Open AccessLegendre Functions, The Hilbert Transform and Surface Waves on a Sphere
Author(s) -
Ansell J. H.
Publication year - 1973
Publication title -
geophysical journal of the royal astronomical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.302
H-Index - 168
eISSN - 1365-246X
pISSN - 0016-8009
DOI - 10.1111/j.1365-246x.1973.tb06522.x
Subject(s) - legendre function , legendre polynomials , surface wave , mathematical analysis , legendre transformation , dispersion (optics) , surface (topology) , attenuation , mathematics , hilbert transform , expression (computer science) , physics , classical mechanics , geometry , optics , spectral density , statistics , computer science , programming language
Summary By the use of analytic expansions of the Legendre function expression P v ‐½ (−cos θ)/cos v π in terms of other Legendre functions, the Clemmow functions, it is possible to give a natural derivation of the expression of a surface wave on a sphere as a sum of travelling waves which have made multiple passages around the sphere. Results on the wave form changes as the waves pass through the axes and on the attenuation and dispersion of the waves as they travel are gained directly from the analytic expansions and do not depend on the use of asymptotic approximations. The method is applied to study the surface waves on a spherical cavity in an elastic medium.