A general asymptotic theory is developed to describe the acoustic response of heavily fluidloaded 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

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... 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. outer field punctuated by sharp fluctuations arising from the inner or resonance field. A straightforward extension of the concept of a rigid background response is not equally rewarding when applied to thin shell structures. •-7 As an example consider an infinitely long cylindrical thin shell in water subjected to a time harmonic acoustic plane wave. The far-field backscattered amplitude of the acoustic pressure field is plotted versus kR in Fig. 1 , where k = co/c, to is the circular excitation frequency, c is the fluid sound speed, and R is the radius of the cylinder. The sharp lines indicate resonances associated with extensional and flexural wave motion on the shell. It seems fairly clear that neither the rigid nor the soft boundary condition is adequate to model the shell response away from resonances, because, if either background were indeed representative of the actual field one would expect the response to drop almost down to zero between resonances, which is not the case from Fig. 1, although it is worth noting that at lower values of kR the soft or pressure release boundary actually performs better than its rigid counterpart. The same failure is found if the matched asymptotic algorithm 3 is used to generate the total response from a thin shell, 8 and the reason can be crudely explained by the presence of another small parameter in the problem, viz., the ratio h/R, where h is a typical shell thickness and R is the radius of curvature. In practice, and in the examples considered in this paper, this ratio is far smaller than the impedance ratio, and, hence, any asymptotic approximation based only upon the latter is doomed to failure. Recently, however, Gaunaurd and 3321 J. Acoust. Soc. Am., Vol. 92, No. 6. December 1992 A. Norris and N. Vasudevan: Scattering from thin shells 3321 112-11, 2, + •.h.(kR) This result corresponds to model I, defined in Sec, III. Model II results from evaluating d,. at to = tom. In order to appreciate the precision in either case, we note that the exact solution can be manipulated into a somewhat similar form by using the identities in Appendix D, to give 3329