The sound pressure field in the ocean due to bottom interacting paths

The pressure field due to sound reflected from the sea-bed is calculated by converting a Normal Mode sum into a sum of integrals, each representing the contribution from a particular ray path, by means of a Poisson sum. Each integral is then evaluated analytically using the stationary phase approximation, first for a simple reflecting bottom, and then for a refracting sediment with constant sound speed gradient. A comparison is made between the resulting expressions and a ray theory calculation, and apart from the phase they are shown to be identical except in the vicinity of caustics where the ray theory breaks down. These simple formulae are shown to be accurate solutions to the wave equation by demonstrating agreement with numerical predictions by the computer program SAFARI. This is done for a number of example problems including the transition through the caustic in the bottom refracted field.