Steep Sharp-Crested Gravity Waves on Deep Water

Abstract

A new type of steady steep two-dimensional irrotational symmetric periodic gravity waves on inviscid incompressible fluid of infinite depth is revealed. We demonstrate that these waves have sharper crests in comparison with the Stokes waves of the same wavelength and steepness. The speed of a fluid particle at the crest of new waves is greater than their phase speed. [5,6], requires knowledge of a form and dynamics of steep water waves. For the first time surface waves of finite amplitude were considered by Stokes [7]. Stokes conjectured that such waves must have a maximal amplitude (the limiting wave) and suggested that a free surface of the limiting wave near the crest forms a sharp corner with the 120• internal angle (the Stokes corner flow). A strict mathematical proof of the existence of small amplitude Stokes waves was given by Nekrasov. Toland [8] proved that Nekrasov's equation has a limiting solution describing a progressive periodic wave train which is such that the flow speed at the crest equals to the train phase speed, in a reference frame where fluid is motionless at infinite depth. Longuet-Higgins and Fox [9] constructed asymptotic expansions for waves close to the 120• -cusped wave (almost highest waves) and showed that the wave profile oscillates infinitely as the limiting wave is approached. Later, in [10], the crest of a steep, irrotational gravity wave was theoretically shown to be unstable.The purpose of the present work is to give evidence that a second branch of two-dimensional irrotational symmetric periodic gravity waves of permanent form exists besides the Stokes waves of the same wavelength. The original motivation is as follows: the Bernoulli equation is quadratic in velocity and admits two values of the particle speed at the crest. The first one corresponds to the Stokes branch of symmetric waves for which the particle speed at the crest is smaller than the wave phase speed. The opposite inequality takes place for the second branch which might correspond to a new type of waves. In the second part of the Letter, we prove this numerically by using two different methods.Consider a symmetric two-dimensional periodic train of waves which propagates without changing a form from left to right along the x-axis with the constant speed c relative to the motionless fluid at infinite depth. The set of equations governing steady potential gravity waves on * Electronic address: lukom@iop.kiev.ua a surface of irrotational, inviscid, incompressible fluid iswhere Φ(θ, y) is the velocity potential, η(θ) is the elevation of a free surface, and y is the upward vertical axis such that y = 0 is the still water level. We have chosen the units of time and length such that the acceleration due to gravity and wavenumber are equal to unity. As it follows from the Bernoulli equation (2), a solution may be not single-valued in the vicinity of the limiting point. Indeed, the particle speed at the crest q(0) is horizontal and is defined as follows:η(0) being the height of the crest above the still water level. Th...

Fractional Fourier approximations for potential gravity waves on deep water

“…Nevertheless, the existence of a set of additional stagnation points, which approach the central stagnation point O as the steepness is increased, makes us expect that a 120 • singularity in the Stokes corner flow is indeed formed from several (probably an infinite number of) coalescing 90 • singularities supporting the conjecture of Grant (1973). Lukomsky et al (2002b) have numerically revealed a new type of flows, where fluid particles move faster than the wave itself in the vicinity of the wave crest due to the stagnation point located inside the flow domain. Because of this such flows and waves were called irregular.…”

“…Note that the collocation method can be used instead but it is less efficient than expansion (18) (see Lukomsky et al, 2002b). The mean level η = η 0 should be zero for exact solutions.…”

“…In the framework of the canonical model of hydrodynamics, where fluid is assumed to be ideal and incompressible, waves are potential, two-dimensional, and symmetric, the authors have recently reported the existence of a new type of gravity waves on deep water besides well studied Stokes waves (Lukomsky et al, 2002b). The distinctive feature of these waves is that horizontal water velocities in the wave crests exceed the speed of the crests themselves.…”

Efficient computation of two‐dimensional steady free‐surface flows

“…A measure of this nonlinearity is that the maximum surface height is max = 0.175 which, for F L = 0.7, is roughly 71% of upper = F 2 L ∕2. While there has been extensive research devoted to studying the shape of a single highly nonlinear gravity wave, [25][26][27][28][29] it is particularly challenging to compute solutions to full free-surface flow problems (including the disturbance that causes the waves) with a train of waves with steepness of s > 0.1. Here, steepness is defined to be the wave amplitude divided by its wavelength.…”

Cnoidal-type surface waves in deep water

“…Indeed, for an exact wave, the streamlines (in the frame of reference where the flow is steady) intersect at a right-angle above the crest, forming a stagnation point (Grant 1973;Longuet-Higgins & Fox 1977, 1978Lukomsky et al 2002b). The stagnation point approches the surface as ka increases.…”