On steady composite capillary-gravitational waves of finite amplitude: PMM vol. 39, n≗2, 1975, pp. 298–305

Abstract

Steady waves of finite amplitude induced by pressure periodically distributed along the fluid surface and disappearing when this pressure becomes constant are called here induced waves. Steady waves of finite amplitude occurring under constant surface pressure at particular velocities of the stream are called free waves. Induced capillary-gravitational waves at the surface of an infinitely deep fluid were investigated in [1, 2], while similar free waves were analyzed in [3,4]. The possibility of concurrent existence of both kinds of such capillary-gravitational waves of small but finite amplitude at some particular velocity of the stream at infinite depth is considered below. These waves are called composite waves. When the varying component of pressure distributed over the surface becomes identically zero, these waves do not disappear but are transformed into free waves. The problem is considered in a rigorous formulation and reduces to the solution of three nonlinear equations one of which is integral and the remaining two transcendental. The pressure at the surface is defined by an infinite trigonometric series whose coefficients are proportional to integral powers of some dimensionless small parameter, which are by two units higher than their index number. The theorem of existence and uniqueness of solution is established, and the method of Its proof is indicated. Derivation of solution in the form of series in powers of the small parameter mentioned above with any approximation is described. The first three approximations are completely calculated. An approximate equation of the wave profile is presented. Purely gravitational composite waves were considered in [5].