Combined volume and surface scattering in a channel using a modal formulation

In previous work, a modal approach was used to study random volume scattering in a shallow channel [M. J. Beran and S. Frankenthal, J. Acoust. Soc. Am. 91, 3203-3211 (1992)]. Here, it is shown how to include the effects of a rough channel surface in the formulation. To include the effects of a rough surface, the modes are taken to be dependent on the range and transverse coordinates in addition to the depth coordinate. The propagation is studied in terms of the ensemble-averaged two-point coherence function and the equation governing the coherence function is derived. The general method is given in the above-cited article on volume scattering. In the present paper, it is shown how the surface scattering terms may be treated using the modal approach. In order to insure energy conservation when the generalized modal field equations are simplified, the parabolic approximation is replaced by a method which includes both forward and backward propagating fields. The two-point coherence function is expressed as the sum over both self-modal and cross-modal coherence functions. The differences between the equations governing the self-modal coherence functions and the cross-modal coherence functions are considered. A numerical example is presented which uses typical shallow water parameters. Figures are presented to show how the mode energies are transferred between the modes as the acoustical field propagates. The difference between the modal transfer of energy for volume scattering and for surface scattering is discussed.