Phase and group velocities of Rayleigh waves in a spherical, gravitating Earth

Abstract

Periods of spheroidal eigen vibrations, with order of spherical harmonic n≥20, have been computed for self-gravitating inhomogeneous spheres corresponding to a variety of earth models. The periods are used to deduce phase and group velocities for the fundamental and first higher modes of Rayleigh waves having periods less than 320 sec. The mathematical methods, program checks and estimations of numerical precision used in the work are presented in some detail. A comparison is made between phase and group velocities for different spherical models and with corresponding flat earth velocities calculated for the same physical parameters using the Thomson-Haskell matrix method for a nongravitating layered half-space. The comparison shows that the inclusion of gravity and sphericity increases the phase velocity by 0.25 km/sec (∼5 per cent) near T = 300 sec and by 0.10 km/sec (∼2.5 per cent) near T = 150 sec, where T is the period of the wave. In striking contrast, for 100 n>40) for a continental model. Extrema of the curve which may be expected to produce relatively large amplitude arrivals occur at a velocity of 4.30 km/sec and period of 60 sec (a minimum) and at a velocity of 4.54 km/sec and period of 25 sec (a maximum). The period and velocity of the latter agree well with the period and velocity of the phase Sa described by Caloi and by Gutenberg and of the phase Sn of Press and Ewing.