Author's personal copy A note on plane-wave approximation Hasan Faik Kara a , Mihailo D. Trifunac b,n a Department of Civil Engineering, Istanbul Technical University, Istanbul, Turkey b Department of Civil Engineering, University of Southern California, Los Angeles CA, USA article info Article history: Received 13 November 2012 Received in revised form 3 March 2013 Accepted 6 April 2013 Keywords: Plane SH waves Cylindrical SH waves Plane-wave approximation. abstract 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. 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 approx- imation 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 inter- ference, 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 densityρ s and shear-wave velocity c s , while the semi-cylindrical valley is described by ρ v and c v . The fault, which radiates periodic SH waves, is located at r ¼ a f , between the angles π þ α f -α fl =2 and π þ α f þ α fl =2. The fault width is a f α fl . 3. Solution To describe radiation from the fault, two displacement fields are defined inside the half space: u s ¼ u s1 for a or oa f and u s ¼ u s2 for a f or o∞. u s1 contains two displacement fields, u s1c represents cylindrical waves propagating toward r ¼ 0, and u s1g represents reflected waves from the valley so that in that region u s1 ¼ u s1g þ u s1c . u s2 represents the waves propagating away from the origin r ¼ 0, and the waves inside the valley are u v . The governing equation for out-of-plane SH waves that are valid in both regions is ∂ 2 ∂r 2 þ 1 r ∂ ∂r þ 1 r 2 ∂ 2 ∂θ 2   Uðr; θ; t Þ¼ 1 c 2 ∂ 2 ∂t 2 Uðr; θ; tÞ: ð1Þ The time dependence of the solution will be taken as harmonic so that Uðr; θ; tÞ¼ uðr; θÞe -iωt ; ð2Þ where ω is the angular frequency. When Eq. (2) is substituted into Eq. (1), there follows ∂ 2 ∂r 2 þ 1 r ∂ ∂r þ 1 r 2 ∂ 2 ∂θ 2   uðr; θÞe -iωt ¼ - ω 2 c 2 uðr; θÞe -iωt : ð3Þ Next, we introduce the wave number, k ¼ ω=c, which gives ∂ 2 ∂r 2 þ 1 r ∂ ∂r þ 1 r 2 ∂ 2 ∂θ 2 þ k 2   uðr; θÞ¼ 0: ð4Þ Contents lists available at SciVerse ScienceDirect journal homepage: www.elsevier.com/locate/soildyn Soil Dynamics and Earthquake Engineering 0267-7261/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.soildyn.2013.04.003 n Corresponding author. Tel.: +1 626 447 9382; fax: +1 213 744 1426. E-mail address: trifunac@usc.edu (M.D. Trifunac). Soil Dynamics and Earthquake Engineering 51 (2013) 9–13