Lamb quasi–modes in curved plates

Abstract

Wave propagation in slowly varying elastic waveguides is analysed in terms of mutually uncoupled quasi–modes. These are a generalization of the Lamb modes that exist in a uniform guide to a weakly non–uniform guide. Quasi–modal propagation is dependent upon the wavelength and two geometrical length–scales, that of the longitudinal variations and the guide thickness. By changing these length–scales one enters different asymptotic regimes. In this paper the emphasis is on the mid–frequency regime, where only a few propagating modes can exist. Our aim is to present an asymptotic theory for quasi–modal propagation in a canonical geometry, an arbitrarily curved two–dimensional plate of constant thickness. We derive practically useful asymptotic expressions of the quasi–modes of a weakly curved plate; these are particularly important since an adiabatic approximation for this problem coincides with the expression for the Lamb modes of a flat plate of the same thickness.