Acoustic waves in a medium bounded by curved surfaces

A method for obtaining quasi-analytic solutions to the three-dimensional (3D) Helmholtz equation for the case of an acoustic medium bounded by two identical curved surfaces is presented. The method can be extended to a semi-infinite medium with a curved boundary for the study of Rayleigh waves on a non-planar surface, albeit the solution procedure entails the numerical matrix method. The formulation of the method is based on the differential-geometry argument employing the curved coordinates (u1,u2,u3u1,u2,u3) where u1u1 and u2u2 are along the local tangent plane of one of the bounding surfaces z=z(x)z=z(x) and u3u3 is perpendicular to the local tangent plane. This choice of coordinates allows the 3D Helmholtz equation, subject to boundary conditions specified on non-planar surfaces, to be solved with relative ease. Normal-mode solutions are shown for the case of a fluid layer with two pressure-release boundaries, where the bounding surfaces are given by the ramp, the Gaussian, and the sinusoidal functions, respectively.