amplitudes in the canyon than isotropic materials, particularly at high ‘I. (d) Not surprisingly, for the vertical incidence of H-waves, symmetry of the displacement amplitudes with respect to the x/A = 0 axis is destroyed in an anisotropic medium compared to an isotropic medium. Acknowledgements-Financial support from the University of Manitoba is gratefully acknowledged. like to convey our thanks to the reviewer for his comments.
Also we would
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