Normal Mode Expansion and Bessel Series

Abstract

The eigenfunctions of the Laplacian operator in function spaces with certain sets of boundary conditions constitute orthogonal sets of functions on the region enclosed by the boundaries. This is developed in Section 6.1 for rectangular boundaries and in Sections 6.2 and 6.3 for circular, sectorial, and annular boundaries in the plane. These are a few of the systems which appear in physics and engineering, where a great variety of operators and boundaries occur. The Laplacian applies mainly to wave and diffusion phenomena, which makes it specially relevant. As for boundary value problems, the above have been chosen for simplicity and because Fourier and Bessel series appear. Bessel series are a family of expansions in terms of orthonormal sets of functions which include those of Fourier as a particular case. In Section 6.4 we give a broad survey of the variants of eigenfunction expansions and some references.