Omni-directional Rayleigh, Stoneley and Scholte waves with general time dependence

Just as uni-directional Rayleigh waves at the traction-free surface of a transversely isotropic elastic half-space and Stoneley waves at the interface between two such media may have arbitrary waveform and may be represented in terms of a single function harmonic in a half-plane, it is shown that surface-guided waves travelling simultaneously in all directions parallel to the surface may be represented, at each instant, in terms of a single function satisfying Laplace's equation in a three-dimensional half-space. That harmonic function is determined so that its normal derivative at the surface equals the normal displacement of the surface (or interface). It is shown, moreover, that the time evolution of that normal displacement may be any solution to the membrane equation with wave speed being equal to that of classical, uni-directional, time-harmonic Rayleigh or Stoneley waves. A similar representation is also shown to exist for Scholte waves at a fluid-solid interface, in the non-evanescent case. Thus, every surface-or interface-guided disturbance in media having rotational symmetry about the surface normal is governed by the membrane equation with appropriate wave speed, provided that the combination of materials allows uni-directional, time-harmonic waves that are non-evanescent. Conversely, each solution to the membrane equation may be used to construct a representation of either a Rayleigh wave, a Stoneley wave or a (non-evanescent) Scholte wave. In each case, the disturbance at all depths may be represented at each instant in terms of a single function harmonic in a half-space.