Amplification of waves by an orthotropic basin: sagittal plane motion

EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICS

Earthquake Engng. Struct. Dyn. 28, 565–584 (1999)

AMPLIFICATION OF WAVES BY AN ORTHOTROPICBASIN: SAGITTAL PLANE MOTION

TAO ZHENG AND MARIJAN DRAVINSKI ∗

Department of Mechanical Engineering; University of Southern California; Los Angeles; CA 90089-1453; U.S.A.

SUMMARY

Scattering of elastic waves by an orthotropic basin of arbitrary shape embedded in a half-space is investigatedfor the sagittal plane motion using an indirect boundary integral equation approach. Steady-state results wereobtained for incident plane harmonic pseudo P-, S-, and Rayleigh waves. Detailed convergence analysis of themethod is presented. Green’s functions are evaluated by using adaptive Newton–Cotes or Filon quadratures.Surface ground motion is presented for semicircular and semielliptical basins with di�erent material propertiesand various angles of incidence. The results show that surface motion strongly depends upon nature of incidentwave, geometry and material properties of the basin, and location of the observation point. Comparison withisotropic basin response demonstrates that anisotropy is very important in ampli�cation of surface groundmotion. Copyright ? 1999 John Wiley & Sons, Ltd.

KEY WORDS: anisotropic basin; site response

1. INTRODUCTION

Scattering of elastic waves by isotropic subsurface inclusions has been the subject of many studiesin non-destructive evaluation of materials and prediction of earthquake induced ground motionatop a sediment-�lled valley (e.g. References 1 and 2). For example, for ampli�cation of elasticwaves by two- and three-dimensional isotropic alluvial valleys, an extensive literature review canbe found in the paper by Bouchon and Coutant.2 These studies provided reasonable understandingof the site ampli�cation for isotropic models. For general anisotropic media studies detailed surveyof literature can be found in Nayfeh.3 As for the scattering of elastic waves problems Rajapakseand Gross4 examined the response of an orthotropic elastic medium with an embedded cavity ofarbitrary shape due to transient pressurization using a boundary integral equation method. Karabulutand Ferguson5 studied SH-wave propagation in transversely isotropic medium by using discretewavenumber boundary integral method. Two semi-in�nite half-spaces and a multi-layered earthmodel were considered.Recently, Zheng and Dravinski6 considered ampli�cation of SH-waves by an orthotropic basin

of arbitrary shape using an indirect boundary integral equation method. The antiplane strain re-sults showed that material anisotropy may signi�cantly change surface response of a basin when

∗ Correspondence to: Marijan Dravinski, Department of Mechanical Engineering, Olin Hall 430, University of SouthernCalifornia, Los Angeles, CA 90089-1453, U.S.A. E-mail: [email protected]

CCC 0098–8847/99/060565–20$17·50 Received 27 January 1998Copyright ? 1999 John Wiley & Sons, Ltd. Revised 23 July 1998

566 T. ZHENG AND M. DRAVINSKI

compared with the corresponding isotropic model. Present paper is a continuation of that investi-gation to include the motion in the sagittal plane.Problem of di�raction of elastic waves by an obstacle embedded within an elastic medium can

be solved analytically or numerically. The analytical solutions apply mainly to linear, isotropic,and homogeneous materials, and simple geometries. The numerical solutions, on the other hand,are often inapplicable to the problems of interest in earthquake engineering and strong groundmotion seismology. Namely, the most commonly used numerical methods, �nite elements and�nite di�erences, require a construction of a computational grid which �lls the solution domain ofthe problem in space and time. This reduces the e�ectiveness of these methods for geotechnicalproblems which involve large dimensions. In addition, for �nite element and �nite di�erencemethods the radiation conditions in the far �eld are not satis�ed exactly.7

Boundary Element Methods (BEM) are e�cient techniques for modelling unbounded mediasince they require discretization only along the boundaries of the model and the radiation in thefar �eld is satis�ed exactly.8 Boundary integral equation methods can be of direct or indirecttype. Direct methods require calculation of the displacement Green’s functions and their spatialderivatives. For indirect methods, only the displacement Green’s functions need to be computedbut care must be taken for the placement of the auxiliary surfaces.The indirect boundary integral equation method used in this paper originates in the works of

Ursell9 and Kupradze.10 Extensions of the method to the wave propagation problems in geophysicsand earthquake engineering can be found in papers by Sanchez-Sesma and Rosenblueth,11 Wong12

and Dravinski.13

2. STATEMENT OF PROBLEM

The objective of this paper is to determine the elastodynamic �eld generated by incident planewaves of di�erent types in an orthotropic basin perfectly embedded within a half-space. Geometryof the problem is depicted by Figure 1. The model consists of a half-space with a sedimentarybasin of arbitrary shape. The interface C between the basin and the half-space is assumed to besu�ciently smooth without any corners. The domains of the half-space and the basin are denoted byD1 and D2, respectively. Materials are assumed to be linearly elastic, homogeneous, and orthotropicwith the symmetry planes coincident with the Cartesian co-ordinate system xi; i=1; 2; 3:The equations of motion for a general anisotropic medium without body forces are given by3

�ij; j + �!2ui=0; (•); j ≡ @(•)=@xj; i; j=1; 2; 3 (1)

where � is the mass density, �ij(x; !), and ui(x; !), are, respectively, the components of stressand displacement, while ! denotes circular frequency. Unless stated di�erently, summation overrepeated indices is understood. Throughout, the time dependence factor e−

√−1!t is understood.It should be noted that if the co-ordinate system is chosen to be in arbitrary direction then

the material behaves as those with lower symmetry and the problem becomes more complicated.However at this stage of investigation the simplest case is considered in order to determine thein uence of material anisotropy on the surface response.The stress–strain relations for a homogeneous anisotropic linearly elastic solid are given by the

generalized Hooke’s law,

�ij = cijklekl; i; j; k; l=1; 2; 3 (2)

Copyright ? 1999 John Wiley & Sons, Ltd. Earthquake Engng. Struct. Dyn. 28, 565–584 (1999)

AMPLIFICATION OF WAVES BY AN ORTHOTROPIC BASIN 567

Figure 1. Problem model

where ekl(x;!) are strain components de�ned by

ekl= 12(uk; l + ul; k) (3)

In equation (2) cijkl are the constants of elasticity. By virtue of the symmetry conditions

cijkl= cjikl= cijlk = cklij (4)

out of 81 components of cijkl only 21 are independent. In particular, for an orthotropic material,the constitutive equations can be written as3

�11�22�33�23�13�12

=

C11 C12 C13 0 0 0C12 C22 C23 0 0 0C13 C23 C33 0 0 00 0 0 C44 0 00 0 0 0 C55 00 0 0 0 0 C66

e11e22e33 23 13 12

(5)

where ij =2eij (with i 6= j) represents the engineering shear components. In order to relate cijkl toCpq (i; j; k; l=1; 2; 3 and p; q=1; 2; : : : ; 6) the following subscript contracting convention has beenused: 1→ 11; 2→ 22; 3→ 33; 4→ 23; 5→ 13, and 6→ 12. Thus, for example, c1122→C12 and so on.Substituting equation (5) into equation (1) results in the following partial di�erential equations

for the displacement components u1; u2 and u3:

C11u1;11 + C55u1;33 + (C13 + C55)u3;13 + �!2u1 = 0 (6)

C66u2;11 + C44u2;33 + �!2u2 = 0 (7)

(C13 + C55)u1;13 + C55u3;11 + C33u3;33 + �!2u3 = 0 (8)



Therefore, the equations of motion are partially decoupled. Equation (7) governs the SH-motionwhile equations (6) and (8) correspond to the motion in the sagittal plane. Thus the displacement�eld vector considered in this paper involve only two components, i.e. u=(u1; 0; u3).The stress-free boundary conditions are given by

�( j)13 = �( j)33 = 0; x3 = 0; x∈Dj; j=1; 2 (9)

where the superscript ( j) denotes the domain Dj. Perfect bonding along the interface C betweenthe half-space and the basin can be stated as

(u1; 0; u3)(1) = (u1; 0; u3)(2); x∈C(t1; 0; t3)(1) = (t1; 0; t3)(2); x∈C

(10)

where ti= �ij j are components of the traction vector and denotes unit normal vector on C (seeFigure 1).As an incident wave strikes the basin it generates scattered waves. These interact with the free-

�eld resulting in ampli�cation (constructive interference) or reduction (destructive interference) ofmotion. The objective of this paper is to determine the unknown scattered waves and subsequentlythe displacement and stress �elds throughout the medium.

3. SOLUTION OF PROBLEM

3.1. Free-�eld

Throughout the paper the incident wave will be assumed to be propagating upwards in the thirdquadrant (see Figure 1). Therefore, following the development of Nayfeh,3 the general incidentand re ected waves are represented by (see Appendix I)

(u1; 0; u3)i =∑q=2;4

(1; 0; Wq)U i1qe

√−1k1(x1+�qx3−ct)

(u1; 0; u3)r =∑q=1;3

(1; 0; Wq)U r1qe

√−1k1(x1+�qx3−ct)(11)

where U i1q are known and U

r1q are to be determined. Here, Wq denotes the polarization directions,

�q de�nes the directions of propagation, and c represents apparent phase velocity along the surfaceof the half-space. Corresponding traction vectors are given by

(t1; 0; t3)i = (�1j j; 0; �3j j)i

(t1; 0; t3)r = (�1j j; 0; �3j j)r(12)

where

(�13; �11; �33)i =∑q=2;4

√−1k1(D3q; D2q; D1q)U i1qe

√−1k1(x1+�qx3−ct)

(�13; �11; �33)r =∑q=1;3

√−1k1(D3q; D2q; D1q)U r1qe

√−1k1(x1+�qx3−ct)(13)

and D1q; D2q, and D3q are given in Appendix I.



The free-�eld can be written as

(u1; 0; u3)� = (u1; 0; u3)i + (u1; 0; u3)r

(t1; 0; t3)� = (t1; 0; t3)i + (t1; 0; t3)r(14)

Stress-free boundary conditions along the surface of the half-space for the free-�eld result inthe following equation:[

D11 D13D31 D33

][U r11

U r13

]=

[−D11 −D13D31 D33

][U i12

U i14

](15)

where it has been used that

D12 =D11; D14 =D13D32 =−D31; D34 = − D33

(16)

Equation (15) can be solved for the unknown re ected wave amplitudes. This topic is discussednext for three di�erent incident waves.

3.1.1. Incident pseudo P-wave. In this case U i14 = 0 and the incident wave follows from equation

(11) to be

(u1; 0; u3)i = (1; 0; W2)U i12e

√−1k1(x1+�2x3−ct) (17)

Corresponding re ected waves are described by the bottom equation (11) with the unknown am-plitudes determined from equation (15) to be

U r11 =

D11D33 + D13D31D13D31 − D11D33U

i12

U r13 =

−2D11D31D13D31 − D11D33U

i12

(18)

3.1.2. Incident pseudo S-wave. With U i12 = 0 the incident wave �eld becomes

(u1; 0; u3)i = (1; 0; W4)U i14e

√−1k1(x1+�4x3−ct) (19)

and the amplitudes of the re ected waves are given by

U r11 =

2D13D33D13D31 − D11D33U

i14

U r13 =−D13D31 + D11D33

D13D31 − D11D33Ui14

(20)

3.1.3. Incident pseudo Rayleigh wave. Here the outgoing wave �eld is of the form

(u1; 0; u3)R =∑q=1;3

(1; 0; Wq)UR1qe

√−1k1(x1+�qx3−ct) (21)



and the stress-free boundary conditions along the surface of the half-space lead to

D11D33 − D31D13 = 0 (22)

which de�nes the surface wave speed c= cR. Based on this phase velocity one can solve forthe directions of propagation �1;3 (see equation (56) in Appendix I). Furthermore, the boundarycondition �33(x1; 0; 0)=0 implies that

UR13 = − D11

D13UR11 (23)

Consequently, the displacement �eld speci�ed by equation (21) can be expressed in terms ofsingle displacement amplitude, say, UR

11: The corresponding stress �eld then can be obtained in astraightforward manner.If the material of the medium is assumed to be isotropic, the free-�eld results derived for pseudo

P-, S-, and Rayleigh waves reduce to standard results for isotropic media.This concludes the analysis of the free-�eld. The scattered waves are considered next.

3.2. Scattered wave �eld

As the incident wave strikes the interface C (Figure 1), it is partially transmitted into the basinand partially re ected into the half-space. Consequently, the waves in the half-space consist of thefree- and the scattered wave �elds, while the waves inside the basin comprise of scattered wavesonly. The displacement vectors in the half-space and the basin can be expressed as

(u1; 0; u3)(1) = (u1; 0; u3)� + (u1; 0; u3)(1)s; x∈D1 (24)

(u1; 0; u3)(2) = (u1; 0; u3)(2)s; x∈D2where the superscripts s and � denote the scattered and free-�eld, respectively.Assuming that the scattered wave �eld can be expressed in terms of single layer potentials,6; 9; 13

it means that the scattered �elds in the half-space and the basin are obtained as a superpositionof the responses to the corresponding line sources. This leads to

u( j)si (x; !)=∫C( j)q( j)k (y)G

( j)ik (x; y;!) dS(y); i; k =1; 3; x∈Dj; j=1; 2 (25)

Here q( j)k ; k =1; 3, are unknown density functions and C(1) and C(2) denote the auxiliary surfaces

de�ned inside and outside of the interface C (Figure 1). G( j)ik (x; y;!) are the half-space Green’sfunctions which must satisfy the equations

cijklGkp; jl + �!2Gip =−�ip�(x− y); p=1; 3

x= (x1; 0; x3); y=(y1; 0; y3)(26)

and stress-free conditions at the surface of the half-space. Thus Gkp(x; y; !) corresponds to theoutgoing waves and it represents the kth component of displacement vector at x due to a harmonicload in the xp-direction applied at y. The corresponding stress �eld,

∑ijp, may be determined



through the use of equation (2):∑ijp(x; y; !)= cijklGkp; l(x; y; !) (27)

The displacement and stress Green’s functions are given by Rajapakse and Wang.14

If the scattered waves are assumed to be represented in terms of discrete line sources, equation(25) becomes

u(1)s1 = amG(1)11 (x; xm) + bmG

(1)13 (x; xm); xm ∈C(1)

u(1)s3 = amG(1)31 (x; xm) + bmG

(1)33 (x; xm); xm ∈C(1)

m = 1; : : : ; M

u(2)s1 = clG(2)11 (x; xl) + dlG

(2)13 (x; xl); xl ∈C(2)

u(2)s3 = clG(2)31 (x; xl) + dlG

(2)33 (x; xl); xl ∈C(2)

l = 1; : : : ; L (28)

where am; bm; cl; and dl are unknown source intensities and summation over repeated indices mand l is understood.Substitution of scattered wave (28) into equation (24) and then using the continuity conditions

(10) leads to the following system of equations

Aa= f (29)

The 4N -by-2(M + L) matrix A and 4N -vector f are known (see Appendix II) while the 2(M +L)-vector aT = [a1; : : : ; aM ; b1; : : : ; bM ; c1; : : : ; cL; d1; : : : ; dL] contains the unknown source intensities.Equation (29) is solved in the least-square sense using QR decomposition.15 Once the sourcemagnitudes are known, the scattered wave �eld can be determined according to equation (28).This completes the solution of the problem. Numerical results are considered next.

4. NUMERICAL RESULTS

4.1. Evaluation of Green’s functions

A typical integral in the numerical calculation of Green’s functions is of the type

I =∫ ∞

0Q(�; z; z′; !)

{cos(�x�)

sin(�x�)

}d� (30)

where

�=�!2

C55; z= x3; z′=y3; x= x1

These integrals can be evaluated numerically provided that the problems which arise from veryslow decay and the oscillatory nature of the integrand when z→ z′ are eliminated (source andobservation points are at the same depth). It can be shown4 that the function Q in equation



(30) behaves asymptotically as e−�|z−z′| and 1=�e−�|z−z

′| in the case of traction and displacementGreen’s functions, respectively. Consequently, direct integration for the displacement Green’s func-tion can be performed when z→ z′, but not for the traction Green’s functions. For that purposeone can use the procedure originally suggested by Luco16 and also used by Rajapakse and Wang.4

In this approach the integral given by equation (30) is replaced by

I =∫ ∞

0[Q(�; z; z′; !)− Q∗(�; z; z′)]

{cos(�x�)

sin(�x�)

}d�+ I∗ (31)

where Q∗ asymptotic value of Q for large � and

I∗=∫ ∞

0Q∗(�; z; z′)

{cos(�x�)

sin(�x�)

}d� (32)

The �rst improper integral in equation (31) can be evaluated by a numerical quadrature andI∗ can be determined analytically. In the present paper the integrals are calculated by using anadaptive Newton–Cotes quadrature as suggested by Van Loan17 or an adaptive Filon quadrature.18

All calculations were performed in a MATLAB environment which facilitates e�cient vectorizationof the algorithms.

4.2. Veri�cation of Green’s functions

In order to verify the algorithms used for numerical evaluation of Green’s functions the problemof an orthotropic half-space loaded uniformly over a width of dimension 2a with intensity q0 andacting at a depth z′=a=1 is considered. For this problem, Rajapakse and Wang14 calculated theGreen’s functions for three materials: ice, layered soil and Cadmium. Here, these calculations arereproduced together with the ones obtained in the present investigation. Calculations are done �rstby using adaptive Newton–Cotes quadrature and then repeated by using adaptive Filon integrationrule. For the sake of briefness only two comparison results are shown here.Figures 2 and 3 depict the Green’s functions for ice and layered soil, respectively as a function

of depth. Although some di�erences can be observed between the two calculations still, the resultsof Figures 2 and 3 con�rm the accuracy of the proposed algorithms used for evaluation of Green’sfunctions.

4.3. Models

Three models will be employed in the analysis of numerical results:Model 1 consists of a semicircular isotropic basin of unit radius embedded within an isotropic

half-space. In that case the material constants reduce to C11 =C33 = �+ 2�; C13 = � and C55 = �:Model 2 consists of a semi-circular orthotropic basin of unit radius while model 3 describes semi-elliptic orthotropic basin with the same material properties as in model 2. The principal axes formodel 3 are assumed to be R1 = 1 and R3 = 0·7. The material properties of the three models aredescribed in Table I.



Figure 2. Testing of Green’s functions for an othotropic half-space subjected to a distributed loading over a width 2aof intensity q0 acting at a depth z′=a=1. Normalized displacement Green’s functions GijC44=(aq0) are presented asfunctions of depth and frequency. Dimensionless frequency is de�ned by a0 = a!

√�=C44 = 1. Solid lines depict the

results of this study while open circles those of Rajapakse and Wang.14 The material properties for ice are assumed to be:C11=C55 = 4·26; C12=C55 = 2·05; C33=C55 = 4·57; C13=C55 = 1·64; C55 = 0·317× 104 N=mm2

4.4. Testing and convergence analysis

The indirect boundary integral equation method requires the following parameters (Figure 1)

• N—number of collocation points along the interface C between the half-space and the basin,• M; L—number of source points along the auxiliary surfaces C(1) and C(2), respectively, and• dr—spacing between the auxiliary surfaces C(1); C(2) and the interface C.The following relationships were suggested by Ding and Dravinski19

ds¡110�inc (33)

dr' 3ds (34)

2NM + L

=2·25∼ 4·0 (35)

M =L (36)



Figure 3. The same as in Figure 2 for layered soil and frequencies a0 = 0·5 and 3. Material properties are assumed to be:C11=C55 = 2·11; C12=C55 = 0·43; C33=C55 = 2·58; C13=C55 = 0·47; C55 = 1·40× 104 N=mm2

Table I. Material properties for three models

Model 1: �(1) �(1) �(1) �(2) �(2) �(2)

1 1 2 1/6 1/2 1

Models 2,3: C (1)11 =C

(1)55 C (1)

13 =C(1)55 C (1)

33 =C(1)55 C (1)

55 �(1)

3·9 1·8 5·1 1 1·5C (2)11 =C

(2)55 C (2)

13 =C(2)55 C (2)

33 =C(2)55 C (2)

55 �(2)

0·65 0·3 0·85 0·1667 1

where ds denotes the spacing between two adjacent collocation points and �inc is the incident wavelength.The basin interface C is speci�ed by

x= R1 cos �z = R3 sin �; 06 �6 �

(37)

where R1 and R3 denote the major and minor principal axes, respectively.



Figure 4. Test for surface response of an isotropic semi-circular basin of unit radius subjected to a verticallyincident P-wave (top) and SV-wave (u= u1 and w= u3). Solid and dash lines denote the results obtained inthis study while stars and open circles correspond to those of Dravinski and Mossessian.7 Material properties:C11 =C22 =C33 = �+2�;C12 =C13 =C23 = �;C44 =C55 =C66 = �: �1 = �1 =

√�=�=1; �1 =

√(� + 2�)=�=2; �1 = 1=3;

�1 = 1; �2 = 1=6; �2 = 1=2; �2 = 1; �2 = 1=3; �2 = 2=3;!= �=2 s−1: Auxiliary surfaces parameters: �i = 0·8152; �o = 1·1848:Incident wave �eld amplitude for P-waves: U12 = 1 for �0 = 0 (vertical incidence) and U12 = − sin(�0) for �0 6=0; for

SV-waves: U14 = 1 for �0 = 0 (vertical incidence) and U14 = cos(�0) for �0 6=0



Figure 5. The same as in Figure. 4 but for incident plane harmonic Rayleigh wave. Incident wave �eld amplitudes:U11 = 2·3

The inner and outer sources are placed along the surfaces C(1) and C(2), respectively, de�nedby

C(1):

{x= R1i cos �

z = R3i sin �; 06 �6 �(38)

C(2):

{x= R1o cos �

z = R3o sin �; 06 �6 �(39)

with the principal axes being de�ned by

{R1;3i = �iR1;3

R1;3o = �oR1;3; 0¡�i¡1; 1¡�o(40)

Here the parameters �i and �o are determined according to equations (33)–(36).Therefore, if the incident wave length is given, then ds can be chosen according to equa-

tion (33) and all other parameters can be determined in accordance with relations equations(34)–(36). Since the spacing ds depends on the number of collocation points N , the key problemis to determine N .



Figure 6. Surface displacement spectral amplitudes for an orthotropic semicircular basin (model 2) subjected to incidentoblique incident pseudo P- and S-waves. != �=2 s−1; N =60; M = L=24; Ns =121: Incident wave �eld amplitudes are the

same as in Figure 4

Lack of analytical solutions for the problem at hand requires that the veri�cation of the numericalcalculations is done indirectly. This is accomplished in two di�erent ways.First, the so-called transparency test is performed where the material properties of the half-

space and the basin are assumed to be the same. The surface response should be that of thefree-�eld. This test allows estimation of the initial parameters, such as the orders of expansionand the number of collocation points. For the sake of briefness the results of this test are omit-ted. Second, the isotropic response test is performed in which the materials of the basin andthe half-space are assumed to be isotropic. These results can be compared with the ones ob-tained for isotropic media which have been con�rmed by several independent studies. Figure 4depicts the results of isotropic response test for a semi-circular basin subjected to a verticalplane harmonic P and SV waves while Figure 5 shows the same test for an incident Rayleighwave. The results of isotropic response test show very good agreement with those of Dravinski



Figure 7. Surface displacement spectral amplitudes for an orthotropic semicircular basin (model 2) subjected to an incidentpseudo Rayleigh wave. != �=2 s−1; N =60; M = L=24; Ns =121: Incident wave �eld amplitude the same as in Figure 5

and Mossessian7 which o�ers further credibility in the numerical result obtained in the presentstudy.

4.5. Steady-state response

Surface displacement spectral amplitudes for a semi-circular basin (model 2) and incident pseudoP- and S-waves are depicted by Figure 6. For the same basin and incident pseudo Rayleigh wavesthe displacement spectra are shown by Figure 7. Finally, the results corresponding to a semi-elliptical basin (model 3) are shown by Figure 8. Apparently, the surface motion is very sensitiveupon the type of incident wave, angle of incidence, location of observation station, and shape ofthe basin. In addition, the resulting ground motion strongly depends upon the frequency of theincident wave �eld (not shown here). Comparison of the response for orthotropic and isotropicbasins (Figures 6 and 7 with Figures 4 and 5) shows that material anisotropy may cause signi�cantchange in the ampli�cation on the surface ground motion. This is especially pronounced for thepeak amplitude of the ground motion for incident pseudo P- and Rayleigh waves where anisotropicbasin produced considerable larger ampli�cation than in the case of the isotropic basin.Therefore, presented results demonstrate the importance of anisotropy of basin in ampli�cation

of surface ground motion.



Figure 8. Surface displacement spectral amplitudes for an orthotropic semi-elliptical basin (model 3) subjected to incidentoblique incident pseudo P- and S-waves. != �=2 s−1; N =60; M = L=24; Ns =121: Incident wave �eld amplitudes are the

same as in Figure 4

5. SUMMARY AND CONCLUSIONS

Scattering of waves in sagittal plane by an orthotropic basin of arbitrary shape has been formulatedin terms of an indirect boundary integral equation approach. The known free-�elds were outlinedfor incident pseudo P-, S-, and Rayleigh plane harmonic incidence. The unknown scattered wave�eld is expressed in terms of single layer potentials which involve line load half-space Green’sfunctions and unknown density functions. The Green’s functions were evaluated numerically byusing adaptive Newton–Cotes or Filon quadratures. The unknown density functions are determinedin the least-square sense.Numerical results for steady-state surface response were presented for three models: model 1

which consists of a semi-circular isotropic basin and models 2 and 3 which incorporate semi-circular and semi-elliptical orthotropic basins, respectively.



Detailed testing and convergence analysis of the method was presented. Subsequently, surfaceresponse was evaluated for incident pseudo P-, S-, and Rayleigh waves. These were comparedwith the ones corresponding to the isotropic basin response. The results demonstrated that surfaceground motion ampli�cation strongly depends upon the nature of incident wave, shape and materialproperties of the basin and the half-space, and the location of the observation point. In addition itwas shown that material anisotropy may signi�cantly change surface ground motion when comparedto the corresponding isotropic case.

APPENDIX I: FREE-FIELD

A.1. General plane waves

According to Nayfeh,3 a bulk plane wave propagating in an anisotropic medium can be repre-sented in the form

ui(x; !)=Uie√−1(k•x−!t) =Uie

√−1(kn•x−!t); i=1; 2; 3 (41)

where Ui denotes displacement amplitude vector components, k=(k1; k2; k3) is the vector wavenum-ber of magnitude k, and n is unit vector de�ning the direction of propagation. Substituting equation(41) into equations of motion (6)–(8) leads to the eigenvalue problem

(�il − v2�il)Ul=0 (42)

where v=!=k is the phase velocity along n, and

�il=1�cijklnknj (43)

Existence of a non-trivial solution of equation (42) requires that

det(�− v2I)= 0 (44)

Due to decoupled nature of equations of motion, equation (44) can be solved for the phasevelocities in the x1x3-plane v2q; q=1; 3 and corresponding eigenvectors Uq: For an orthotropicmaterial the eigenvalues are found to be

v21 =A2�+

√B2�

(45)

v23 =A2�

−√B2�

(46)

A = (C11 + C55)n21 + (C33 + C55)n23 (47)

B = (C11 − C55)2n41 + (C55 − C33)2n43 (48)

+ 2[(C13 + C55)(2C13 + C55) (49)

+C55(C11 + C13 + C33)− C11C33]n21n23 (50)



Phase velocities v1 correspond to pseudo P-waves and v3 to pseudo S-waves. The eigenvectors Uqlead to the polarization directions

(1; 0; rq)=√1 + rq2; q=1; 3 (51)

where

rq = U3q=U1q (52)

=�v2q − (C11n21 + C55n23)(C13 + C55)n1n3

; q=1; 3

Therefore, the phase velocity v and the polarization of the plane wave depend upon its directionof propagation n and the material properties cijkl:

A.2. Re ection from a free surface

In this case it is convenient to recast the equation of the plane wave in the following form:

ui(x; !)=Uie√−1k1(x1+�x3−ct); i=1; 3 (53)

where

�= k3=k1 = n3=n1 (54)

and

c=!=k1 = v=n1 (55)

Here c denotes apparent phase velocity along the surface of the half-space and � characterizesthe direction of propagation. Once an incident plane wave strikes the surface of the half-spaceit generates a re ected wave. Both the amplitude and direction of the re ected wave have to bedetermined for a particular incidence. The horizontal phase velocity of the re ected wave is knownsince it has to match the corresponding phase velocity of the incident wave (see equation (55)).Substitution of equation (53) into equations of motion (equations (6) and (8)) leads to

A1�4 + B1�2 + C1 = 0 (56)

where

A1 = C33C55 (57)

B1 = C33(C11 − �c2) + C55(C55 − �c2)− (C13 + C55)2C1 = (C11 − �c2)(C55 − �c2)

Equation (56) results in four roots

�1 = n3=n1 = − �2 (58)

�3 = |n1=n3|√(C11 − �c2)(C55 − �c2)

C33C55= − �4



and the general displacement �eld in the sagittal plane can be represented by

(u1; 0; u3)=4∑q=1(1; 0; Wq)U1qe

√−1k1(x1+�qx3−ct) (59)

where

Wq=U3qU1q

=�c2 − C11 − C55�2q(C11 + C55)�q

(60)

Therefore, �1 and �3 describe the waves propagating in the positive x3-direction while �2 and�4 are associated with the waves propagating in the negative x3-direction. Corresponding tractionvector is given by

(t1; 0; t3)= (�1j j; 0; �3j j) (61)

where

(�13; �11; �33)=4∑q=1

√−1k1(D3q; D2q; D1q)U1qe√−1k1(x1+�qx3−ct) (62)

and

D1q=C13 + C33�qWq; D2q=C11 + C13�qWq; D3q=C55(�q +Wq) (63)

APPENDIX II

Matrix A and vector f in equation (29) are de�ned as follows:

A=

G(1)11 G(1)13 −G(2)11 −G(2)13G(1)31 G(1)33 −G(2)31 −G(2)33X(1)1 X(1)3 −X(2)1 −X(2)3Z(1)1 Z(1)3 −Z(2)1 −Z(2)3

(64)

where

G(1)ij = [G(1)ij (xn; xm)]; xn ∈C; xm ∈C(1);

G(2)ij = [G(1)ij (xn; xl)]; xn ∈C; xl ∈C(2);

X(1)i = [G(1)11i(xn; xm) 1 + G(1)13i(xn; xm) 3]; xn ∈C; xm ∈C(1); i=1; 3

X(2)i = [G(2)11i(xn; xl) 1 + G(2)13i(xn; xl) 3]; xn ∈C; xl ∈C(2); i=1; 3

Z(1)i = [G(1)13i(xn; xm) 1 + G(1)33i(xn; xm) 3]; xn ∈C; xm ∈C(1); i=1; 3

Z(2)i = [G(2)13i(xn; xl) 1 + G(2)33i(xn; xl) 3]; xn ∈C; xl ∈C(2); i=1; 3

n=1; : : : ; N; m=1; : : : ; M; l=1; : : : ; L; i; j=1; 3 (65)



Similarly,

f = −

u�

w�

t�1t�3

(66)

where

u� =

[ ∑q=2;4

U i1qe

√−1k1(x1n+�qx3n−ct) +∑q=1;3

U r1qe

√−1k1(x1n+�qx3n−ct)]

(67)

w� =

[ ∑q=2;4

Ui1qWqe

√−1k1(x1n+�qx3n−ct) +∑q=1;3

U r1qWqe


t�1 =

[ ∑q=2;4

√−1k1(D2q 1 + D3q 3)U i1qe

√−1k1(x1n+�qx3n−ct)

+∑q=1;3

√−1k1(D2q 1 + D3q 3)U r1qe


t�3 =

[ ∑q=2;4

√−1k1(D3q 1 + D1q 3)U i1qe

√−1k1(x1n+�qx3n−ct)

+∑q=1;3

√−1k1(D3q 1 + D1q 3)U r1qe


(x1; 0; x3)n ∈ C; n=1; : : : ; N (68)

ACKNOWLEDGEMENTS

The support to T. Zheng from All University Predoctoral Merit Fellowship and Teaching Assis-tantship at University of Southern California is greatly appreciated.

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Amplification of waves by an orthotropic basin: sagittal plane motion

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