Stationary phase evaluation of the bottom interacting field in isovelocity water

Abstract

The near surface sound pressure field in an isovelocity ocean is calculated analytically using ray theory and then normal mode theory for two different bottom types: first, a simple reflecting bottom with constant density and sound speed for which the two methods give identical solutions; then, a refracting bottom with constant sound-speed gradient. In the latter case the two solutions differ only (a) by a cumulative π/2 phase difference in the transition through each caustic, and (b) in the vicinity of the caustics themselves where the ray theory is inapplicable. The normal mode sum is evaluated by means of a Poisson sum, each term of which is then integrated using the stationary phase approximation, giving closed form expressions for the contribution from each ray path (or group of ray paths) that are identical to the ray approximation (apart from the phase) away from caustics, and provide a uniformly valid solution through the caustics. The resulting formulas are used to simplify the interpretation of complicated interference effects often predicted by numerical models. Specifically, comparisons are made with numerical predictions using the fast field method at 25 and 250 Hz, showing close agreement through caustics as well as in shadows for a 1000-m water depth.