A note on the dispersion of Love waves in layered monoclinic elastic media
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A note on the dispersion of Love waves in layered monoclinic elastic media SARVA JIT SINGH, N E E L A M SACHDEVA and SANDHYA KHURANA Department of Mathematics, Maharshi Dayanand University, Rohtak 124001, India E-mail: sj singh @mdul.ernet.in MS received 12 November 1998; revised 5 June 1999 Abstract. The dispersion equation for Love waves in a monoclinic elastic layer of uniform thickness overlying a monoclinic elastic half-space is derived by applying the traction-free boundary condition at the surface and continuity conditions at the interface. The dispersion curves showing the effect of anisotropy on the calculated phase velocity are presented. The special cases of orthotropic and transversely isotropic media are also considered. It is shown that the well-known dispersion equation for Love waves in an isotropic layer overlying an isotropic half-space follows as a particular case. Keywords.
Dispersion; half-space; Love waves; monoclinic media.
1. Introduction The study of surface wave dispersion in an isotropic half-space containing anisotropic layers is important in seismology for determining the presence or absence of anisotropic layers within the Earth. Such studies play a significant role in in-seam seismic exploration as well. The propagation of surface waves in an anisotropic half-space has been considered by many investigators. Wave propagation in a half-space with cubic symmetry has been discussed by Buchwald and Davies [3], and with orthorhombic symmetry by Stoneley [10]. Elastic wave propagation in transversely isotropic media has been reviewed by Payton [8]. Van der Hijden [12] discussed in great detail the propagation of transient elastic waves in stratified anisotropic media. Recent investigations on the propagation of elastic waves through anisotropic media include, among others, papers by Mench and Rasolofosaon [7], Savers [9] and Thomsen [11]. In an isotropic medium, SH type motion is decoupled from the P-SV type motion [6]. Surface waves of the SH type are known as Love waves and surface waves of the P-SV type are known as Rayleigh waves. The dispersion relation for Love waves in an isotropic elastic layer of uniform thickness H overlying an isotropic elastic half-space can be written in the form ([2], Sec. 3.6.2) tan
kH
-
1
1
= tz2
,1
where k denotes the wave number, c the phase velocity,/Zl and/~2 the rigidities of the layer and the half-space, respectively, and 31 and 32 the shear-wave velocities of the layer and the half-space, respectively (/3a < c H) is designated as medium (2) with displacement ul2)(xz, x3, t), density P2 and elastic constants d O. A monoclinic medium has one plane of elastic symmetry [4]. We assume that the plane of symmetry is par~lel to the x2x3-plane. For Love waves propagating in the positive x2-direction with phase velocity c, we assume Ul 1) = f(x3) exp[ik(x2 - ct)].
(2)
The horizontal displacement ul 1) satisfies the equation
02Ul . 02Ul O~Ul 02Ul C66 "-~"-X~-'b 2C56 ~ "~--C55 "-~3 = Pl Ot2 9
(3)
Equations (2) and (3) yield c55ftt(x3) d- 2c56ikft(x3)
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