Shallow‐Water Waves. I: Theory

Based on a generalized treatment of geometric scaling laws inherent in the Boltzmann integral for nonlinear wave‐wave interactions, a theoretical framework for the characteristic form of equilibrium spectra in water of arbitrary depth is developed. A theoretical consequence of this spectral equilibrium range formulation is that a strong, constant flux of wave energy exists through the equilibrium range of the spectrum, analogous to the Komolgorov range in a turbulence spectrum. One possibly important conclusion from the theoretical material presented here is that the equilibrium range of a wind wave spectrum should be a depth‐independent constant in terms of a representation of the spectral action density, rather than the spectral energy density. Another point of particular interest to wave generation in shallow water is the apparent existence of a natural limit to the evolution of the frequency of the spectral peak into lower frequencies. This removes the necessity of requiring that only bottom friction can limit wave growth in shallow water.