Richard B. Evans
Sci. Appl. Internatl. Corp., New London, CT 06320
An example is given of a horizontally stratified acoustic waveguide that supports generalized eigenfunctions in addition to the usual eigenfunctions or normal modes. Generalized eigenfunctions occur when the characteristic equation has a zero with multiplicity greater than one. Both eigenfunctions and generalized eigenfunctions are required to provide a complete representation of a wide class of functions. The separation of variables solution for a point source in a waveguide, based only on the usual eigenfunctions or normal modes, is not valid for the example presented here. The correct solution, based on a combination of eigenfunctions and generalized eigenfunctions, contains modes that are simultaneously exponentially damped and algebraically amplified in the direction of propagation.