Standing waves in deep water: Their stability and extreme form

Abstract

A stable and accurate numerical method to calculate the motion of an interface between two fluids is used to calculate two-dimensional standing water waves. The general method calculates arbitrary time-dependent motion of an interface, possibly including interfacial tension and different density ratios between the fluids. Extremely steep standing waves are determined, significantly steeper than has been previously reported. The peak crest acceleration is used as the determining parameter rather than the wave steepness as the wave steepness is found to have a maximum short of the most extreme wave. Profiles with crest accelerations up to 98% of gravity are calculated (a sequence of raster images of this profile as it evolves in time over one period may be obtained upon application to the authors: e-mail gmercer@spam.ua.oz.au or aroberts@spam.ua.oz.au), and the shape of these extreme standing wave profiles are discussed. The stability of the standing waves is examined and growth rates of the unstable modes are calculated. It is found that all but very steep standing waves are generally stable to harmonic perturbations. However, standing waves are typically unstable to subharmonic perturbations via a sideband-type instability.

Different forms for nonlinear standing waves in deep water

“…The present investigation is restricted to standing waves of small to moderate heights. Mercer & Roberts (1992) also studied steep standing waves using a distribution of vortices on the water surface, in a semi-Lagrangian approach. All of the above authors restricted themselves to pure standing waves with a Stokes ordering of harmonic amplitudes.…”

Computation of three-dimensional standing water waves

“…4(b). By the time that the (1, 1) amplitude has been increased to 0.28, the wave heights clearly deviate from the linearized form, with the wave crests becoming sharper and wave troughs becoming wider, in a similar manner to the trends seen in two-dimensional computations [14,48]. Figure 4(c) shows a plot of the initial wave surface for the case when the (1, 1) amplitude is 0.24.…”

“…11(a), this is in close agreement with the computed results although some deviations become visible at higher amplitudes. In studies of two-dimensional standing waves [14,20], it has been useful to examine the relationship between the initial downward acceleration of the crest, and the wave steepness, defined as half of the crest-to-trough height difference. The dashed line in Fig.…”

Analytical solutions for steep standing waves

“…This is expected to underpredict limit wave conditions. As discussed in relation to Figure 8a, Mercer and Roberts (1992) predict a deep water limit standing wave of ω 2 H/g ≈ 1.14. This point is included on Figure 10, again as a box marker on the right.…”

“…This situation is reached at wave heights ω 2 H/g approaching 1 in deep water and at progressively lower wave heights in shallower water. For deep water (theoretically ω 2 h/g = ∞, but pragmatically ω 2 h/g 2.5), Mercer and Roberts (1992) predict that the limit standing wave steepness (defined as the present expansion parameter) is 0.6202, which corresponds to a deep water limit wave of ω 2 H/g ≈ 1.14. This limit wave prediction is the box marker in Figure 8a.…”