Ray treatment of wave propagation on thin-walled curved elastic plates with truncations

Abstract

Thin elastic plates form the building blocks in whole, or in part, for many structures of current interest. The acoustic response of such plates can be found from thin plate or shell equations. It has recently been shown (A. D. Pierce, IUTAM Meeting, Galway, Ireland, 1988) how time-harmonic point force excitation of a thin circular cylindrical elastic tube of uniform thickness and composition in vacuum can be modeled equivalently as line source excitation in a homogeneous anisotropic medium, and how the exact Fourier spectral integral representation of the field in the radiation zone can be reduced by asymptotics to distinct ray contributions that account for the anisotropy. For the cylindrical geometry, the thin shell equations yield three fundamental wave types established, at high frequencies, by compression, bending, and shear. By appealing to concepts of spectral localization, these results are extended here to allow for weak deviations brought about by variable radius of curvature, variations in thickness and(or) material properties, and edge truncations or joints. The treatment is approximate, based on localization of wave phenomena that occur at sufficiently high frequencies. Modifications for fluid loading have also been considered. The tools are adiabatic spectral transforms as well as uniform and nonuniform asymptotics with their ray or mode interpretations, as performed within the context of an equivalent inhomogeneous anisotropic medium with boundaries or interfaces. In essence, the aim of this study is to formulate for the acoustic response of a class of curved sections of thin plates a geometrical theory of diffraction.