Validity of the sonar equation and Babinet's principle for scattering in a stratified medium

The sonar equation rests on the assumption that received sound pressure level after scattering can be written in decibels as a sum of four terms: source level, transmission loss from the source to the target, target strength, and transmission loss from the target to the receiver. This assumption is generally not valid for scattering in a shallow water waveguide and can lead to large errors and inconsistencies in estimating a target's scattering properties as well as its limiting range of detection. By application of coherent waveguide scattering theory, the sonar equation is found to become approximately valid in a shallow water waveguide when the object's complex scatter function is roughly constant over the equivalent horizontal grazing angles +/- delta psi spanned by the dominant waveguide modes. This is approximately true (1) for all objects of spatial extent L and wavelength lambda when 2delta psi