Variational nature of dispersion equations revisited

Abstract

Corrections and clarifications are made of some past treatments of the variational nature of the eigenfrequency calculation for dispersion equations and new results are presented. The main conclusions are the following: (1) Any relation between a normal mode and its dual must be consistent with the fact that the boundary conditions satisfied by the normal mode may differ from the adjoint boundary conditions satisfied by the dual. This will affect whether or not a given bilinear form will yield a variational result for the eigenfrequency. (2) If a dispersion matrix is constructed from the dispersion operator by using left and right basis functions that satisfy homogeneous boundary conditions on the dual eigenfunction and the eigenfunction, respectively, then generally a second-order accurate eigenfrequency is obtained by solving the matrix form of the dispersion equation. (3) When solving for the normal modes in terms of perturbation potentials, the adjoint boundary conditions are gauge dependent. For cases where the adjoint boundary conditions allow only the trivial solution for the dual eigenfunction, it may be possible to obtain variational results for the eigenfrequency by requiring that the trial functions for the normal mode and its dual satisfy variational boundary conditions.