A note on three-dimensional scattering and diffraction by a hemispherical canyon–I: Vertically incident plane P-wave

Soil Dynamics and Earthquake Engineering 61-62 (2014) 197–211

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A note on three-dimensional scattering and diffraction by a hemispherical canyon–I: Vertically incident plane P-wave Vincent W. Lee n,1, Guanying Zhu n,2 Civil Engineering Department, University of Southern California, Los Angeles, CA 90089, USA

art ic l e i nf o

a b s t r a c t

Article history: Received 9 November 2013 Received in revised form 3 February 2014 Accepted 8 February 2014 Available online 19 March 2014

The three-dimensional scattering by a hemi-spherical canyon in an elastic half-space subjected to seismic plane and spherical waves has long been a challenging boundary-value problem. It has been studied by earthquake engineers and strong-motion seismologists to understand the amplification effects caused by surface topography. The scattered and diffracted waves will, in all cases, consist of both longitudinal (P-) and shear (S-) shear waves. Together, at the half-space surface, these waves are not orthogonal over the infinite plane boundary of the half-space. Thus, to simultaneously satisfy both zero normal and shear stresses on the plane boundary numerical approximation of the geometry and/or wave functions were required, or in some cases, relaxed (disregarded). This paper re-examines this boundary-value problem from the applied mathematics point of view, and aims to redefine the proper form of the orthogonal sphericalwave functions for both the longitudinal and shear waves, so that they can together simultaneously satisfy the zero-stress boundary conditions at the half-space surface. With the zero-stress boundary conditions satisfied at the half-space surface, the most difficult part of the problem will be solved, and the remaining boundary conditions at the finite canyon surface will be easy to satisfy. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Scattering Stress-free elastic half-space Spherical canyon Free-field amplification

1. Introduction It has long been observed that topographies can have significant influence on ground displacements during strong earthquake shaking. It is thus important to understand the theoretical aspects of the effects of such scattering in both two- (2-D) and threedimensional (3-D) geometries [1]. In this note, we analyze the 3-D scattering problem of a hemi-spherical canyon in an elastic halfspace subjected to plane longitudinal waves of specified frequency. The proper coordinate system for this problem is the 3-D-spherical coordinate system. In simpler 2-D, Lamb [2] presented a paper titled “On the propagation of tremors over the surface of an elastic solid”. In that paper, Lamb solved the problem of wave motion generated at the surface of an elastic half-space subjected to a concentrated load at the surface or inside the half-space. Both time harmonic and impulsive loads were considered. Some three decades ago, we started to develop analytic solutions for scattering problems for both surface and subsurface topographies. Trifunac [3] solved the scattering and diffraction problem for incident plane SH-waves in a semi-circular canyon. n Correspondence address: Sonny Astani Civil & Environmental Engineering, University of Southern California, Los Angeles, California, 90089-2531 U.S.A. 1 Professor. 2 PhD candidate.

http://dx.doi.org/10.1016/j.soildyn.2014.02.010 0267-7261 & 2014 Elsevier Ltd. All rights reserved.

Lee [4] extended the work to an underground, circular, unlined tunnel (cavity). Later Lee and Trifunac [5] generalized the solution to the case of an underground, circular, elastic tunnel. In all such SH-wave solutions, it was possible to use the method of images, which created a line of symmetry on the half-space surface resulting in zero anti-plane shear stresses. Such a method, however, failed for in-plane longitudinal P- and shear S-waves. Without the method of images, approximations for the reflections from the flat surface were proposed. Lee and Cao [6] and Cao and Lee [7,8] used a large, circular, almost-flat surface to approximate the half-space surface and presented solutions to scattering and diffraction problems of surface circular canyons of various depths for incident plane P- and SV-waves. Todorovska and Lee [9–11] used the same method for anti-plane SH-waves and for incident Rayleigh-waves on circular canyons. Later, Lee and Karl [12,13] extended the method to scattering and diffraction of P- and SV-waves by circular underground cavities. Lee and Wu [14,15] used the same method to study arbitrary-shaped 2-D canyons. Davis et al. [16] used such an approximate method in their studies of the failure of underground pipes involving transverse incidence of SV-waves. Liang et al. [17–33] continued the analyses by the same method for problems involving circulararc canyons and valleys and underground pipes in elastic and poro-elastic half-space. All of the above studies considered only 2-D problems.

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V.W. Lee, G. Zhu / Soil Dynamics and Earthquake Engineering 61-62 (2014) 197–211

and there are no reflected S-waves. Together the incident and reflected plane waves form the input free-field P-wave potentials

Φðff Þ ¼ ΦðiÞ þ ΦðrÞ ¼ Φ0 expð  ikα zÞ  Φ0 expð þ ikα zÞ;

ð6Þ

which, in spherical coordinates, will simplify to 1

Φðff Þ ¼ Φ0 ∑ a2n þ 1 j2n þ 1 ðkα rÞP 2n þ 1 ðuÞ n¼0

with a2n þ 1 ¼ Fig. 1. Three-dimensional hemispherical canyon with vertical plane incidence.

2. Previous work in three dimensions Many studies [34–36] have aimed to find the analytical solution of wave scattering by 3-D topographies on the half-space surface. One fundamental solution was developed by Lee [34]. Lee's same geometry will be used in this paper, but we will consider only the special case of plane vertically incident longitudinal P-waves to illustrate the new approach and new method. Fig. 1 shows the plane vertically incident P-waves on the halfspace and hemispherical canyon with radius a.

μ and λ are Lame

constants of the elastic half-space and ρ is its density. The longitudinal (compressional) wave speed in the half-space is cα ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi ðλ þ 2μÞ=ρ, and its shear (transverse) wave velocity is cβ ¼ μ=ρ.

All waves are assumed to be harmonic with frequency ω and the pffiffiffiffiffiffiffiffi corresponding time-dependent term e  iωt , i ¼  1, will be assumed to be present in all wave terms and omitted. Both the Cartesian ðx; y; zÞ (as shown) and spherical ðr; θ; ϕÞ coordinate systems will be used. All waves will satisfy the Helmholtz-wave equation, which in spherical coordinates takes the form, for the wave function Fðr; θ; ϕÞexpð  iωtÞ, as     1 ∂ 2 ∂F 1 ∂ ∂F 1 ∂2 F 2 r þ þ sin θ þ k F ¼ 0; ð1Þ 2 2 2 2 ∂r r ∂r r sin θ ∂θ ∂θ r sin θ ∂ϕ2 where k is the wave number, such that k ¼ kα ¼ ω=cα for P-waves and k ¼ kβ ¼ ω=cβ for S-waves. The two coordinate systems are related by x ¼ r sin θ cos θ;

y ¼ r sin θ sin ϕ;

z ¼ r cos θ;

ð2Þ

and r ¼ ðx2 þ y2 þz2 Þ1=2 ;

θ ¼ cos  1 ðz=rÞ;

ϕ ¼ cos  1 ðx=ðx2 þy2 Þ1=2 Þ ð3Þ

The incident plane P-wave is axis-symmetric with respect to the z-axis. Its potential will be expressed in both the rectangular and spherical coordinate systems as

ΦðiÞ ¼ Φ0 expð  ikα zÞ 1

¼ Φ0 ∑ ð2n þ 1Þð  iÞn jn ðkα rÞP n ðuÞ;

ð4Þ

n¼0

where jn ðkα rÞ is the spherical Bessel function of the 1st kind of order n, Pn(u) is the (0th-order) Legendre polynomial of degree n, with u ¼ cos θ, 0 ru r 1 for 0 r θ r π =2 in the half-space. Φ0 is the amplitude of the P-wave potential to be scaled, Φ0 ¼ 1=ikα , so that it will have a displacement amplitude of “1” at the half-space surface. The presence of the half-space surface z¼0 results in ðrÞ reflected plane P-wave potential Φ propagating vertically downwards and of the form

ΦðrÞ ¼  Φ0 expð þ ikα zÞ 1

ð7Þ

ðff Þ

The free-field P-potential, Φ , will have a displacement amplitude of “2” at the half-space surface. It will also satisfy the zero-stress boundary conditions at the half-space surface. For all r Za

sz ∣z ¼ 0 ¼ τzx ∣z ¼ 0 ¼ τzy ∣z ¼ 0 ¼ 0

ð8Þ

or

sθ ∣θ ¼ π=2 ¼ τθr ∣θ ¼ π=2 ¼

¼0

ð9Þ

For the axisymmetric case here, the waves are not dependent of

ϕ, hence the stress term τθϕ ∣θ ¼ π =2 ¼ 0 is trivial.

In the presence of the hemispherical canyon, scattered and diffracted waves are produced and diffracted from the canyon, which takes the following form [34,35,47]: 1

ΦðsÞ ¼ ∑ An hð1Þ n ðkα rÞP n ðuÞ n¼0 1

χ ¼ ∑ C n hð1Þ n ðkβ rÞP n ðuÞ ðsÞ

ð10Þ

n¼0

ð1Þ

respectively for the P- and SV-waves. hn ðÞ are the spherical Hankel functions of the 1st kind, representing outgoing P- and ðff Þ ðsÞ S- waves. Combining all the waves Φ , Φ , and χ ðsÞ the stresses everywhere take the form: 1

ð3Þ srr ¼ ∑ ½a2n þ 1 Eð1Þ 11 ð2n þ 1; kα rÞP 2n þ 1 ðuÞ þðAn E 11 ðn; kα rÞ n¼0

þ C n Eð3Þ 13 ðn; kβ rÞÞP n ðuÞ ¼ 0 1

ð3Þ sθθ ¼ ∑ ½a2n þ 1 Eð1Þ 21 ð2n þ 1; kα rÞP 2n þ 1 ðuÞ þ ðAn E21 ðn; kα rÞ n¼0

þ C n Eð3Þ 23 ðn; kβ rÞÞP n ðuÞ ¼ 0  1 dP 2n þ 1 ðuÞ τrθ ¼ ∑ a2n þ 1 Eð1Þ þ ðAn Eð3Þ 41 ð2n þ 1; kα rÞ 41 ðn; kα rÞ dθ n¼0  dP n ðuÞ ¼ 0: þ C n Eð3Þ 43 ðn; kβ rÞÞ dθ

n

ð5Þ

ð11Þ

ði ¼ 1; 2; 3; 4; j ¼ 1; 2; 4; k ¼ 1; 2; 3Þ Here the stress equations EðiÞ jk are used to denote the stress–potential relationships that involve only the spherical Bessel/Hankel functions corresponding to various waves. Those are given in Mow and Pao [37]. The superscript i is used to denote the type of spherical Bessel functions and/or Hankel functions used. Here i ¼1, 2, 3, 4 are respectively for ð1Þ ð2Þ the functions j; y; h and h . The subscript j is used to denote the particular type of stress functions. Here the subscripts j ¼ 1, 2, 4 are the stress components for srr, sθθ and τrθ respectively. Stresses are due to Φ and X respectively when k ¼1 and 3. Detailed expressions of EðiÞ are given in Appendix A. jk At the half-space surface, where ϕ ¼ π =2, u ¼ cos θ ¼ 0, we need the following: 1

ð3Þ sθθ ∣θ ¼ π =2 ¼ ∑ ½ðAn Eð3Þ 21 ðn; kα rÞ þ C n E 23 ðn; kβ rÞÞP n ð0Þ ¼ 0 n¼0

¼  Φ0 ∑ ð2n þ 1Þð þ iÞ jn ðkα rÞP n ðuÞ; n¼0

ð8n þ 6Þi2n þ 1 2ð4n þ 3Þð  1Þn ¼ : ikα kα

1



ð3Þ τrθ ∣θ ¼ π=2 ¼ ∑ ðAn Eð3Þ 41 ðn; kα rÞ þ C n E 43 ðn; kβ rÞÞ n¼0

 dP n ð0Þ ¼0 dθ

ð12Þ

V.W. Lee, G. Zhu / Soil Dynamics and Earthquake Engineering 61-62 (2014) 197–211

The free-field wave terms are not included here because they already satisfy the half-space zero-stress boundary conditions. Here the term P 1n ð0Þ is the associated Legendre polynomial (order 1 and degree n) at u ¼ cos θ ¼ 0 from the identity d P n ðuÞ∣θ ¼ π =2 ¼ P 1n ð0Þ: dθ

ð13Þ

2-D half space [24,27,38]: 1

P 2m þ 1 ðuÞ ¼ ∑ γ mn P 2n ðuÞ

1

where

γ mn ¼

n¼0

There are a total of four boundary conditions in (12) and (14) to be satisfied by the two sets of unknown coefficients An and Cn. Additional P- and S-wave functions were thus proposed to be present, in addition to the waves in (10) in Lee [34,35]. They will have spherical Bessel functions of the 1st kind instead of Hankel functions. On the half-space surface, Lee [34,35] replaced the spherical Bessel and Hankel wave functions by their Laurent series representations in radial distance r, and the independence properties of the power series are again used to satisfy the zero-stress boundary condition in (12). However, the resulting system of equations is numerically complicated and difficult to solve in that the coefficients EðiÞ are made up jk of two sets of (P- and S-) waves that are not orthogonal to each other. This is the only known numerical-implementation step that has ever been used for the wave function-expansion solution method.

3. The “stress-free” analytic-wave functions: a new approach In this section, we will use a new formulation to account for the scattered and diffracted waves that will satisfy the zero-stress boundary conditions at the half-space surface. We will start with the same model of the hemispherical canyon that we used in the previous section. Beginning with the scattered and diffracted wave potentials in the form derived (see Appendix B) as 1

ΦðsÞ ¼ ∑ A2n þ 1 hð1Þ 2n þ 1 ðkα rÞP 2n þ 1 ðuÞ n¼0 1

χ ¼ ∑ C 2n þ 1 hð1Þ 2n þ 1 ðkβ rÞP 2n þ 1 ðuÞ ðsÞ

〈P 2m þ 1 ðuÞ; P 2n ðuÞ〉 ; 〈P 2n ðuÞ; P 2n ðuÞ〉

ð15Þ

n¼0

where the wave potentials, as in the case of the free-field plane waves (7), are expressed in terms of the odd and only odd degree Legendre functions. In contrast to (10), they are not expressed in the full set of Legendre functions, fP n ðuÞ; n ¼ 0; 1; 2; …g, of both even and odd degrees. It is known that this full set forms an orthogonal set of basis functions in the full space where u  cos θ, 0 r θ r π . However, here in the half-space where 0 r θ r π =2 ð0 r u r1Þ, both the odd-degree fP 2n þ 1 ðuÞg and even-degree fP 2n ðuÞg Legendre polynomials (and similarly for the 1st order associated Legendre polynomials fP 1n ðuÞg each by themselves form an orthogonal set of basis functions, and the odd- and even-degree Legendre polynomials together are not orthogonal in the half-space. Moreover every function defined in the half-space can be expanded either in terms of a series of odd degree fP 2n þ 1 ðuÞg and/or even degree fP 2n ðuÞg Legendre polynomials. In fact, in the half-space, each odddegree Legendre polynomial P 2n þ 1 ðuÞ can be expanded as a series of even-degree fP 2n ðuÞg Legendre polynomials. This is equivalent to expressing the sine function as a series of cosine functions in the

ð17Þ

in terms of

〈f ; g〉 ¼

the inner product

þ C n Eð3Þ 13 ðn; kβ aÞÞP n ðuÞ ¼ 0  1 dP 2n þ 1 ðuÞ τrθ ∣r ¼ a ¼ ∑ a2n þ 1 Eð1Þ þ ðAn Eð3Þ 41 ð2n þ 1; kα aÞ 41 ðn; kα aÞ dθ n¼0  dP n ðuÞ þ C n Eð3Þ ¼ 0: ð14Þ 43 ðn; kβ aÞÞ dθ

ð16Þ

n¼0

At the surface of the spherical canyon, where r ¼a, we need ð3Þ srr ∣r ¼ a ¼ ∑ ½a2n þ 1 Eð1Þ 11 ð2n þ1; kα aÞP 2n þ 1 ðuÞ þ ðAn E 11 ðn; kα aÞ

199

Z

1

f ðuÞgðuÞ du:

ð18Þ

0

It is known that [39] Z

1

P 2m þ 1 ðuÞP 2n ðuÞ du ¼

0

and Z 1 ðP 2n ðuÞÞ2 du ¼ 0

ð  1Þðm þ n þ 1Þ ð2mþ 1Þ!ð2nÞ! 2ðm þ nÞ ð2n  2mþ 1Þð2n þ 2mþ 2Þðn!Þ2 ðm!Þ2

1 4n þ1

;

ð19Þ

ð20Þ

Using (16), the wave potentials in (15) (of odd degrees) can be expressed in terms of the Legendre functions of even degrees as 1

ð1Þ ΦðsÞ ¼ ∑ A2n þ 1 h2n þ 1 ðkα rÞP 2n þ 1 ðuÞ n¼0 1

1

ð1Þ

¼ ∑ A2m þ 1 h2m þ 1 ðkα rÞ ∑ γ mn P 2n ðuÞ m¼0 1

χ ¼ ∑ ðsÞ

n¼0 1

n¼0

ð1Þ C 2n þ 1 h2n þ 1 ðkβ rÞP 2n þ 1 ðuÞ 1

ð1Þ

¼ ∑ C 2m þ 1 h2m þ 1 ðkβ rÞ ∑ γ mn P 2n ðuÞ; n¼0

or

 1 ð1Þ ∑ A2m þ 1 h2m þ 1 ðkα rÞγ mn P 2n ðuÞ n¼0 m¼0  1  1 ð1Þ ðsÞ χ ¼ ∑ ∑ C 2m þ 1 h2m þ 1 ðkβ rÞγ mn P 2n ðuÞ 1

ΦðsÞ ¼ ∑

n¼0

ð21Þ

n¼0



ð22Þ

m¼0

Eqs. (15) and (22) thus show that the scattered P- and S-wave potentials in the half-space can be expressed as a series of Legendre polynomials of both odd and even degrees, each being a set of wave functions that are complete in the half-space. This is equivalent to, and an extension of, the 2-D semi-circular canyon problem [38] using the concept of “half-space expansion” of trigonometric functions, where a function, f ðθÞ, defined in a half-range ½0; π , can be expressed both as a sine series and a cosine series, obtained respectively by the odd and even extensions of the function to the full range ½  π ; π . Here the sine and cosine functions are replaced by Legendre polynomials of odd and even degree, and the same concept of “half-range expansion” is used. The longitudinal and shear-wave potential functions as represented by (15) and (22) can now be shown to be the wave functions that implicitly satisfy the zero-stress boundary conditions on the half-space surface, namely, (9): 1. The zero normal stress boundary condition on the surface of the half-space sz ∣z ¼ 0 ¼ sθ ∣θ ¼ π =2 ¼ 0: With (15) for the P- and S-wave scattered potentials each as an odd-degree series in u  cos θ, the normal stress will only be computed for the scattered and diffracted waves because the freefield incident and reflected plane P-waves already satisfy the zero-stress boundary conditions. From (12), it takes the form as

sθθ ∣θ ¼ π=2 ¼

1

∑

ð3Þ ½An Eð3Þ 21 ðn; kα rÞ þ C n E23 ðn; kβ rÞP n ð0Þ:

ð23Þ

n ¼ 1;3;5…

Eq. (23) is summed over all the odd integers, n ¼ 1; 3; 5…, where P n ð0Þ ¼ 0 for all odd n-degree Legendre polynomials and thus for all r Z a, sz ∣z ¼ 0 ¼ sθ ∣θ ¼ π =2 ¼ 0. 2. The zero shear stress boundary condition on the surface of the half-space τzx ∣z ¼ 0 ¼ τzy ∣z ¼ 0 ¼ τθr ∣θ ¼ π =2 ¼ 0:

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V.W. Lee, G. Zhu / Soil Dynamics and Earthquake Engineering 61-62 (2014) 197–211

This time we write both the P- and S-wave potentials as a Legendre series of even degrees in u  cos θ, using (22), then the shear stress will take the form, from (12) as  1 1 τθr ∣θ ¼ π=2 ¼ ∑ ∑ ðA2 m þ 1 Eð3Þ 41 ð2 m þ 1; kα rÞ n¼0 m¼0 i ð3Þ ð2 m þ 1; kβ rÞÞγ mn P 12n ð0Þ: ð24Þ þ C 2 m þ 1 E43 Eq. (24) is summed over all the even 2n ¼ 0; 2; 4; …, where P 12n ð0Þ ¼ 0 for all 1st order Legendre polynomials of even degrees in n, and thus for all r Z a, τzx ∣z ¼ 0 ¼ τzy ∣z ¼ 0 ¼ τθr ∣θ ¼ π=2 ¼ 0 is satisfied. ðsÞ The scattered and diffracted waves Φ and χ ðsÞ of (15) are a complete set of cylindrical-wave functions that each by themselves and together both satisfy the zero-stress boundary conditions of (12) at the half-space surface. The canyon zero-stress boundary conditions are next considered: For 0 r θ r π =2, at r¼a

sr jr ¼ a ¼ 0 τrθ jr ¼ a ¼ 0

ð25Þ

The P- and SV-wave potentials will now include both the free-field waves and scattered and diffracted waves. Together they will satisfy the zero-stress boundary conditions at the surface of the canyon, so that we need

sr jr ¼ a ¼ srðff Þ jr ¼ a þ sðsÞ r jr ¼ a ¼ 0 ðff Þ ðsÞ τrθ jr ¼ a ¼ τrθ jr ¼ a þ τrθ jr ¼ a 0

ð26Þ

3. Zero normal stress boundary condition on the canyon surface sr ∣r ¼ a ¼ 0 : From above 1

ð1Þ Þ sðff r ∣r ¼ a ¼ ∑ a2n þ 1 E11 ð2n þ 1; kα aÞP 2n þ 1 ðuÞ

s

ðsÞ r ∣r ¼ a

n¼0 1

¼ ∑ ðA2n þ 1 Eð3Þ 11 ð2n þ 1; kα aÞ n¼0

þ C 2n þ 1 Eð3Þ 13 ð2n þ 1; kβ aÞÞP 2n þ 1 ðuÞ

ð27Þ

Þ ðsÞ sr ∣r ¼ a ¼ sðff r ∣r ¼ a þ sr ∣r ¼ a ¼ 0 gives, using the orthogonality of the Legendre polynomials, for each n

from which the coefficients An and Cn are given by, for n ¼ 1; 3; 5; …, 9 2 38 ( ) ð3Þ ð3Þ =  E13 ðn; kβ aÞ < Eð1Þ An  an 4 E43 ðn; kβ aÞ 11 ðn; kα aÞ 5 ; ¼ ð1Þ ð3Þ : Cn DetðnÞ  Eð3Þ E ðn; k aÞ E11 ðn; kα aÞ β ; 41 41 ðn; kα aÞ ð32Þ ð3Þ ð3Þ DetðnÞ ¼ Eð3Þ 11 ðn; kα aÞE 43 ðn; kβ aÞ  E 13 ðn; kβ aÞ

with the determinant Eð3Þ 41 ðn; kα aÞ. Eq. (32) is a simple and explicit close-form analytic expression for the coefficients of the P- and S-scattered and diffracted wave functions. This is very uncommon for both 2-D and 3-D elasticwave propagation problems in a homogeneous half-space. For the 2-D half-space cases, Trifunac [40] was the first to present an explicit and analytic expression for the coefficients for the scattered and diffracted wave functions of the out-of-plane SH-waves. The wave functions there, namely, the cylindrical Hankelfunction series, are expressed as half-range, Fourier-cosine series, which also explicitly satisfy the half-space stress-free boundary condition for out-of-plane SH-waves. The same technique was applied to the corresponding semicircular, cylindrical alluvial valley [41] and semi-circular, cylindrical rigid foundation in a soil–structure interaction problem [42]. Later, Trifunac's work on 2-D cylindrical coordinate problems was extended to 2-D elliptical-coordinate problems. Wong and Trifunac [43] also presented an explicit and analytic expression for the coefficients for the scattered and diffracted wave functions in elliptical coordinates for out-of-plane SH- waves. They later extended their work to a semi-elliptical, alluvial valley [44], and to semi-elliptical, rigid foundation in a soil–structure interaction problem [45]. All of the above-cited work was for out-of-plane SH-waves. Most of the other work using wave-function series often resulted in non-analytic equations that needed to be solved numerically. No explicit, analytic expressions for the coefficients of the corresponding series wave functions existed for in-plane P- and SV-waves for both 2-D and 3-D diffraction problems until the writing of this note, where we show that explicit, analytic expressions for the coefficients of 3-D spherical waves exist for the case of normal P-wave incidence. We will next use these results to calculate the displacements on and around the half-space canyon.

ð3Þ A2n þ 1 Eð3Þ 11 ð2n þ 1; kα aÞ þ C 2n þ 1 E 13 ð2n þ 1; kβ aÞ

¼  a2n þ 1 Eð1Þ 11 ð2n þ1; kα aÞ

ð28Þ

4. Zero shear stress boundary conditions on the canyon surface τrθ ∣r ¼ a ¼ 0: 1

ð1Þ τrðffθ Þ ∣r ¼ a ¼ ∑ a2n þ 1 E41 ð2n þ 1; kα aÞP 12n þ 1 ðuÞ

τ

ðsÞ ∣ rθ r ¼ a

n¼0 1

¼ ∑ ðA2n þ 1 Eð3Þ 41 ð2n þ 1; kα aÞ n¼0

1 þ C 2n þ 1 Eð3Þ 43 ð2n þ 1; kβ aÞÞP 2n þ 1 ðuÞ

ð29Þ

Þ τrθ ∣r ¼ a ¼ τðff ∣ þ τrðsÞθ ∣r ¼ a ¼ 0 gives, again using orthogonality, rθ r ¼ a

for each n

ð3Þ A2n þ 1 Eð3Þ 41 ð2n þ 1; kα aÞ þ C 2n þ 1 E 43 ð2n þ 1; kβ aÞ

¼  a2n þ 1 Eð1Þ 41 ð2n þ1; kα aÞ

ð30Þ

4

Eð3Þ 11 ð2nþ 1; kα aÞ

Eð3Þ 13 ð2nþ 1; kβ aÞ

Eð3Þ 41 ð2nþ 1; kα aÞ

Eð3Þ 43 ð2nþ 1; kβ aÞ

(a) Vertical displacements: The displacement amplitudes at various points along the surface of the half-space and around the hemispherical canyon will help our understanding of amplification of ground motion. The precise description of the amplitudes of surface-ground motion will allow determination of the space-dependent transfer functions of ground motion at and near the hemispherical canyon. Complete knowledge of the displacement at each point will also enable us to calculate all components of t-strains and stresses. With the scattered and diffracted wave functions in spherical coordinates, the components of displacement are first calculated in spherical coordinates and then transformed back to rectangular coordinates U z ¼ U r cos θ  U θ sin θ

ð33Þ

where

Combining (28) and (30) 2

4. Surface displacements

3( 5

A2n þ 1 C 2n þ 1

) ¼ a2n þ 1

9 8 < Eð1Þ ð2n þ1; kα aÞ = 11 : Eð1Þ ð2n þ1; kα aÞ ; 41

ð31Þ

Uθ ¼ Ur ¼

P n ðuÞ ð3Þ ð3Þ ½an Dð1Þ 21 ðn; kα aÞ þ An D21 ðn; kα aÞ þ C n D23 ðn; kβ aÞ dθ

1

∑

n ¼ 1;3;5;…; 1

∑

ð3Þ ð3Þ ½an Dð1Þ 11 ðn; kα aÞ þ An D11 ðn; kα aÞ þ C n D13 ðn; kβ aÞP n ðuÞ

n ¼ 1;3;5;…;

ð34Þ

V.W. Lee, G. Zhu / Soil Dynamics and Earthquake Engineering 61-62 (2014) 197–211

201

Fig. 2. Vertical displacement amplitudes along a radial line for normal incidence (axisymmetric case) with η ¼ 4, 8, 15 and 50.

DðiÞ the displacement-potential expressions are given in jk Appendix A at the end of the paper. For each of the complex components of U, its modulus along the plane stress-free surface is defined as the “displacement amplitude” of that component as jU z j ¼ ½Re2 ðU z Þ þ Im2 ðU z Þ1=2

ð35Þ

For consistency with previous work [40], the following dimensionless frequency is defined as

η¼

ωa 2fa kβ a 2a ¼ ¼ ¼ π λβ πβ β

ð36Þ

Fig. 2 presents the vertical displacement amplitudes along the radial line for selected dimensionless frequencies η ¼ 4, 8, 15 and 50, where the case of η ¼ 50 is the highest frequency we attempted to consider. The far-field displacement amplitudes at the half-space surface are equal to two at all frequencies. Eq. (36) shows that given a half-space with a shear wave speed of β ¼ 2 km/s and a hemispherical canyon of radius a¼ 1 km, this would correspond to elastic waves at given cyclic frequencies of f ¼ 4, 8, 15 and 50 Hz. In terms of wavelengths, they would correspond to shear wavelengths 0.5, 0.25, 0.133 and 0.04 km, or 1/4, 1/8, 1/15 and 1/50 of the diameter of the hemispherical canyon. The dimensionless frequencies used in this paper are all in a frequency range much higher than those calculated in the related previous work [35,36,46]. This is because the new analytic wave functions automatically satisfy the zero-stress boundary conditions on the half-space surface, and the explicit expressions for the

wave coefficients are much simpler than that before. Back then, the use of power-series expansions of the wave functions resulted in complicated matrix equations that were much higher in order and numerically more difficult to solve. Considering that the highest-dimensionless frequencies we could present in the 1970s and 1980s where only η ¼0.5 and η ¼ 1.0 are the results in this paper represent a significant improvement. In Fig. 2, with a as the hemispherical-canyon radius, the displacement amplitudes are plotted along the dimensionless horizontal-radial distances from x/a¼0 to x/a¼10. Being axisymmetric, this is also the same amplitudes along r/a¼0 to r/a¼ 10. The range x=a 4 1 corresponds to the half-space surface measured from the center of the canyon. In each of the four graphs for the four dimensionless frequencies, the displacement amplitudes all oscillate about the free-field amplitude of two. The periods of oscillation correspond to those of the input periods of the waves. Thus, for example, in the 2nd graph from the bottom, at a dimensionless frequency of η ¼ 2a=λβ ¼ 15, or a wavelength at 0.133a km, there are almost eight positive peaks, or almost eight wavelengths ða=8 ¼ 0:125aÞ. The amplitudes plotted at distances in the range x/a¼0 to x/a¼1 are the amplitudes on the hemispherical canyon. The point x/a¼1 is the rim of the canyon. The displacement amplitudes also oscillate on the canyon surface. At the rim of the canyon surface, it is observed that the displacement amplitude exhibits a “dip and spike” in amplitude. This is the cornerpoint phenomena, which is also observed for the semicylindrical canyon case for SH-wave incidence in [40]. It means that the displacement amplitude on the side of the canyon close

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to the rim will decay to almost zero, but beyond the rim on the half-space surface, the amplitudes will shoot up to above the free-field amplitude. At much lower frequencies ðη r 1Þ, in the previous work of Lee [35], these amplitudes were not as prominent because the waves at lower frequencies tend “not to see” these corner points. As pointed out in [38], this corner point in wave-propagation theory is often known to act as a secondary wave source point. Figs. 3–8 show the corresponding 3-D plots of the displacement amplitude along the same horizontal radial line r=að ¼ x=aÞ. The points are for the 6 ranges of η: η ¼ 0 5; 5 10; 10  15; 15  20; 20  23; 23  25. The points are plotted, from left to right, vs. the radial distance x/a from the center of the canyon outward, and from front to back, vs. the dimensionless frequency η. For 0 r x=a r1, the displacement amplitudes are those on the hemispherical surface of the canyon. For x=a 41, they are displacement amplitudes on the half-space surface away from the canyon. For the frequency range considered, η, from 0 to 25 in the six graphs, the amplitudes up to three are observed at the flat surface close to the canyon, and the same “dip and spike” behavior is observed for all frequencies at the corner point, the rim between the half-space and the spherical-canyon's surface. A displacement amplitude equal to a constant of two everywhere along the half-space surface is observed when the frequency η is low and approaches zero; i.e., the long waves do not “see” the

inhomogeneity. As the dimensionless frequency increases through the five graphs, the oscillations of the displacement amplitudes increase as the wavelengths get shorter. (b) Radial displacements: The radial-displacement amplitudes jU r j of (34) in Figs. 9–13 are 3-D plots plotted along radial distance x/a from the center of the canyon outward with dimensionless frequencies η ¼ 0–5, η ¼ 5  10; …; η ¼ 20  25. Note that jU r j is zero on the half-space surface because the Legendre polynomial of the odd order is zero at z ¼ 0, where u ¼ cos θ ¼ 0 when θ ¼ π =2. For the free-field incident and reflected P-waves there are only vertical motions. For the scattered and diffracted waves, considered as the superposition of waves from Models 0 and 1 (Appendix B), the radical motions exist for waves from each model, but they cancel each other at the half-space surface. The motions in the figures are thus seen only on the canyon surface, where x=a r 1. Further, it is noted that jU r j is zero everywhere on the canyon surface at zero frequency, η ¼0 (Fig. 9), corresponding to an infinitely long period, the static case. As the dimensionless frequency η increases from Figs. 9 to 13, the radial motions are more and more oscillatory along the canyon surface. Fig. 14 shows the 2-D plots of radial displacements for dimensionless frequency of η ¼ 4, 8, 15, 50, where the case of η ¼ 50 is the highest frequency we considered. They are plotted along x=a ¼ r=a from 0 to 1.5 with x=a r 1 again for points on the canyon surface. It shows the same feature as the 3-D curves in

Fig. 3. Vertical displacement jU z j along x/a for normal incidence, η ¼ 0–5.

Fig. 4. Vertical displacement jU z j along x/a for normal incidence, η ¼ 5–10.

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203

Fig. 5. Vertical displacement jU z j along x/a for normal incidence, η ¼ 10–15.

Fig. 6. Vertical displacement jU z j along x/a for normal incidence, η ¼15–20.

Fig. 7. Vertical displacement jU z j along x/a for normal incidence, η ¼ 20–23.

Figs. 9–13; namely, that the oscillations increase with increasing frequency with maximum amplitudes of 2 or less. Also, the oscillations on the canyon surface are all with amplitudes between 0.0 and 2.0, with the oscillation amplitudes closer to 2 at the bottom of the canyon where x=a ¼ 0, and oscillating with

decreasing amplitudes down to zero as it goes up to the canyon surface, where x=a Z 1. (c) Discussion of convergence: Figs. 15–19 show the numerical implementation of calculating the displacements at every point using (33)–(35).

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Fig. 8. Vertical displacement jU z j along x/a for normal incidence, η ¼23–25.

Fig. 9. Radial displacement jU r j along x/a for normal incidence, η ¼0–5.

Fig. 10. Radial displacement jU r j along x/a for normal incidence, η ¼ 5–10.

Even though the coefficients of all wave terms are given in explicit form, the waves are given as infinite series. Thus, the infinite sum has to be truncated to a finite series in numerical calculations. As in all previous works, at each given value of frequency, a sufficient number of terms have to be used to make sure that the sum from the finite series converges to that of the infinite series

at each frequency. Figs. 15–19 are all for dimensionless frequencies η ¼ 1, 4, 8, 15, and 50 respectively. At each frequency, the plots consist of four graphs of displacement amplitudes along radial distance of x/a from 0 to 5. The total number of terms used, N, is labeled as Nterm in the figures. Fig. 15, for η ¼1, has Nterm ¼ 2, 4, 6, 8 when it already

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205

Fig. 11. Radial displacement jU r j along x/a for normal incidence, η ¼ 10–15.

Fig. 12. Radial displacement jU r j along x/a for normal incidence, η ¼ 15–20.

Fig. 13. Radial displacement jU r j along x/a for normal incidence, η ¼20–25.

shows convergence. Note that this was the highest frequency we could achieve in the older, related work from the 1970s and 1980s. Fig. 16, for η ¼ 4, has Nterm ¼2, 4, 8,15. Fig. 17, for η ¼8, has Nterm ¼4, 8,15, 20 and will not converge for Nterm o 15. Fig. 18, for η ¼15, needs Nterm around 25 to guarantee convergence. Finally, Fig. 19, for η ¼50 needs Nterm around 60–80 to guarantee convergence.

5. Conclusions Based on the plots and discussion above for Figs. 3–19, the following can be concluded: 1. By using Legendre polynomials (of odd orders) and half-range expansions, this paper presented the spherical-wave functions

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Fig. 14. Radial displacement amplitudes with η ¼ 4, 8, 15 and 50.

Fig. 15. Vertical displacement jU z j along x/a for various Nterms for η ¼1.

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207

Fig. 16. Vertical displacement jU z j along x/a for various Nterms for η ¼4.

Fig. 17. Vertical displacement jU z j along x/a for various Nterms for η ¼ 8.

that can satisfy the stress-free boundary conditions at the surface of the half-space. In the case of normal incident waves, explicit, analytic expressions were derived for each coefficient of the spherical-wave functions. As a result, the numerical calculations are simpler and results for much higher frequencies are obtained. 2. In Lee [34,35], it was stated that the amplification of surfacedisplacement amplitudes around the hemispherical canyon can be high. Although the same conclusion now holds, it is for much higher frequencies as well, as shown in this paper.

3. Spikes in displacement amplitudes are observed at the rim of the canyon in all figures. We could not observe these spike-displacement amplitudes in our previous work because we were not able then to get results at any dimensionless frequencies beyond η ¼1. Earlier, calculations were at lower frequencies, which tend not to “see” these corner points. Now we can get frequency as high as η ¼ 50. These corner points in wave-propagation theory are often known to create secondary wave sources. 4. Previously, it was stated that, in [34,35], “The dimensionless frequency η plays an important role in determining the

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Fig. 18. Vertical displacement jU z j along x/a for various Nterms for η ¼ 15.

Fig. 19. Vertical displacement jU z j along x/a for various Nterms for η ¼ 50.

displacement patterns. Larger values of η will result in higher complexity of displacements and in higher amplifications”. The same conclusion holds for the results in this paper. 5. The concept of using odd-degree Legendre polynomials to represent both P- and S-wave functions that are stress-free at the half-space surface for P-wave normal incidence can also be

extended to the cases of: (i) (ii) (iii) (iv) (v)

Axisymmetric P-wave point-source incidence, Vertical plane SV-wave incidence, Vertical SV-wave point-source incidence, Vertical plane SH-wave incidence, and Vertical SH-wave point-source incidence.

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209

This means that this concept can be extended to all cases of normal incidences of all types of waves. 6. The above concept can and will next be extended to elastic plane and point-sources with arbitrary-oblique incidence, which will be non-axisymmetric, and both the free-field waves and scattered P- and S-waves will be of the form zn ðkrÞP m n ðuÞ

cos mϕ sin mϕ

;

where zn(kr) is the Bessel or Hankel functions of the 1st and/or 2nd kind of order n, k ¼ kα for P-waves and k ¼ kβ for S-waves; Pm n ðuÞ is the associated Legendre polynomials of order m and degree n with u ¼ cos θ. The right form of P- and S-wave functions will again have to be derived and chosen so that they can satisfy the zero-stress boundary conditions at the half-space surface.

Fig. B1. Model 0: vertically upward incident plane P-wave onto spherical cavity.

Appendix A. Pao and Mow (1973) The stress and displacement functions used in (14) and (34): " # ! 2 kβ r 2 ðiÞ 2μ ðiÞ ðiÞ 2 E11 ðn; kα rÞ ¼ 2 n n  ðA:1Þ zn ðkα rÞ þ 2kα rzn þ 1 ðkα rÞ 2 r Fig. B2. Model 1: vertically downward incident plane P-wave onto spherical cavity.

EðiÞ 13 ðn; kβ rÞ ¼

o 2μn ðiÞ nðn þ 1Þ½ðn  1ÞzðiÞ n ðkβ rÞ  kβ rzn þ 1 ðkβ rÞÞ r2

ðA:2Þ

EðiÞ 41 ðn; kα rÞ ¼

2μ ðiÞ ½ðn  1ÞzðiÞ n ðkα rÞ kα rzn þ 1 ðkα rÞ r2

ðA:3Þ

EðiÞ 43 ðn; kβ rÞ ¼

2μ r2

"

2 2

n2  1 

kβ r 2

!

as in (4) above. The presence of the spherical cavity resulted in scattered and diffracted waves of the form

#

ðiÞ zðiÞ n ðkβ rÞ þ kβ rzn þ 1 ðkβ rÞ

ðA:4Þ

ðiÞ ðiÞ DðiÞ 11 ðn; kα rÞ ¼ nzn ðkα rÞ  kα rzn þ 1 ðkα rÞ

ðA:5Þ

ðiÞ DðiÞ 13 ðn; kβ rÞ ¼ nðn þ 1Þzn ðkβ rÞ

ðA:6Þ

ðiÞ DðiÞ 21 ðn; kα rÞ ¼ zn ðkα rÞ

ðA:7Þ

ðiÞ ðiÞ DðiÞ 23 ðn; kβ rÞ ¼ ðn þ1Þzn ðkβ rÞ  kβ rzn þ 1 ðkβ rÞ

ðA:8Þ

zðiÞ n ðÞ,

where i¼ 1,3 are separately referred to the first kind of spherical-Bessel function with order n jn ðÞ and first kind of Hankel ð1Þ function with order n hn ðÞ.

Appendix B. Derivation of scattered and diffracted waves for vertical P-wave incidence The scattered and diffracted waves will be derived in what follows as the superposition of waves from two full space models: Model 0 and Model 1. Consider the full-space model, Model 0, in Fig. B1, where a plane P-wave is moving vertically upward in the negative z-direction, propagating in the x  z plane, incident onto the spherical cavity (expð  iωtÞ is assumed to be present on all wave terms in what follows): The potential of the incident P-wave is given by

ΦðiÞ ¼ Φ0 expð  ikα zÞ 1

¼ Φ0 ∑ ð2n þ 1Þð iÞn jn ðkα rÞP n ðuÞ n¼0

ðB:1Þ

1

Φs0 ¼ Φ0 ∑ An hð1Þ n ðkα rÞP n ðuÞ X s0

n¼0 1

ð1Þ

¼ Φ0 ∑ C n hn ðkβ rÞP n ðuÞ

ðB:2Þ

n¼0

as in (10) above. All waves here are independent of the spherical coordinate ϕ. Φs0 and Xs0 form a complete solution of this wave problem. The coefficients An and Cn in the potentials can be solved in terms of the coefficients of the incident wave exactly by the boundary conditions

sr ¼ τrθ ¼ 0;

ðB:3Þ

at r ¼a the surface of the spherical cavity. In exactly the same way, consider the 2nd full space model, Model 1, in Fig. B2, where a z plane P-wave is moving vertically downward in the position z-axis, propagating on the x  z plane, incident onto the spherical cavity. With respect to the ðx1 ; y1 ; z1 Þ rectangular-coordinate system, and the corresponding spherical-coordinate system ðr 1 ; θ1 ; ϕ1 Þ with ðiÞ the z1 axis pointing upward, the incident P-wave potential Φ1 in ðiÞ Fig. B2 would have the same form as the Φ potential in Fig. B1, as it would be going in the negative z1 direction. With u1 ¼ cos θ1

ΦðiÞ 1 ¼ Φ1 expð þ ikα zÞ ¼ Φ1 expð  ikα z1 Þ 1

¼ Φ1 ∑ ð2n þ1Þð  iÞn jn ðkα r 1 ÞP n ðu1 Þ;

ðB:4Þ

n¼0

with amplitude Φ1. The corresponding scattered and diffracted waves Φs1 and Xs1, with respect to the ðr 1 ; θ1 ; ϕ1 Þ coordinate, would also take the same form as Φs0 and Xs0 in Fig. B1: 1

Φs1 ¼ Φ1 ∑ An hð1Þ n ðkα r 1 ÞP n ðu1 Þ X s1

n¼0 1

ð1Þ

¼ Φ0 ∑ C n hn ðkβ r 1 ÞP n ðu1 Þ

ðB:5Þ

n¼0

As in the first case, the potentials Φs1 and Xs1 form a complete set of solutions for this model in Fig. B2. The two spherical

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Eqs. (B.11) and (B.12) are now the complete solution for the scattered and diffracted waves to be used in the paper, which we will show to satisfy the zero-stress boundary conditions at the half-space surface. They are the superposition of the complete solutions from Model 0 and Model 1. References

Fig. B3. Half-space model: vertically plane P-wave onto hemispherical canyon.

coordinate systems, ðr; θ; ϕÞ of Fig. B1 and ðr 1 ; θ1 ; ϕ1 Þ of Fig. B2 are related by r 1 ¼ r θ1 ¼ π  θ ϕ1 ¼ 2π  ϕ so that u1 ¼ cos θ1 ¼  cos θ ¼  u and

Φ and s 1

Φ ¼Φ s 0

X s0

s 0 ðr;

¼ X s0 ðr;

Xs1

ðB:6Þ

in the ðr; θ; ϕÞ coordinates would take the form 1

n θÞ ¼ Φ0 ∑ An hð1Þ n ðkα rÞð  1Þ P n ðuÞ n¼0 1

n θÞ ¼ Φ0 ∑ C n hð1Þ n ðkβ rÞð  1Þ P n ðuÞ

ðB:7Þ

n¼0

as P n ðu1 Þ ¼ P n ð  uÞ ¼ ð  1Þn P n ðuÞ. Consider now a Half-Space Model of an hemispherical canyon in an elastic half-space subjected to vertically incident plane P-wave of potential Φi. The presence of the half-space will result in a reflected plane P-wave of potential Φr propagating vertically downwards, as shown in Fig. B3. The incident and reflected plane P-waves are

Φi ¼ Φ0 expð  ikα zÞ Φr ¼ Φ1 expð þ ikα zÞ;

ðB:8Þ

which together satisfy the zero-stress boundary conditions at the half-space surface. Take the incident-potential amplitude of Φ0 ¼ 1. It is then known that the reflected potential has amplitude Φ1 ¼  1. The half-space model in Fig. B3 can now be considered as a superposition of the models in Figs. B1 and B2 such that

Φi of Half  Space Model ¼ Φi0 of Model 0 Φr of Half  Space Model ¼ Φi1 of Model 1:

ðB:9Þ

In other words, the half-space model is the superposition of Model 0 and Model 1. The presence of the hemispherical canyon will result in scattered and diffracted waves Φs and Xs, which can be taken as the superposition of the scattered and diffracted waves of Model 0 and Model 1:

Φs ¼ Φs0 þ Φs1 X s ¼ X s0 þ X s1 ;

ðB:10Þ

which takes the form, with Φ0 ¼ 1 and Φ1 ¼  1. From the above equations 1

1

n Φs ¼ Φs0 þ Φs1 ¼ ∑ An hnð1Þ ðkα rÞP n ðuÞ  ∑ An hð1Þ n ðkα rÞð  1Þ P n ðuÞ n¼0 1

n¼0

ð1Þ

¼ ∑ A02n þ 1 h2n þ 1 ðkα rÞP 2n þ 1 ðuÞ;

ðB:11Þ

n¼0

with only the odd-order terms and A02n þ 1 ¼ 2A2n þ 1 . Similarly 1

1

ð1Þ

ð1Þ

X s ¼ X s0 þ X s1 ¼ ∑ C n hn ðkβ rÞP n ðuÞ  ∑ C n hn ðkβ rÞð  1Þn P n ðuÞ n¼0 1

n¼0

ð1Þ

¼ ∑ C 02n þ 1 h2n þ 1 ðkβ rÞP 2n þ 1 ðuÞ; n¼0

again with only the odd-order terms and C 02n þ 1 ¼ 2C 2n þ 1 .

ðB:12Þ

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