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We apply the idea of choosing new variables that are nonlinear functions of the old in order to simplify calculations of irrotational, surface gravity waves. The usual variables consist of the surface elevation and the surface potential, and the transformation to the new variables is a canonical (in Hamilton's sense) one so as to maintain the Hamiltonian structure of the theory. We further consider the approximation of linear dynamics in these new variables. This approximation scheme exactly reproduces the effects of the lowest-order nonlinearities in the usual variables, does well at higher orders, and also captures important features of short waves interacting with longer waves. We provide a physical interpretation of this transformation which is correct in the one-dimensional case, and approximately so in the two-dimensional case.

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We present an analysis of acoustic wave propagation in a random shallow-water waveguide with an energy absorbing sub-bottom, in which deviations of the index of refraction are a stochastic process. The speci c model we study is motivated by the oceanic waveguide in shallow waters, in which the sub-bottom sediment leads to energy loss from the acoustic eld, and the stochastic process results from internal (i.e. density) waves. In terms of the normal modes of the waveguide, the randomness leads to mode coupling while the energy loss results from di erent attenuation rates for the various modes (i.e. mode stripping). The distinction in shallow water is that there exists a competition between the mode coupling terms, which redistribute the modal energies, and mode stripping, which results in an irreversible loss of energy. Theoretically, we formulate averaged equations for both the modal intensities and uctuations (the second and fourth moments of acoustic pressure, respectively), similar to previous formulations which, however, did not include the e ects of sub-bottom absorption of acoustic energy. The theory developed here predicts that there is a mismatch between decay rates between the second and fourth moments, implying that the scintillation index (which is a measure of the strength of the random scattering) grows exponentially in range. Thus the usual concept of equilibrium or saturated statistics must be modi ed. We present Monte Carlo simulations of stochastic coupled-mode equations to verify these theoretical conclusions.

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