Surface Waves over a Slope

Surface waves motion over a slope is studied within the shallow water approximation and solutions of the equation are expanded in series in wave amplitude. It is shown that if a linear solution is known, the nonlinear corrections can be obtained in the terms of it’s derivatives. If the amplitude is sufficiently small, the constructed sequence converges to an exact solution of the shallow water equation for a wide class of problems. The quadratic problem is considered and it is shown that the nonlinear wave is of the bigger amplitude and steepness than the linear one.