A computer study of finite-amplitude water waves

Modern applications of water-wave studies, as well as some recent theoretical developments, have shown the need for a systematic and accurate calculation of the characteristics of steady, progressive gravity waves of finite amplitude in water of arbitrary uniform depth. In this paper the speed, momentum, energy and other integral properties are calculated accurately by means of series expansions in terms of a perturbation parameter whose range is known precisely and encompasses waves from the lowest to the highest possible. The series are extended to high order and summed with Padé approximants. For any given wavelength and depth it is found that the highest wave is not the fastest. Moreover the energy, momentum and their fluxes are found to be greatest for waves lower than the highest. This confirms and extends the results found previously for solitary and deep-water waves. By calculating the profile of deep-water waves we show that the profile of the almost-steepest wave, which has a sharp curvature at the crest, intersects that of a slightly less-steep wave near the crest and hence is lower over most of the wavelength. An integration along the wave profile cross-checks the Padé-approximant results and confirms the intermediate energy maximum. Values of the speed, energy and other integral properties are tabulated in the appendix for the complete range of wave steepnesses and for various ratios of depth to wavelength, from deep to very shallow water.

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ESTIMATION OF WATER PARTICLE VELOCITIES OF SHALLOW WATER WAVES BY A MODIFIED TRANSFER FUNCTION METHOD

Coastal Engineering Proceedings ◽

10.9753/icce.v20.33 ◽

1986 ◽

Vol 1 (20) ◽

pp. 33 ◽

Cited By ~ 2

Author(s):

Hirofumi Koyama ◽

Koichiro Iwata

Keyword(s):

Transfer Function ◽

Shallow Water ◽

Approximate Method ◽

Stream Function ◽

Water Waves ◽

Function Method ◽

Finite Amplitude ◽

Water Particle ◽

Irregular Wave ◽

Particle Velocities

This paper Is intended to propose a simple, yet highly reliable approximate method which uses a modified transfer function in order to evaluate the water particle velocity of finite amplitude waves at shallow water depth in regular and irregular wave environments. Using Dean's stream function theory, the linear function is modified so as to include the nonlinear effect of finite amplitude wave. The approximate method proposed here employs the modified transfer function. Laboratory experiments have been carried out to examine the validity of the proposed method. The approximate method is shown to estimate well the experimental values, as accurately as Dean's stream function method, although its calculation procedure is much simpler than that of Dean's method.

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On solitary water-waves of finite amplitude

Archive for Rational Mechanics and Analysis ◽

10.1007/bf00250799 ◽

1981 ◽

Vol 76 (1) ◽

pp. 9-95 ◽

Cited By ~ 106

Author(s):

C. J. Amick ◽

J. F. Toland

Keyword(s):

Water Waves ◽

Finite Amplitude

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KINEMATICS AND RETURN FLOW IN A CLOSED WAVE FLUME

Coastal Engineering Proceedings ◽

10.9753/icce.v21.31 ◽

1988 ◽

Vol 1 (21) ◽

pp. 31 ◽

Cited By ~ 3

Author(s):

Jerald D. Ramsden ◽

John H. Nath

Keyword(s):

Water Waves ◽

Water Surface ◽

Large Scale ◽

Wave Motion ◽

Finite Amplitude ◽

Return Flow ◽

Order Stream ◽

Wave Flume ◽

Seventh Order ◽

Surface Measurements

Stokes (1847) showed that finite amplitude progressing waves cause a net drift of fluid, in the direction of wave motion, which occurs in the upper portion of the water column. In a closed wave flume this drift must be accompanied by a return flow toward the wave generator to satisfy the conservation of mass. This study presents Eulerian velocity and water surface measurements soon after the onset of wave motion from 12 locations in a large scale flume. Waves with .67 < kh < 2.29 and .09 < H/h < .39 were produced in a water depth of 3.5 meters. Superimposing the return flow theory of Kim (1984) with seventh order stream function theory is shown to improve the velocity predictions. The measured return flows are a function of time and depth and agree with Kim's theory as a first approximation. The mean water surface set-down agrees with the theory of Brevik (1979) except for the nearly deep water waves.

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Water waves of finite amplitude on a sloping beach

Journal of Fluid Mechanics ◽

10.1017/s0022112058000331 ◽

1958 ◽

Vol 4 (1) ◽

pp. 97-109 ◽

Cited By ~ 477

Author(s):

G. F. Carrier ◽

H. P. Greenspan

Keyword(s):

Shallow Water ◽

Water Waves ◽

Finite Amplitude ◽

Explicit Solutions ◽

Shallow Water Theory ◽

Non Linear ◽

Wave Forms ◽

Time Histories ◽

Sloping Beach

In this paper, we investigate the behaviour of a wave as it climbs a sloping beach. Explicit solutions of the equations of the non-linear inviscid shallow-water theory are obtained for several physically interesting wave-forms. In particular it is shown that waves can climb a sloping beach without breaking. Formulae for the motions of the instantaneous shoreline as well as the time histories of specific wave-forms are presented.

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Finite-amplitude deep-water waves on currents

Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences ◽

10.1098/rsta.1979.0066 ◽

1979 ◽

Vol 292 (1392) ◽

pp. 371-390 ◽

Cited By ~ 31

Keyword(s):

Wave Propagation ◽

Deep Water ◽

Water Waves ◽

Large Scale ◽

Finite Amplitude ◽

The Other ◽

Wave Trains ◽

Deep Water Waves

Accurate integral properties of plane periodic deep-water waves of amplitudes up to the steepest are tabulated by Longuet-Higgins (1975). These are used to define an averaged Lagrangian which, following Whitham, is used to describe the properties of slowly varying wave trains. Two examples of waves on large-scale currents are examined in detail. One flow is that of a shearing current, V ( x ) j , which causes waves to be refracted. The other flow, U ( x ) i , varies in the direction of wave propagation and causes waves to either steepen or become more gentle. Some surprising features are found.

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A model for the two-way propagation of water waves in a channel

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s030500410005218x ◽

1976 ◽

Vol 79 (1) ◽

pp. 167-182 ◽

Cited By ~ 71

Author(s):

Jerry L. Bona ◽

Ronald Smith

Keyword(s):

Water Waves ◽

Open Channel ◽

Boussinesq Equations ◽

Coupled System ◽

Finite Amplitude ◽

Regularity Of Solutions ◽

Initial Value ◽

One Dimensional ◽

Image Position ◽

Continuous Dependence Of Solutions

Global existence, uniqueness and regularity of solutions and continuous dependence of solutions on varied initial data are established for the initial-value problem for the coupled system of equationsThis system has the same formal justification as a model for the two-way propagation of (one-dimensional) long waves of small but finite amplitude in an open channel of water of constant depth as other versions of the Boussinesq equations. A feature of the analysis is that bounds on the wave amplitude η are obtained which are valid for all time.

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On a class of non-breaking finite amplitude water waves

Zeitschrift für angewandte Mathematik und Physik ◽

10.1007/bf01593893 ◽

1967 ◽

Vol 18 (1) ◽

pp. 57-65 ◽

Cited By ~ 7

Author(s):

Alan Jeffrey

Keyword(s):

Water Waves ◽

Finite Amplitude

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Instability of periodic wavetrains in nonlinear dispersive systems

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1967.0123 ◽

1967 ◽

Vol 299 (1456) ◽

pp. 59-76 ◽

Cited By ~ 201

Keyword(s):

Deep Water ◽

Water Waves ◽

Second Harmonic ◽

Finite Amplitude ◽

Dynamical Equations ◽

Side Band ◽

General Terms ◽

Constant Form ◽

Harmonic Components ◽

State Of Rest

The defining property of the class of physical systems under consideration herein is that, by striking a balance between nonlinear and frequency-dispersive effects, they can transmit periodic waves of finite amplitude but constant form. For any such system, therefore, in respect of propagation in the x direction relative to a state of rest, the dynamical equations have exact periodic solutions of the form η ( x, t ) = H ( x - ct ), say, where c is a constant phase velocity depending on wave amplitude as well as on frequency or wavelength. This paper is concerned with the proposition that in many cases these uniform wavetrains are unstable to small disturbances of a certain kind, so that in practice they will disintegrate if the attempt is made to send them over great distances. The outstanding example only recently brought to light is that finite gravity waves on deep water are unstable: unmistakable experimental evidence of this property is now available, and it has also been demonstrated analytically. In §2 the essential factors leading to instability are explained in general terms. A disturbance capable of gaining energy from the primary wave motion consists of a pair of wave modes at side-band frequencies and wavenumbers fractionally different from the fundamental frequency and wavenumber. In consequence of a nonlinear effect on these modes counteracting the detuning effect of dispersion on them, they are forced into resonance with second-harmonic components of the primary motion and thereafter their amplitudes grow mutually at a rate that is exponential in time or distance. In §3 a detailed stability analysis is presented for wavetrains on water of arbitrary depth h , and it is shown that they are unstable if the fundamental wavenumber k satisfies kh > 1·363, but are otherwise stable. Finally, in §4, some experimental results regarding the instability of deep-water waves are discussed, and a few prospective applications to other specific systems are reviewed.

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