Stationary Nonlinear Potential Waves on the Surface of a Layer of Finite-Thickness Ideal Homogeneous Fluid. The First Stokes Method

A two-dimensional problem of stationary nonlinear waves on the surface of a layer of finite-thickness ideal fluid is considered. The solution to the problem using the proposed technique includes the following steps. Firstly, the stream function trace is used to change the kinematic condition on the free surface. Secondly, the Bernoulli–Cauchy integral is applied to present the dynamic condition in a new form. Thirdly, an integral operator of the convolution type is introduced, which allows one to simplify the nonlinear boundary-value problem of determining four functions of one variable, the main ones of which are a wave profile shape and a stream function trace at the surface level. This technique allows reducing the two-dimensional problem to a one-dimensional one. Two forms of the nonlinear dispersion relation are obtained: the dependence of the wave velocity on the amplitude of the fundamental harmonic of the wave and the dependence of the wave velocity on the wave amplitude. The cases of short and long waves are considered.