DIFFERENTIAL METHOD OF MODELING WEAKLY NO NLINEAR-WAVES ON WATER OF VARIABLE DEPTH

The starting equations of hydrodynamics are reduced to two equations f or a three-dimensional perturbation of the free ocean surface and the horizontal component of the fluid-speed vector averaged over the water depth. This system forms a mathematical basis for studying waves of a rbitrary length in an ocean with a gently sloping bottom. The evolutio n equation for the fluid level is derived under the condition that the perturbation varies slowly in a reference frame moving with the wave (quasi-stationary approximation for perturbations). The linearized for m of this equation has only neutrally stable solutions. The characteri stic speed of propagation can vary over a wide range. In the case of a horizontal bottom, the model equation possesses plane solutions of th e Stokes-wave type, which agree with the results previously obtained f or deep and shallow water. Sufficiently long perturbations can also be cnoidal waves. In particular, solitary solutions are in close agreeme nt with the solutions of the Korteweg-de Vries and Boussinesq equation s. Two equilibrium states are found, and their stability is discussed from first (linear) approximations. From this, the upper boundary of t he solitary-perturbation. amplitude is estimated. The classic problem on the smooth passage of a linear monochromatic wave from deep water i nto shallow water is considered as a test example. It is shown that th e error in determining the amplitude does not exceed 4%.