Acoustic wave scattering from thin shell structures

Abstract

A general asymptotic theory is developed to describe the acoustic response of heavily fluid-loaded thin shells in the midfrequency regime between the ring and coincidence frequencies. The method employs the ideas of matched asymptotic expansions and represents the total response as the sum of an outer, or background response, plus an inner or resonant contribution. The theory is developed for thin shells with smoothly varying material and geometrical properties. First, a suitable background field is found which satisfies neither the rigid nor the soft boundary conditions that have been typically employed, but corresponds to an impedance boundary condition. The background field is effective throughout the midfrequency as well as the strong bending regimes. The corresponding inner or resonance field is also valid in the same range. The approach taken is to represent these fields as inverse power series in the asymptotically small parameter 1/kR, where R is a typical radius of curvature of the shell and k is the fluid wave number. The leading-order terms in the series differ in the inner and outer expansions, in such a way that the displacement tangential to the surface is negligible in the outer (background) region, but dominates the scattering near resonances. The resonances can therefore be associated with compressional and shear waves in the shell. A uniform asymptotic solution is derived from the combined outer and inner fields. Numerical results are presented for the circular cylinder and the sphere and comparisons are made with exact results for these canonical geometries. The results indicate that the method is particularly effective in the midfrequency range. The strong bending regime is also well represented, especially for cylindrical scatterers.