Legendre Functions, The Hilbert Transform and Surface Waves on a Sphere

By the use of analytic expansions of the Legendre function expression Pν-½(−cos θ)/cos νπ 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.