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.

A prony algorithm for shallow water waveguide analysis

“…An interesting trait of the algorithm is the limited ability to track modes past cutoff. These virtual modes may be modelled as highly damped modes [19,20,67]. The energy from the virtual modes is "dumped" into the penetrable bottom; the detectable amplitude will significantly decrease for each range block until that mode can no longer be detected.…”

A simple expression for sound attenuation due to surface duct energy leakage in low-latitude oceans

“…The energy leakage is important at frequencies lower than the cutoff frequency since it accounts for signals observed in the geometrical shadow zone beneath the duct. [8][9][10][11][12] In addition, because of the leakage, the sound energy in the duct decays much quicker with range than geometric spreading. This paper focuses on the leakage attenuation and aims to present a simple expression for it.…”

“…Finally, the attenuation monotonically decreases with frequency until a duct-thickness-related critical frequency, above which the attenuation slowly increases. 8,9 The critical frequency is generally a little higher than the cut-off frequency for the duct.…”

“…However, the possible form of an expression for the attenuation is not explicit from these simulations. To determine this, we combine the two parameters into one parameter, as inspired by analysis of the cut-off frequency, 8 and then the form of the expression for the single-parameter-dependent attenuation is clear. First, we note that the cut-off frequency f c ðhÞ below which no mode is trapped in the duct is proportional to the thickness, h, to the power À1.5 for a bilinear SSP.…”

Mathematical Foundations for Normal Mode Modelling in Waveguides

“…The Hankel functions are related to the Bessel function as follows, In order to establish the behavior of the integrand in the 7-plane, we recall that the definitions of 91 and 02 from equations (18) lead to the branch points given below, …”