The green function of an infinite, fluid loaded membrane

Abstract

In this paper the response of a fluid loaded plane structure (a membrane) to a concentrated line force excitation is considered in great detail. The normalized velocity response—here called the Green function G—depends upon a dimensionless range x 0= k m | x|, where k m is the free wavenumber on the membrane in a vacuum, on the Mach number M= k 0 k m , the ratio of wave phase speed ω/ k m on the unloaded membrane to the sound speed ω/ k 0, and on a parameter ϵ which can be regarded as a measure of fluid loading at the “coincidence” condition M=1. In the analogous problem involving a thin elastic plate, the corresponding parameter is independent of frequency and plate thickness and may be regarded as an intrinsic measure of fluid loading; moreover, in cases of common interest (steel in water, aluminium in air) that parameter is small. In the present paper, the asymptotic structure of G( x 0, M, ϵ) is therefore sought in the limit ϵ → 0. Naturally, no single asymptotic expansion can be expected to be valid throughout the ( x 0, M) plane, and the programme therefore involves the delineation of regions of that plane in which distinct asymptotic results apply, the construction and discussion of those results, and the asymptotic matching (according to the procedures of the method of matched asymptotic expansions) of results holding in adjoining regions. The Fourier integral for G is broken into surface wave and acoustic components, and the asymptotic structure obtained for each. Previously obtained results for the behaviour at large distances are recovered, with a demonstration that very large distances indeed ( x 0 ⪢ ϵ −2) may be needed for their validity for some ranges of M; and the drive point behaviour, of G( x 0=0, M, ϵ) as ϵ → 0 qua function of M, is shown to correspond to that already discussed in the literature. Elsewhere, in the covering of the whole ( x 0, M) plane by different asymptotic expressions, a wide variety of analytical results is found, reflecting the achievement in different regions of different balances among the five competing physical mechanisms represented in the model: namely, structural stiffness, structural inertia, fluid pressures, fluid compressibility and fluid inertia. These different balances give rise to a wide variety of expressions for the phase and amplitude of the surface wave and acoustic components which can now be used to isolate the dominant structural and acoustic mechanisms at any point in the ( x 0, M) plane.