Diffraction of Elastic Waves by a Spherical Surface

To clarify the effect of the curvature of a surface on the propagation of elastic waves generated at a point source, as alluded to by Lamb in the case of a plane surface, we investigate the properties of waves in an infinite elastic medium enclosing a spherical cavity. Waves of any kind are shown to be represented as linear combinations of \(\), A ϕ , d ϕ , the scalar potential, the ϕ-component of the vector potential and the ϕ-component of the displacement, if the phenomena are independent of ϕ, the longitudinal angle in spherical polar coordinates with the polar axis joining the source and the center of the cavity. Taking as the initial waves \(\), A ϕ , d ϕ with monochromatic dipole character respectively, we obtain, in the ordinary way of mathematical physics, the solutions satisfying the boundary conditions in series forms, which are transformed into more rapidly convergent ones. In the first two cases we find the waves representing \(\), A ϕ in three separated groups, corresponding to three types in the case of a plane surface respectively, and in the third we find only group of solutions representing d ψ . The angular and distance dependence of their amplitudes and phases are investigated in detail.