Interction of a short-wave field with a dominant long wave in deep water: derivation form Zakharov's spectral formulation

Craik, Alex D. D.

doi:10.1017/s0334270000005920

Cited by 6 publications

(10 citation statements)

References 7 publications

(20 reference statements)

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“…(3.23) This last result agrees with that recently obtained by Craik [4] who used a completely different derivation, based on long-short wave interactions described in spectral space. While this latter approach can also, in principle, obtain results for wide-band spectrums as well, it appears to be immensely more complicated algebraically than the present theory.…”

Section: Basic Flow Is a Deep-water Current: Applicationssupporting

confidence: 87%

“…The free-surface boundary conditions (4.7a, b) then give Hence we obtain the local dispersion relation <7 2 =(0 + 7/c 2 )Ktanh«;Z), (4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14) which, together with the kinematic equation (4.11) determines the variation of the phase ©. The amplitude variation is now determined by considering the higher-order terms in (4.12) and (4.13a, b).…”

Section: Basic Flow Is a Shallow-water Currentmentioning

confidence: 87%

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The modulation of short gravity waves by long waves or currents

Grimshaw

1988

J. Aust. Math. Soc. Series B, Appl. Math

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The modulation of short gravity waves by long waves or currents is described for the situation when the flow is irrotational and when the short waves are described by linearised equations. Two cases are distinguished depending on whether the basic flow can be characterised as a deep-water current, or a shallow-water current. In both cases the basic flow has a current which has finite amplitude, while in the first case the free surface slope of the basic flow can be finite, but in the second case is small. The modulation equations are the local dispersion relation of the short waves, the kinematic equation for conservation of wave crests and the wave action equation. The results incorporate and extend the earlier work of Longuet-Higgins and Stewart [10,11].

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Section: Basic Flow Is a Deep-water Current: Applicationssupporting

confidence: 87%

Section: Basic Flow Is a Shallow-water Currentmentioning

confidence: 87%

The modulation of short gravity waves by long waves or currents

Grimshaw

1988

J. Aust. Math. Soc. Series B, Appl. Math

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“…Thus the iteration is actually Furthermore, on the centre manifold y and z are actually of order x 2they should be counted as quantities of order 2 in r rather than order 1. Doing this, I observe that if / is of order 3 and | f is of order 2, as occurs in (5)(6)(7), then the centre manifold of the reduced system will approximate the actual centre manifold by two more orders of accuracy at each iteration, as seen in Section 2.…”

Section: ( a K + / ( X Y3* K \X Y)))=by + G(x Y3r {K \X ?)mentioning

confidence: 96%

“…However, a very useful, if extremely complicated, approximation for the four-wave resonant interaction in a continuous spectrum of deep water waves has been derived by Zakharov [33]. The Zakharov equation has been used in a number of studies of wave evolution, see [32,7] for example, and I have commented later on its relation to the main thrust of this paper.…”

Section: )mentioning

confidence: 99%

Planform evolution in convection–an embedded centre manifold

Roberts

1992

J. Aust. Math. Soc. Series B, Appl. Math

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The new motion of embedding a centre manifold in some higher-dimensional manifold leads to a practical approach to the rational low-dimensional approximation of a wide class of dynamical systems; it also provides a simple geometric picture for these approximations. In particular, I consider the problem of finding an approximate, but accurate, description of the evolution of a two-dimensional planform of convection. Inspired by a simple example, the straightforward adiabatic iteration is proposed to estimate an embedding manifold and arguments are presented for its effectiveness. Upon applying the procedure to a model convective planform problem I find that the resulting approximations perform remarkably well-much better than the traditional Swift-Hohenberg approximation for planform evolution.

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“…One of them was the work by Longuet-Higgins and Stewart (1960). The work by Craik (1987) was based on the Zakharov's (1968) spectral formulation and gives more general results. Both of them take into account wave-wave interaction.…”

Section: Introductionmentioning

confidence: 99%

On the Doppler distortion of the sea-wave spectra

Korotkevich¹

2008

Physica D: Nonlinear Phenomena

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Discussions on a form of a frequency spectrum of wind-driven sea waves just above the spectral maximum continue during the last three decades. In 1958 Phillips made a conjecture that wave breaking is the main mechanism responsible for the spectrum formation. That leads to the spectrum decay ∼ ω −5 , where ω is the frequency of the waves. There is a contradiction between the numerous experimental data and this spectrum. The experiments show decay ∼ ω −4 . There are two general ways of the explanation of this phenomenon. The first one (proposed by Banner (1990)) takes into account the Doppler effect due to surface circular currents generated by long waves in the Phillips model. The second approach ascends to the work by Zakharov and Filonenko (1968). It is based on four-wave interactions in the kinetic equation and gives good agreement with the experimental data.In this article the contribution to the Phillips model due to the Doppler effect was examined.We considered the cases of isotropic and strongly anisotropic spectra. Both analytical results and numerical computations give us an evidence to state that the Doppler effect cannot explain the divergence of Phillips model with experimental data.

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Interction of a short-wave field with a dominant long wave in deep water: derivation form Zakharov's spectral formulation

Cited by 6 publications

References 7 publications

The modulation of short gravity waves by long waves or currents

The modulation of short gravity waves by long waves or currents

Planform evolution in convection–an embedded centre manifold

On the Doppler distortion of the sea-wave spectra

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