A TIME-DEPENDENT NONLINEAR MILD-SLOPE EQUATION FOR WATER-WAVES

A weakly nonlinear and dispersive water wave equation, which in linear ized form yields a new version of the time-dependent mild-slope equati on of Smith & Sprinks (1975), is derived. The applicable spectral widt h of the new wave equation for random waves is found to be more satisf actory than that of Smith and Sprinks (1975). For very shallow depths the equation reduces to the combined form of Airy's nonlinear non-disp ersive wave equations; if the lowest-order dispersion is retained it p roduces the combined form of Boussinesq's equations. In the deep-water limit the equation admits the second-order Stokes waves as analytical solutions. Furthermore, by introducing a right-moving coordinate tran sformation, the equation is recast into a unidirectional form, renderi ng the KdV equation in one limit while reproducing the second-order St okes waves in the other.