On the orthogonality of surface wave eigenfunctions in cylindrical coordinates

Abstract

SUMMARY 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.