Echo and reverberation in a Pekeris waveguide by convolution and by the product rule

The detection performance of an active sonar depends on the intensity of the signal (target echo) relative to that of a background of reverberation plus noise. The echo is calculated for a standard test problem by convolving the time-domain impulse response at the target position with itself. The same approach is applied to a closely related test problem for reverberation by integrating over scatterers at all ranges. The result is compared with a widely used rule whereby the reverberation intensity is approximated by integrating the product of the source, propagation, and scattering factors over grazing angle. The error resulting from this approximation, which increases with increasing grazing angle and tends to infinity as the upper limit of integration tends to π/2, can be corrected by including a simple trigonometric multiplying factor in the integrand.