On the orthogonality of surface wave eigenfunctions in cylindrical coordinates

Abstract

The orthogonality of the Rayleigh wave eigenfunctions in laterally homogeneous, plane‐stratified media is guaranteed by the structure of the ordinary differential equations describing elastic wave motion in cylindrical coordinates. This coupled first‐order system is identical to that which characterizes 2‐D plane wave propagation in a Cartesian coordinate reference frame. The orthogonality relation in 2‐D can also be derived from energy considerations; however, an analogous argument in cylindrical coordinates has not hitherto been made. We derive the orthogonality relations for Rayleigh waves in 3‐D from energy considerations and demonstrate that the standard (2‐D) expression is, in fact, the generalization of a slightly more specific form. In addition, the cylindrical coordinate formulation permits the derivation of a functional orthogonality relation between Love and Rayleigh waves. The normalization of Love and Rayleigh wave eigenfunctions in cylindrical coordinates is shown to be related to the energy transport of a given outgoing Fourier–Bessel component across any surface which wholly encompasses the z‐axis. Copyright © 1990, Wiley Blackwell. All rights reserved