Coupled mode analysis of multiple rough surface scattering

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Abstract
The range dependent boundary condition imposed by a rough surface serves to couple the normal modes of an underwater acoustic field. Beginning with the parabolic wave equation, the solution is expanded in terms of normal mode depth functions modulated by random amplitudes having a quasistatic time dependence. A system of master equations for the quantities 〈an(r,t) a*n(r,t+τ) 〉 is derived, where the an(r,t) are the normal mode amplitudes. It is found that for τ≠0, the sideband spectra (analytically obtained by Fourier transformation of the temporal coherence function with respect to τ) has information with regard to the directionality of the wind driving the rough surface. For τ=0 we obtain a system of equations for the mean power in each mode describing the transfer of energy between modes. Explicit formulas for the coupling coefficients (transition probabilities) are obtained in terms of the spectrum of the rough surface. The formalism is applied to multiple ocean surface scattering to long ranges. Transmission loss, intensity redistribution, and the approach to equipartition of energy as a function of range and depth are obtained numerically. Relaxation ranges characterizing equipartition are predicted for various physical conditions.

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