Normal modes, virtual modes, and alternative representations in the theory of surface-duct sound propagation

Abstract

This paper presents a theoretical study of sound propagation in an ocean-surface duct. It deals with several aspects of the theory from a point of view which has not heretofore been taken in the analyses of this problem. The model used to describe a duct assumes the ocean surface to be smooth and the square of the refractive index to be bilinear. Alternative representations of the sound field excited by a point source are derived, the two playing the most significant role in this paper being the residue series and the normal-mode representation. It is shown that the depth functions of the residue series do not form a complete set, as those of a normal-mode representation must, and that the normal-mode spectrum is continuous, rather than discrete. The completeness properties of the normal-mode functions are then utilized in a study of the energy-trapping capabilities of the duct. In this connection, virtual modes are introduced and shown to lead naturally to the derivation of a leakage coefficient characteristic of the exponential leakage of energy out of the duct with increasing range. In addition, a cutoff-frequency criterion, useful in determining when a surface duct can trap energy, is derived.