Ray acoustics models for sound radiation from vibrations of fluid‐loaded circular shells

Abstract

A theory of sound radiation by shells is developed using the canonical model of an elastic coaxial cylinder, with inner radius R − 12h and outer radius R + 12h immersed in a compressible fluid. As can be constructed from the earlier work of Pochhammer, Mindlin, Herrmann, Miklowitz, Greenspon, Veksler, and others, this dynamical system admits guided wave solutions of constant angular frequency ω, in which all of the appropriate field variables vary with azimuthal angle θ and axial coordinate xthrough a common factor Eih einθ Admissible solutions that do not grow at large positive x or at large r are governed by a dispersion relation of the form F(ω,kv, n) = 0, where F is a highly transcendental function involving Bessel and Neumann functions of various orders and with various arguments. In the present paper, n is not required to be an integer and kv = n/R is identified as another wavenumber component. In various limits, all with h/R ⩽ 1, simpler analytic approximations for possible dispersion relations result, which can be cast in the form k = k (ω,θλ), where wavenumber k for phase propagation in direction θλ has a positive non‐zero imaginary part, which is interpreted in terms of sound radiation into the surrounding fluid. For length and frequency scales relevant to structural acoustics, the most important dispersion relations are those that correspond to longitudinal, shear, and flexural waves on plates, only with corrections to account for fluid loading and curvature. All modes radiate sound. [Work supported by ONR and by the William E. Leonhard endowment to Penn State Univ.]