Integral model of the propagation of stationary potential waves in fluids

Abstract

Much attention in fluid mechanics is traditionally devoted to the study of potential waves. For these waves, the constructive nature of the classical analytical models of fluid flows was originally demonstrated in [1] and definitions of the main properties of waves (in particular, the concepts of the group and phase velocities) were introduced [2, 3]. The important role played by nonlinear effects (which, together with dispersion, determine the disturbed surface shape) was established [4], and effective methods for the investigation of complex equations were developed [5]. The current interest in studying potential waves is related to the search for a mathematical description of the complicated pattern of the disturbed sea surface [7], including the waves of an anomalously large amplitude (rogue or freak waves), which cause considerable damage to seaside structures and ships. The problems of stationary wave propagation are conventionally analyzed using both differential [6] and integral methods. The first integral equation was obtained in 1921 by Nekrasov [8, 9] for determining the eikonal dependence of the slope of fluid particle velocities under the assumption of a symmetric shape of these waves. The properties of this equation were investigated in detail in many studies and used as a basis for the proof of the existence of stationary nonlinear waves [10, 11]. Later, a more complicated integral equation describing the fluid velocity in a wave was derived and approximately analyzed [12]. Recently, a system that includes an integral equation for the wave height and a differential equation for the velocity potential on the free surface was obtained by Ablowitz et al. [13] in solving the main problem of wave theory, namely, determining the disturbed liquid surface shape. In this study, we present a new nonlinear integral equation governing the displacements of a heavy fluid surface, which allows one to investigate the propagation of a wide class of perturbations on this surface. Let us consider the propagation of two-dimensional waves over the surface of a homogeneous, ideal incompressible fluid of depth h . The system under consideration includes the Euler equation, the continuity equation, and the potentiality condition together with the conventional boundary conditions imposed on the disturbed surface and the plane bottom [14]: