Некоторые особенности решения уравнений гидродинамики для уединенных волн в открытом водном канале

Abstract

The opportunity of use of an impulse equation special form for the solving of a problem of solitary waves (solitons) occurrence in the open water channel is considered. It is shown that the used of an impulse equation allows take into account a role of surface tension and gravitational forces in formation of waves. Using of the continuity equation expansion into series on Rayleigh’s method the system of the differential equations is received, one of which is nonlinear. Application of Dalembert’s method for running waves for the solving of the nonlinear differential equation in a hydrodynamic problem of solitary waves spreading in the open water channel is considered. It is shown that as against Dalembert’s theory for the linear hyperbolic equations where initial conditions completely determine the form of arising waves, for the nonlinear equations the form of waves is determined by character of the equation nonlinearity. Thus during the solution of equations the sum of the functions describing linear waves extending in opposite directions, in the Dalembert’s method for nonlinear waves is replaced with the sum of the nonlinear differential equations.