Fixed time versus fixed range reverberation calculation: analytical solution.

Reverberation is commonly calculated by estimating the propagation loss to and from an elementary area, defined by transmitted pulse length and beam width, and treating the resulting backscatter from the area as a function of its range. In reality reverberation is strictly a function of time and contributions for a given time come from many ranges. Closed-form solutions are given for reverberation calculated both at fixed range and at fixed time isovelocity water and some variants of Lambert's law and linear reflection loss with an abrupt critical angle. These are derived by considering the shape of the two-way scattered multipath pulse envelope from a point scatterer. The ratio of these two solutions is shown to depend on the dominant propagation angle spread for the particular range or time. The ratio is largest at intermediate ranges (though typically less than 1 dB) and depends explicitly on the critical angle. At longer ranges mode-stripping reduces the propagation angle spread and the ratio reduces ultimately to unity. At short range the ratio is also close to unity although interpreting it requires care.