Twenty-First Symposium on Naval Hydrodynamics (1997) / Chapter Skim
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Progress Toward Understanding How Waves Break
Progress Toward Understanding How Waves Break
Pages 5-28
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TwenW-First Symposium on NAVAL HYDRODYNAMICS Technical Sessions
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The simplest time-dependent model of gravity-wave breaking is the lowest superharmonic instability of a steep Stokes wave. Recent calculations for both deep-water waves and for solitary waves in shallow water show that as the wave steepness increases, the region of instability is increasingly confined to the wave crest, and takes the limiting form of a "crest instability".
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The instabilities of this crest flow have been recalculated in three recent papers, as described in Section 3, and in Section 4 it is shown that these "crest instabilities" do indeed correspond to limiting forms of the superharmonic instabilities. Similar results for solitary waves in shallow water are described in Section 5.
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A descriptive theory is also given for the nonlinear features known as "capillary jumps;" see Section 8. The effect of surface tension on crest instabilities is de scribed in Section 9.
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How are these crest instabilities related to the known instabilities of a steep Stokes wave in deep water?
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[14] that the simplest superharmonic normal modes of a deep-water Stokes wave became unstable at a wave steepness ah cor responding to the first maximum of the en ergy density E; (see Section 2~.
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5. Instabilities in solitary waves Historically there has been some interplay between studies of solitary waves in shallow water and waves in deep water.
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to ET. For many years it had been thought, on the basis of the approximate KortewegdeVries equation, that aD solitary waves were essentially stable to small perturbations.
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The profile resulting from the "negative" initial disturbance developed as in Figure 17, with a transition to a ware of lower amplitude and almost the same total energy. Unlike the solitary wave, however, the wave profile then returned almost to its original form, in a kind of nonlinear recurrence; see Figure 18 where the crest-to-trough wave height is plotted as a function of the time t.
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Nonlinear development of the surface profile corresponding to Figure 14. From Longuet-Higgins and Dommermuth t124.
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Nonlinear development of the surface profile resulting from a negative perturbation. From Longuet-Higgins and Dommermuth [124.
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A typical photograph of short gravity waves (Figure 19a) shows a train of ripples riding just ahead of the wave crests.
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The calculated surface slope dy/d~ in a wave of length 8.0 cm (b) Subcritical wave steepness, (b)
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. Figure 25 shows a one-parameter family of solitary waves each determined uniquely by its maximum surface slope Ctmax or by the depth H of the central trough relative to the level at infinity.
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A common situation in which to find a velocity gradient such as described is on the forward face of a progressive gravity wave; see Figure 27c. For steep Stokes waves, 20
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If surface tension is applied artificially to the unstable wave in Figure 15 at time t = 0 and if the wavelength ~ is 132 cm, the wave develops not as in Figure 15 but as in Figure 28a. The crest does not overturn, but instead forms a capillary jump, with a few ripples or parasitic capillaries "upstream".
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:2 . dye/ 3 2 dy/dx -1 With the same time-stepping technique we are able to carry out a fully nonlinear calculation of the growth of parasitic capillary waves on a regular, periodic train of Stokes waves.
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[1~. Without surface tension, the focussed wave overturns and breaks, as in a.
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The second important limiting case is when the wave spectrum is sufficiently narrow that the surface can be thought of as a more or less slowly modulated wave train. The best-known example, in two dimensions, is the subharmonic Benjam~n-Feir instability, in which the modulation of the waves gradually increases and may lead You —~ ~ ~—0.2 ~ ~ ~ _]
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especially in deep water. We may note that for solitary waves, where the group-velocity must presumably be defined as equal to the phase-speed c, the onset of crest instabilities actually occurs in waves of less than the limiting height (see Section 5~.
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(6) A nonlinear theory for parasitic capillary waves shows that a "capillary jump" can occur near the crest of a steep gravity wave.
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(8) Breaking in modulated wave trains (wave grouped occurs generally at lower wave steepnesses than in a uniform wave train.
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and Dias, F 1992 Gravity-capiDary solitary waves in water of infinite depth and related freesurface flows.
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