Open AccessOn the orthogonality of surface wave eigenfunctions in cylindrical coordinates
Author(s) -
Bostock M. G.
Publication year - 1990
Publication title -
geophysical journal international
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.302
H-Index - 168
eISSN - 1365-246X
pISSN - 0956-540X
DOI - 10.1111/j.1365-246x.1990.tb05688.x
Subject(s) - bipolar coordinates , eigenfunction , cartesian coordinate system , orthogonal coordinates , rayleigh wave , orthogonality , mathematical analysis , coordinate system , cylindrical coordinate system , mathematics , physics , classical mechanics , spherical coordinate system , geometry , wave propagation , eigenvalues and eigenvectors , optics , quantum mechanics
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.