On eigenfunction expansions associated with wave propagation along ducts with wave-bearing boundaries

Abstract

A class of boundary value problems, that has application in the propagation of waves along ducts in which the boundaries are wave-bearing, is considered. This class of problems is characterised by the presence of high order derivatives of the dependent variable(s) in the duct boundary conditions. It is demonstrated that the underlying eigenfunctions are linearly dependent and, most significantly, that an eigenfunction expansion representation of a suitably smooth function, say $f(y)$, converges point-wise to that function. Two physical examples are presented. It is demonstrated that, in both cases, the eigenfunction representation of the solution is rendered unique via the application of suitable edge conditions. Within the context of these two examples, a detailed discussion of the issue of completeness is presented.