Closed form bistatic reverberation and target echoes with variable bathymetry and sound speed

Some existing closed-form expressions for target echo and reverberation, assuming classical mode-stripping and Lambert's law in isovelocity water, are extended to the case of uniform slope bathymetry combined with range-independent linear sound speed and bistatic geometry. The analytical, or extremely fast numerical, calculation of signal-to-reverberation-ratio provides insight useful for sonar design, and operational research. There are three types of path, each of which may be important at some range: Those that only interact with one boundary, those that always interact with both boundaries and those that sometimes interact with both. Each provides a separate closed-form contribution. The earlier finding, that with Lambert's law the reverberation tends to follow the same range trend as the target echo, is slightly modified by the addition of refraction. The stronger reverberation from the low sound speed side of the duct tends to fall off slightly less slowly than the target echo according to the strength of the refraction. Some graphical examples show that the very large parameter space can be handled very easily