Rayleigh Waves in a Two-Layer Heterogeneous Medium

A method is developed for the solution of the equations for Rayleigh waves in a semi-infinite incompressible medium in which a crust of rigidity varying linearly with depth lies on top of a uniform elastic medium of great depth. The method is extended to deal with compressible media and in both cases is applied to a model Earth consisting of a crust of depth 37.5 km. in which the rigidity increases linearly from 2.3 × 1011 to 4.53 × 1011 dynes/cm.2 over the ultrabasic material of infinite depth in which the rigidity is constant and equal to 6.47 × 1011 dynes/cm2. The variation of wave velocity with wavelength is obtained numerically, the same features emerging as in the problem of two homogeneous layers, i. e. a distinct minimum group velocity when the disturbance is of wave-length approximately twice the depth of the upper layer and, in the compressible case, an additional less pronounced minimum for wave-length about six times the depth of the layer. Two sets of readings taken at Göttingen are chosen for comparison. With the second at least there is no real agreement, so that the effect of adjusting the constants of our trial model arises. The applicability of the theory to the problem of microseisms is considered and it seems that it may be found useful in the study of those microseisms of period a fraction of a second. Finally, X(1), X(2), those solutions of a particular equation which it is found convenient to use here, are related to the appropriate Whittaker confluent hypergeometric function, already known to be a solution.