Nonstandard Integral Equations for the Harmonic Oscillations of Thin Plates

Abstract

In [ThCo97] and [ThCo09a] the problems of high frequency harmonic oscillations of thin elastic plates with Dirichlet, Neumann, and Robin boundary conditions were investigated by means of a classical indirect boundary integral equation method. This method was not entirely satisfactory since, for the exterior problems, it produced integral equations with nonunique solutions for certain values of the oscillation frequency, although the actual boundary value problems always had at most one solution. When a direct method was employed (see [ThCo99] and [ThCo10]), it was found that uniqueness could be guaranteed only if a pair of integral equations was derived for each exterior problem. The classical techniques did not seem to offer any answer to the question of whether the solutions could be obtained from single, uniquely solvable equations. Below we propose a modified indirect boundary integral equation method, based on constructing a matrix of fundamental solutions satisfying a dissipative (or Robin-type) condition on a curve interior to the scatterer, which answers the above question in the affirmative.