Harmonic Green's functions for flexural waves in semi-infinite plates with arbitrary boundary conditions and high-frequency approximation for convex polygonal plates

Abstract

This paper provides a method for obtaining the harmonic Green's function for flexural waves in semi-infinite plates with arbitrary boundary conditions and a high frequency approximation of the Green's function in the case of convex polygonal plates, by using a generalised image source method. The classical image source method consists in describing the response of a point-driven polygonal plate as a superposition of contributions from the original source and virtual sources located outside of the plate, which represent successive reflections on the boundaries. The proposed approach extends the image source method to plates including boundaries that induce coupling between propagating and evanescent components of the field and on which reflection depends on the angle of incidence. This is achieved by writing the original source as a Fourier transform representing a continuous sum of propagating and evanescent plane waves incident on the boundaries. Thus, the image source contributions arise as continuous sums of reflected plane waves. For semi-infinite plates, the exact Green's function is obtained for an arbitrary set of boundary conditions. For polygonal plates, a high-frequency approximation of the Green's function is obtained by neglecting evanescent waves for the second and subsequent reflections on the edges. The method is compared to exact and finite element solutions and evaluated in terms of its frequency range of applicability.