A general theory of scattering by thin interface layers, plates, and shells.

Abstract

In recent years there have been a number of attempts to model the effect of thin interface layers on the scattering and transmission of sound by bodies submerged in a continuous medium. Interface layers or coatings occur in numerous engineering acoustic applications. For example, when two solid bodies are bonded together, the adhesive or the imperfect fusion region will generally introduce an acoustic mismatch with both partners. Similarly, the use of austenitic cladding as a protection against corrosion creates an analogous problem, except that here the layer should be thought of as being fused to the surface of the material. In underwater and aeroacoustics the fluid–solid interactions with a shell or plate are often modeled using the Euler–Bernoulli thin plate theory. As far as the scattering problem here is concerned, the thin plate equation appears as an ‘‘effective boundary condition’’ relating the transverse displacement to the fluid pressure. Similarly, a thin interface layer between two solids is also modeled by some effective boundary condition. A typical approach is to assume that the layer is characterized by interfacial mass, spring, and damping constants. The latter are estimated by comparing the predictions of the model with the exact solutions to certain statical canonical problems. The difficulty with this approach is that, generally, it is not self-consistent. In other words if the spring constant is calculated so that the predictions of the model agree well with certain exact solutions at ‘‘normal incidence,’’ say, then it is usually not the case that they give agreement under different loading conditions. Similarly, in the fluid–solid interaction problem, the Euler–Bernoulli theory gives a correct approximation to the dispersion relation for leaky antisymmetric Rayleigh–Lamb waves in a fluid-loaded thin plate but incorrectly predicts the transmission and reflection of a plane wave by such a plate in the fluid! In this paper these problems are addressed by developing a rational approximation scheme for an arbitrary thin layer immersed in a contrasting matrix. a)The author is currently on leave from The Department of Mathematics, University of Manchester, Manchester M13 9PL, U.K.