ABSTRACT
Inhomogeneous plane waves as solutions of the (Fourier transformed) wave equation for isotropic nondissipative materials (Section 8.2) have found physical realizations concerning reflection and transmission of plane waves at the planar boundary between such materials or at their planar (stress-free) surface, respectively (for electromagnetic waves, this topic is called Fresnel’s reflection): The existence of up to three critical angles yields an equal number of evanescent plane waves (Sections 9.1.2, 9.2.1, and 9.2.2). Plane waves on one selected side of the boundary exhibit at most two critical angles [Figures 9.19 (SVc)–(SVa′)], one for the pressure and one for the shear wave, where the amplitudes of the resulting evanescent waves are given by complex transmission coefficients. However, a stress-free surface only allows for one evanescent pressure wave with an amplitude given by a complex mode conversion coefficient. The question leading to Rayleigh surface waves is the following: Does the homogeneous wave equation allow for a solution in terms of the superposition of two evanescent plane waves along the stress-free surface of an isotropic nondissipative material, namely in terms of the superposition of an inhomogeneous pressure and an inhomogeneous shear wave? As for the case of a boundary separating two materials, both waves should propagate with the same phase velocity along the surface, however with different attenuation constants. What we already know: Neither an incident plane pressure nor an incident plane shear wave can excite such a “free” surface wave!
