Three‐dimensional Green's function for fluid‐loaded thin elastic cylindrical shell: Formulation and solution

Abstract

This paper treats sound radiation from a time‐harmonic point pressure source located either inside or outside a thin, homogeneous, infinitely long circular cylindrical elastic shell, which is immersed in different interior and exterior fluid media. This Green's function problem is attacked by a combination of the method of separation of variables and the method of images applied to an infinitely extended azimuthal (φ) domain. The reduced one‐dimensional problems in the cylindrical (r, φ, z) coordinates are solved by general spectral techniques in terms of one‐dimensional characteristic Green's functions gr, gφ, gz, which depend on one or both of the two complex spectral separation parameters (spatial wavenumbers) λ1 and λ2. While the one‐dimensional problems in the φ and z domains are straightforward, the presence of the shell in the radial domain introduces substantial complexity. The solution is obtained by defining the discontinuities in the pressure and normal displacement across the shell via recourse to the dynamical equations of motion inside the shell. The synthesis problem is made unique through a complete analysis of the spectral singularities gr,φ,z in their respective complex planes, which permits selection of appropriate integration contours. A host of alternative representations, whose choice (concerning utility) is motivated by the parameter range of interest, can be derived from the fundamental spectral form, and asymptotic reductions lead to a variety of wave processes that have a cogent ray acoustic interpretation. [Work supported by ONR.]