A note on plane-wave approximation

Soil Dynamics and Earthquake Engineering 51 (2013) 9–13

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Soil Dynamics and Earthquake Engineering journal homepage: www.elsevier.com/locate/soildyn

A note on plane-wave approximation Hasan Faik Kara a, Mihailo D. Trifunac b,n a b

Department of Civil Engineering, Istanbul Technical University, Istanbul, Turkey Department of Civil Engineering, University of Southern California, Los Angeles CA, USA

art ic l e i nf o

a b s t r a c t

Article history: Received 13 November 2012 Received in revised form 3 March 2013 Accepted 6 April 2013 Available online 4 May 2013

It is shown that the plane-wave assumption for incident SH waves is a good approximation for cylindrical waves radiated from a finite source even when it is as close as twice the size of inhomogeneity. It is concluded that for out-of-plane SH waves the plane-wave approximation should be adequate for many earthquake engineering studies. & 2013 Elsevier Ltd. All rights reserved.

Keywords: Plane SH waves Cylindrical SH waves Plane-wave approximation.

1. Introduction Many earthquake engineering studies of amplification of incident seismic waves by internal inhomogeneities and by surface topography assume excitation by plane harmonic waves [1–18]. It is assumed in these studies that when the spherical and cylindrical wave fronts are sufficiently far from the earthquake source the plane-wave approximation may represent an adequate approximation. Most studies assume periodic excitation and present the results in terms of transfer-function amplitudes, usually along the ground surface and in the vicinity of inhomogeneity. The significance of these studies has been (1) in showing how the two- and three-dimensional interference, focusing, scattering, and diffraction of linear plane waves by inhomogeneities lead to changes in the amplitudes, frequencies, and locations of the observed peaks of transfer functions; and (2) in comparing the relative significance of surface topography and interior material inhomogeneities (sedimentary valleys) [18]. A review of these studies is presented in [11]. The purpose of this brief note is to show, by using elementary examples of SH waves, that the plane-wave approximation does indeed provide reasonable and useful approximation. We will show this by comparing the transfer functions for incident plane waves with the transfer functions for excitation by cylindrical waves emanating from a periodic finite source of SH waves.

2. Model The model we consider consists of a semi-circular sedimentary valley, with radius a, surrounded by the elastic homogeneous and

isotropic half-space (Fig. 1). The half-space is characterized by densityρs and shear-wave velocity cs , while the semi-cylindrical valley is described by ρv and cv . The fault, which radiates periodic SH waves, is located at r ¼ af , between the angles π þ αf −αf l =2 and π þ αf þ αf l =2. The fault width is af αf l . 3. Solution To describe radiation from the fault, two displacement fields are defined inside the half space: us ¼ us1 for a o r o af and us ¼ us2 for af o r o ∞. us1 contains two displacement fields, us1c represents cylindrical waves propagating toward r ¼ 0, and us1g represents reflected waves from the valley so that in that region us1 ¼ us1g þ us1c . us2 represents the waves propagating away from the origin r ¼ 0, and the waves inside the valley are uv . The governing equation for out-of-plane SH waves that are valid in both regions is
2  ∂ 1∂ 1 ∂2 1 ∂2 þ þ ð1Þ Uðr; θ; tÞ ¼ 2 2 Uðr; θ; tÞ: c ∂t ∂r 2 r ∂r r 2 ∂θ2 The time dependence of the solution will be taken as harmonic so that Uðr; θ; tÞ ¼ uðr; θÞe−iωt ;

Next, we introduce the wave number, k ¼ ω=c, which gives  ∂ 1∂ 1 ∂2 2 þ 2 2 þ k uðr; θÞ ¼ 0: ð4Þ þ 2 r ∂r r ∂θ ∂r


n

Corresponding author. Tel.: +1 626 447 9382; fax: +1 213 744 1426. E-mail address: [email protected] (M.D. Trifunac).

0267-7261/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.soildyn.2013.04.003

ð2Þ

where ω is the angular frequency. When Eq. (2) is substituted into Eq. (1), there follows
2  ∂ 1∂ 1 ∂2 ω2 þ uðr; θÞe−iωt ¼ − 2 uðr; θÞe−iωt : þ ð3Þ 2 2 2 r ∂r r ∂θ ∂r c 2

10

H.F. Kara, M.D. Trifunac / Soil Dynamics and Earthquake Engineering 51 (2013) 9–13

as follows: N

∑ An C n ðkrÞeinθ :

uðr; θÞ ¼

ð9Þ

n ¼ −N

The solutions for each sub-space in Fig. 1 are written below in such a way that the solutions that do not satisfy the Sommerfeld radiation condition are omitted. The displacement fields are N

∑ Av;n J n ðkβv rÞeinθ

uv ¼

When we assume that separation of the variables solves the problem uðr; θÞ ¼ RðrÞΘðθÞ, there follows
2  ∂ 1∂ 1 ∂2 2 þ þ þ k RðrÞΘðθÞ ¼ 0 ð5aÞ ∂r 2 r ∂r r 2 ∂θ2 ∂2 RðrÞ 1 ∂RðrÞ 1 ∂2 ΘðθÞ 2 þ RðrÞ 2 þ ΘðθÞ þ RðrÞΘðθÞk ¼ 0 r ∂r ∂r 2 r ∂θ2

us1g ¼

us1c ¼

N

∑ As1g;n H n ð1Þ ðkβs rÞeinθ

N

∑ As1c;n H n ð2Þ ðkβs rÞeinθ

r 2 ∂2 RðrÞ r ∂RðrÞ 1 ∂2 ΘðθÞ 2 þ þ þ r2 k ¼ 0 2 RðrÞ ∂r ΘðθÞ ∂θ2 RðrÞ ∂r

ð5cÞ

us2 ¼

r 2 ∂2 RðrÞ r ∂RðrÞ 1 ∂2 ΘðθÞ 2 þ r2 k ¼ − þ : RðrÞ ∂r ΘðθÞ ∂θ2 RðrÞ ∂r 2

ð5dÞ

r 2 ∂2 RðrÞ r ∂RðrÞ 1 ∂2 ΘðθÞ 2 þ r2 k ¼ − þ ¼ n2 RðrÞ ∂r ΘðθÞ ∂θ2 RðrÞ ∂r 2

ð5eÞ

∂2 ΘðθÞ þ ΘðθÞn2 ¼ 0 ¼ 4 ΘðθÞ ¼ beinθ þ be−inθ ∂θ2

ð5f Þ

r2

∂2 RðrÞ ∂RðrÞ 2 þ RðrÞðr 2 k −n2 Þ ¼ 0; þr ∂r ∂r 2

ð5gÞ

∂RðrÞ ∂RðrÞ ∂ξ ∂RðrÞ ¼ ¼ k ∂r ∂ξ ∂r ∂ξ

ð6aÞ




 ∂ ∂RðrÞ ∂ ∂RðrÞ ∂ ∂RðrÞ ∂ξ ∂2 RðrÞ 2 ¼ k ¼ k¼ k ∂r ∂r ∂r ∂ξ ∂ξ ∂ξ ∂r ∂ξ2

ð6bÞ

∂2 RðrÞ 2 ∂RðrÞ 2 k þ RðrÞðr 2 k −n2 Þ ¼ 0 k þr r2 ∂ξ ∂ξ2

ð6cÞ

ξ2

∂2 Rðξ=kÞ ∂Rðξ=kÞ þ Rðξ=kÞðξ2 −n2 Þ ¼ 0: þξ ∂ξ ∂ξ2

ð6dÞ

∞

∑ An C n ðkrÞeinθ :

ð10eÞ

n ¼ −N

where kβs and kβv are wave numbers in the half-space and in the valley, respectively. In terms of η, these wave numbers will be kβs ¼ ηπ=a, and kβv ¼ ðkβs cs Þ=cv . η is the ratio of valley width over wavelength in the half-space ðη ¼ 2a=λs Þ. Lamé's second parameters (shear moduli) for half-space and valley are μs ¼ cs 2 ρs and μv ¼ cv 2 ρv , respectively. The angular frequency is ω ¼ kβs cs ¼ kβv cv . For convenience, we introduce α1 ¼ π þ αf −αf l =2; α2 ¼ π þ αf þ αf l =2 as new variables. The boundary conditions are as follows: Zero stress on a flat surface is   μv ∂  μ ∂ μ ∂   uv  θ ¼ 0 ¼ s us1  θ ¼ 0 ¼ s us2  θ ¼ 0 ¼ 0: ð11aÞ r ∂θ r ∂θ r ∂θ θ¼π

θ¼π

The continuity of stress and displacement on the interface between the valley and the half-space is  ∂  ∂  ¼ μs us1  ð11bÞ μv uv  ∂r ∂r r¼a r¼a ð11cÞ

The continuity of stress in divided regions of half-space is   ∂ ∂   ¼ μs us2  ð11dÞ μs us1  ∂r ∂r r ¼ af r ¼ af The displacement difference in divided regions of half-space is

ð7Þ

Changing the variables again, RðrÞ ¼ C n ðkrÞ, the solution function will be C n ðkrÞeinθ . Since the solution has to be periodic in θ, n has to be an integer. The solution is valid for all integer values of n, from minus infinity to plus infinity. Therefore, the general solution is a linear combination, as follows: uðr; θÞ ¼

N

uv jr ¼ a ¼ us1 jr ¼ a :

This is a Bessel differential equation, and its solution is Rðξ=kÞ ¼ C n ðξÞ:

ð10dÞ

∑ As2;n H n ð1Þ ðkβs rÞeinθ ;

θ¼π

and after variable transformation, ξ ¼ rk,

ð10cÞ

n ¼ −N

us1 ¼ us1g þ us1c

This equation holds if both sides are equal to a constant. If this constant is chosen to be n2 , then

ð10bÞ

n ¼ −N

ð5bÞ

ΘðθÞ

ð10aÞ

n ¼ −N

Fig. 1. Geometry of the problem.

ð8Þ

n ¼ −∞

here, An are complex constants to be determined by boundary conditions. Cn is a Bessel function with order n, which can be either Jn,Y n , Hn(1), or Hn(2) depending on the physical conditions of the problem. Hn(1) describes the outgoing waves, while Hn(2) describes the incoming waves. Because the series above is convergent, it is possible to truncate it into a finite sum with N terms,

us1 jr ¼ af −us2 jr ¼ af ¼ f ðθÞ:

ð11eÞ

Where f ðθÞ is a function that satisfies the relative displacement difference (on the fault surface) between angles α1 and α2 (here assumed to be a constant), and the continuity of displacement elsewhere. To satisfy the zero-stress condition on a flat surface, we introduce another fault, symmetric with regard to the x axis—that is, we employ the imaging method. With this imaginary fault, the f ðθÞ function will take the following form: f ðθÞ ¼ fH½θ−α1 −H½θ−α2 g þ fH½θ−ð2π−α2 Þ−H½θ−ð2π−α1 Þg

ð12Þ

and its finite Fourier transform becomes fn ¼

1 2π

Z

2π 0

f ðθÞe−inθ dθ

ð13aÞ

H.F. Kara, M.D. Trifunac / Soil Dynamics and Earthquake Engineering 51 (2013) 9–13

9 8 i > > ðHð2π−α1 Þðe−2iπn −e−iα1 n Þ þ Hð−α1 Þðe−iα1 n −1Þþ > > > > 2πn > > > > −2iπn > > Hðα1 ÞðHðα1 −2πÞðe2iπn −eiα1 n Þ þ eiα1 n −1Þþ > > > > þe > > = < −2iπn −iα2 n −iα2 n −Hð2π−α2 Þðe −e Þ−Hð−α2 Þð−1 þ e Þþ fn ¼ ; n¼0 : −2iπn iα n 2iπn > > þe 2 −e Hðα ÞððHðα −2πÞ−1Þe Hðα −2πÞ þ 1ÞÞ; n≠0 > > 2 2 2 > > > > > > > > 1 ðHð2π−α1 Þð−α1 þ α1 Hð−α1 Þ þ 2πÞ−Hðα1 Þðα1 þ ð2π−α1 ÞHðα1 −2πÞÞþ > > > > 2π > > ; : −Hð2π−α2 Þð−α2 þ α2 Hð−α2 Þ þ 2πÞ þ Hðα2 Þðα2 þ ð2π−α2 ÞHðα2 −2πÞÞÞ

ð13bÞ since π o α1 o α2 o 2π, f n becomes

8 9 i > > ððe−2iπn −e−iα1 n Þ−ðe−iα1 n −1Þ þ e−2iπn −ðe2iπn −eiα1 n Þ þ eiα1 n −1Þþ > > < 2πn = f n ¼ −ðe−2iπn −e−iα2 n Þ þ ð−1 þ e−iα2 n Þ þ e−2iπn ð−2eiα2 n þ e2iπn þ 1ÞÞ; n≠0 ; > > > > : 1 ð2ð−α1 þ πÞ−ðα1 −ð2π−α1 ÞÞ þ 2ðα2 −πÞ þ ðα2 −ð2π−α2 ÞÞÞ; n ¼ 0 ; 2π

ð13cÞ where H is the Heaviside function, such that ( ) 0; θ o0 HðθÞ ¼ : 1; θ≥0

ð14Þ

f ðθÞ ¼

inθ

∑ f ne

n ¼ −N

:

ð15Þ

Substitution of the general solution into the boundary conditions then gives the following coefficients: 
  N ∂  ∂ 1  μv uv  ¼ μs us1  ¼4 ∑ Av;n kβv μv ðJ n−1 ðakβv Þ−J nþ1 ðakβv ÞÞ r¼a r¼a ∂r ∂r 2 n ¼ −N


 1 ð1Þ kβs μs ðH ð1Þ n−1 ðakβs Þ−H nþ1 ðakβs ÞÞ 2
 1 ð2Þ kβs μs ðH ð2Þ ðak Þ−H ðak ÞÞ einθ ¼ 0 −As1c;n βs βs n−1 nþ1 2

r¼a

¼ us1 jr ¼ a ¼ 4

N

n ¼ −N

inθ −As1c;n ðH ð2Þ n ðkβs aÞÞge

μs

∑ fAv;n ðJ n ðkβv aÞÞ−As1g;n ðH ð1Þ n ðkβs aÞÞ ¼0

  ∂ ∂   us1  ¼ μs us2  ∂r ∂r r ¼ af r ¼ af 
 N 1 ð1Þ ð1Þ kβs μs ðH n−1 ¼4 ∑ As1g;n ðaf kβs Þ−H nþ1 ðaf kβs ÞÞ 2 n ¼ −N
 1 ð2Þ kβs μs ðH ð2Þ ða k Þ−H ða k ÞÞ þAs1c;n βs βs f f n−1 nþ1 2
 1 ð1Þ kβs μs ðH n−1 ðaf kβs Þ−H ð1Þ ða k ÞÞ einθ ¼ 0 −As2;n βs f nþ1 2

ð16aÞ

ð16cÞ

ð2Þ ð1Þ inθ ∑ fAs1g;n ðHð1Þ n ðkβs af ÞÞ þ As1c;n ðH n ðkβs af ÞÞ−As2;n ðH n ðkβs af ÞÞ−f n ge : ¼ 0

ð16dÞ

n ¼ −N

Since the above four equations have to hold for all θ, coefficients of einθ have to be equal to zero independently for all integer values of n from –N to +N so that 8N+4 linear equations are obtained to calculate 8N+4 unknown constants, as follows:
Av;n


 1 1 ðakβs Þ−H ð1Þ ðakβs ÞÞ kβv μv ðJ n−1 ðakβv Þ−J nþ1 ðakβv ÞÞ −As1g;n kβs μs ðH ð1Þ n−1 nþ1 2 2


−As1c;n

−As1g;n

ð16bÞ

us1 jr ¼ af −us2 jr ¼ af ¼ f ðθÞ ¼ 4 N

The amplitude of the fault dislocation is assumed to be one. The inverse of the finite complex Fourier transform in Eq. (13a) is then N

  uv 

11

 1 ð2Þ ð2Þ kβs μs ðH n−1 ðakβs Þ−H nþ1 ðakβs ÞÞ ¼ 0 2

ð2Þ Av;n ðJ n ðkβv aÞÞ−As1g;n ðH ð1Þ n ðkβs aÞÞ−As1c;n ðH n ðkβs aÞÞ ¼ 0

ð17aÞ

ð17bÞ

Fig. 2. Comparison of normalized transfer functions for harmonic fault dislocations at af ¼2a; 3a; 4a; 8a; and 16a, αf ¼ π/16, 301, 601, and 901, for ρs =ρv ¼ 1.5, cs =cv ¼2, and η ¼ 1, with the transfer functions for surface displacements for incident plane SH waves (solid black lines) [13] and the same incidence angles.

12

H.F. Kara, M.D. Trifunac / Soil Dynamics and Earthquake Engineering 51 (2013) 9–13




 1 ð1Þ kβs μs ðH n−1 ðaf kβs Þ−H ð1Þ ða k ÞÞ βs f nþ1 2
 1 ð2Þ kβs μs ðH ð2Þ þ As1c;n ða k Þ−H ða k ÞÞ βs βs f f n−1 nþ1 2
 1 ð1Þ kβs μs ðH ð1Þ −As2;n n−1 ðaf kβs Þ−H nþ1 ðaf kβs ÞÞ ¼ 0 2

As1g;n

ð17cÞ

As1g;n ðH ð1Þ n ðkβs af ÞÞ ð1Þ þ As1c;n ðH ð2Þ n ðkβs af ÞÞ−As2;n ðH n ðkβs af ÞÞ−f n ¼ 0:

ð17dÞ

4. Results To evaluate the plane-wave approximation, we compare the above solution with the solution for incident plane waves [13]. This comparison is shown in Figs. 2–4for η ¼1, 2, and 4, and for af ¼ 2a; 3a; 4a; 8a; and 16a. We consider only one set of material properties with ρs =ρv ¼1.5, cs =cv ¼2, and one fault width with αf l ¼ π=8. We are interested only in the relative shapes of the transfer functions, not in their actual amplitudes. For this reason, we normalize all transfer functions for excitation by the fault in order to have the same average amplitudes as the transfer functions for surface displacements associated with incident plane waves (shown by solid continuous lines in Figs. 2–4). In each figure, we show transfer functions for waves radiated from the fault at af ¼2a; 3a; 4a; 8a;and 16a by dotted, dash-dot, dashed, solid, and dash-dot-dot lines, respectively. It is seen that the shapes of the transfer functions of surface displacements for excitation by cylindrical waves are very close to

those for incident plane SH waves for intermediate and small η. As η increases (Fig. 4), the differences increase, but the overall details in the shapes of the transfer functions remain very close to those for the plane-wave excitation. In Figs. 2–4, we illustrate the comparison between the transfer functions only for one fault width, which is equal to αf l ¼ π=8. We also calculated the corresponding transfer functions for αf l ¼ π=4 and found essentially the same trends as those seen in Figs. 2–4. We note that the static surface displacements for the faults, which cut the free surface, will show a jump with normalized displacement amplitude equal to one. It can be shown that these displacements are larger and decay slower with distance from the fault, on the side of the fault adjacent to softer geologic materials. On the side, which is adjacent to the edge of the valley and beyond, the displacements are smaller and die out faster due to the increasing rigidity of the rocks outside the sedimentary basin. In Figs. 2–4 these jumps cannot be seen because we show the transfer function amplitudes only for –2 ox=ao 2, while af ¼2a; 3a; 4a; 8a;and 16a, and because the transfer function amplitudes only represent the Fourier components in the series representation of the complete steady state dynamic solution. 5. Conclusions In this note, we showed only a limited number of comparisons between (1) the plane SH wave, and (2) cylindrical SH wave excitations of semi-cylindrical inhomogeneity, and we considered only ρs =ρv ¼1.5, cs =cv ¼ 2, and αf l ¼ π=8. Nevertheless, the results suggest that for the frequencies of interest in earthquake

Fig. 3. Comparison of normalized transfer functions for harmonic fault dislocations at af ¼2a; 3a; 4a; 8a; and 16a, αf ¼ π/16, 301, 601, and 901, for ρs =ρv ¼ 1.5, cs =cv ¼ 2, and η ¼ 2, with the transfer functions for surface displacements for incident plane SH waves (solid black lines) [13] and the same incidence angles.

H.F. Kara, M.D. Trifunac / Soil Dynamics and Earthquake Engineering 51 (2013) 9–13

13

Fig. 4. Comparison of normalized transfer functions for harmonic fault dislocations at af ¼2a; 3a; 4a; 8a; and 16a, αf ¼ π/16, 301, 601, and 901, for ρs =ρv ¼ 1.5, cs =cv ¼2, and η ¼ 4, with the transfer functions for surface displacements for incident plane SH waves (solid black lines) [13] and the same incidence angles.

engineering, which correspond roughly to 0 oη o5, the planewave representation of incident SH waves will lead to reasonably realistic predictions of amplification of strong-motion amplitudes on the ground surface. Whether the same conclusions can be extended to in-plane motions associated with P and SV waves will be investigated in our future studies. Acknowledgments During the course of this work, the first author was on leave from the Istanbul Technical University and visiting the University of Southern California. The financial support for his stay that was provided by TUBITAK is gratefully acknowledged. References [1] Moeen-Vaziri N, Trifunac MD. Scattering of plane SH-waves by cylindrical canals of arbitrary shape. Soil Dynamics and Earthquake Engineering 1985;4:18–23. [2] Moeen-Vaziri N, Trifunac MD. Scattering and diffraction of plane SH-waves by two-dimensional inhomogeneities. Earthquake Engineering and Structural Dynamics 1988;7:179–88. [3] Moeen-Vaziri N, Trifunac MD. Scattering and diffraction of plane SV-waves by two-dimensional inhomogeneities. Earthquake Engineering and Structural Dynamics 1988;7:189–200. [4] Sánchez-Sesma FJ. Diffraction of elastic waves by three-dimensional surface irregularities. Bulletin of the Seismological Society of America 1983;73: 1621–36. [5] Sánchez-Sesma FJ. Diffraction of elastic SH waves by wedges. Bulletin of the Seismological Society of America 1985;75:1435–46.

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