Closed-form expressions for ocean reverberation and signal excess with mode stripping and Lambert's law

Closed-form expressions for two-way propagation and reverberation in variable depth ducts are derived for isovelocity water by using ray invariants and acoustic flux. These expressions include the transition to single mode propagation at long range. Three surface scattering laws are considered: Lambert, Lommel-Seeliger, and angle independent, and these are compared with a point target to give explicit signal-to-reverberation ratios. In particular, there is interesting and sometimes surprising behavior when the propagation obeys mode-stripping (the high angles are preferentially attenuated by bottom losses) whilst the scattering obeys Lambert's law (high angles are preferentially back-scattered). There may be conditions where the signal-to-reverberation ratio is independent of range so that there is no reverberation range limit. Bottom slope dependence of both target echo and reverberation is surprisingly weak. The implications of refraction are discussed. The angle dependence for a point or surface scatterer at a given range can be translated into arrival time, so it is possible to calculate the received pulse shape for one-way or two-way paths. Because the tail is exponential with a range-independent half-life that only depends on bottom reflection properties there is scope for extracting geoacoustic information from the pulse shape alone. This environmental time spread is also of use to sonar designers.