Steep gravity waves in water of arbitrary uniform depth

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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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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Integral properties of periodic gravity waves of finite amplitude

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

10.1098/rspa.1975.0018 ◽

1975 ◽

Vol 342 (1629) ◽

pp. 157-174 ◽

Cited By ~ 128

Keyword(s):

Solitary Wave ◽

Gravity Waves ◽

Deep Water ◽

Water Waves ◽

Wave Theory ◽

Finite Amplitude ◽

Cnoidal Wave ◽

The Mean ◽

Exact Relations ◽

Kinetic And Potential Energies

A number of exact relations are proved for periodic water waves of finite amplitude in water of uniform depth. Thus in deep water the mean fluxes of mass, momentum and energy are shown to be equal to 2T(4T—3F) and (3T—2V) crespectively, where T and V denote the kinetic and potential energies and c is the phase velocity. Some parametric properties of the solitary wave are here generalized, and some particularly simple relations are proved for variations of the Lagrangian The integral properties of the wave are related to the constants Q, R and S which occur in cnoidal wave theory. The speed, momentum and energy of deep-water waves are calculated numerically by a method employing a new expansion parameter. With the aid of Padé approximants, convergence is obtained for waves having amplitudes up to and including the highest. For the highest wave, the computed speed and amplitude are in agreement with independent calculations by Yamada and Schwartz. At the same time the computations suggest that the speed and energy, for waves of a given length, are greatest when the height is less than the maximum. In this respect the present results tend to confirm previous computations on solitary waves.

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Some new highest-wave solutions for deep-water waves of permanent form

Journal of Fluid Mechanics ◽

10.1017/s0022112080001413 ◽

1980 ◽

Vol 100 (4) ◽

pp. 801-810 ◽

Cited By ~ 15

Author(s):

D. B. Olfe ◽

James W. Rottman

Keyword(s):

Series Expansion ◽

Gravity Waves ◽

Deep Water ◽

Water Waves ◽

Wave Solutions ◽

Expansion Procedure ◽

Wave Profiles ◽

Permanent Form ◽

Kinetic And Potential Energies ◽

Deep Water Waves

The classical series expansion procedure of Michell is used to calculate some new highest-wave solutions. These solutions are shown to correspond to the types of gravity waves studied recently by Chen & Saffman (1980). Results are presented for wave profiles, phase speeds, and kinetic and potential energies.

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Refraction of finite-amplitude water waves: deep-water waves approaching circular caustics

Journal of Fluid Mechanics ◽

10.1017/s0022112081000931 ◽

1981 ◽

Vol 109 ◽

pp. 63-74 ◽

Cited By ~ 5

Author(s):

D. H. Peregrine

Keyword(s):

Wave Field ◽

Deep Water ◽

Water Waves ◽

Finite Amplitude ◽

Ray Theory ◽

Wave Approximation ◽

Comparison With Experiment ◽

Straight Lines ◽

The Waves ◽

Deep Water Waves

The ‘numerically exact’ properties of plane periodic deep-water waves are used in a slowly-varying-wave approximation for a steady axisymmetric wave field. The linear ‘ray’ theory for such a wave field corresponds to waves approaching a circular caustic. A parameter, C, characterizes each solution. If C is smaller than 20 the wave behaviour is dominated by the convergence of wave energy and waves are expected to break. Comparison with experiment for C = 0 indicates that breaking may be accurately predicted. If C is greater than 50 then the waves propagate closer to the caustic and, since it is of Peregrine & Smith's (1979) type R, it is likely that the waves do not break. These solutions show that wave action does not flow along the straight lines of the linear rays.

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Bound Coherent Structures Propagating on the Free Surface of Deep Water

Fluids ◽

10.3390/fluids6030115 ◽

2021 ◽

Vol 6 (3) ◽

pp. 115

Author(s):

Dmitry Kachulin ◽

Sergey Dremov ◽

Alexander Dyachenko

Keyword(s):

Free Surface ◽

Deep Water ◽

Coherent Structures ◽

Water Waves ◽

Nonlinear Models ◽

Ideal Incompressible Fluid ◽

Equation Model ◽

System Of Nonlinear Equations ◽

Potential Flows ◽

Deep Water Waves

This article presents a study of bound periodically oscillating coherent structures arising on the free surface of deep water. Such structures resemble the well known bi-soliton solution of the nonlinear Schrödinger equation. The research was carried out in the super-compact Dyachenko-Zakharov equation model for unidirectional deep water waves and the full system of nonlinear equations for potential flows of an ideal incompressible fluid written in conformal variables. The special numerical algorithm that includes a damping procedure of radiation and velocity adjusting was used for obtaining such bound structures. The results showed that in both nonlinear models for deep water waves after the damping is turned off, a periodically oscillating bound structure remains on the fluid surface and propagates stably over hundreds of thousands of characteristic wave periods without losing energy.

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Effective Power and Speed Loss of Underwater Vehicles in Close Proximity to Regular Waves

Volume 7A: Ocean Engineering ◽

10.1115/omae2017-62056 ◽

2017 ◽

Author(s):

Stefan Daum ◽

Martin Greve ◽

Renato Skejic

Keyword(s):

Free Surface ◽

Deep Water ◽

Water Waves ◽

Underwater Vehicles ◽

Total Resistance ◽

Wave Load ◽

Regular Waves ◽

Effective Power ◽

Close Proximity ◽

Deep Water Waves

The present study is focused on performance issues of underwater vehicles near the free surface and gives insight into the analysis of a speed loss in regular deep water waves. Predictions of the speed loss are based on the evaluation of the total resistance and effective power in calm water and preselected regular wave fields w.r.t. the non-dimensional wave to body length ratio. It has been assumed that the water is sufficiently deep and that the vehicle is operating in a range of small to moderate Froude numbers by moving forward on a straight-line course with a defined encounter angle of incident regular waves. A modified version of the Doctors & Days [1] method as presented in Skejic and Jullumstrø [2] is used for the determination of the total resistance and consequently the effective power. In particular, the wave-making resistance is estimated by using different approaches covering simplified methods, i.e. Michell’s thin ship theory with the inclusion of viscosity effects Tuck [3] and Lazauskas [4] as well as boundary element methods, i.e. 3D Rankine source calculations according to Hess and Smith [5]. These methods are based on the linear potential fluid flow and are compared to fully viscous finite volume methods for selected geometries. The wave resistance models are verified and validated by published data of a prolate spheroid and one appropriate axisymmetric submarine model. Added resistance in regular deep water waves is obtained through evaluation of the surge mean second-order wave load. For this purpose, two different theoretical models based on potential flow theory are used: Loukakis and Sclavounos [6] and Salvesen et. al. [7]. The considered theories cover the whole range of important wavelengths for an underwater vehicle advancing in close proximity to the free surface. Comparisons between the outlined wave load theories and available theoretical and experimental data were carried out for a submerged submarine and a horizontal cylinder. Finally, the effective power and speed loss are discussed from a submarine operational point of view where the mentioned parameters directly influence mission requirements in a seaway. All presented results are carried out from the perspective of accuracy and efficiency within common engineering practice. By concluding current investigations in regular waves an outlook will be drawn to the application of advancing underwater vehicles in more realistic sea conditions.

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