Calculation of resonant short-crested waves in deep water (original) (raw)
A new limiting form for steady periodic gravity waves with surface tension on deep water
Physics of Fluids, 1996
The method developed by Longuet-Higgins [J. Inst. Math. Appl. 22, 261 (1978)] for the computation of pure gravity waves is extended to capillary-gravity waves in deep water. Surface tension provides an additional term in the identities between the Fourier coefficients in Stokes' expansion. This term is then reduced to a simple function of the slope of the local tangent to the profile of the free surface. A set of nonlinear algebraic equations is derived and solved by using the Newton's method. A new family of limiting profiles of steady gravity waves with surface tension is found.
Wave resistance for capillary gravity waves: Finite-size effects
EPL (Europhysics Letters), 2011
We study theoretically the capillary-gravity waves created at the water-air interface by an external surface pressure distribution symmetrical about a point and moving at constant velocity along a linear trajectory. Within the framework of linear wave theory and assuming the fluid to be inviscid, we calculate the wave resistance experienced by the perturbation as a function of its size (compared to the capillary length). In particular, we analyze how the amplitude of the jump occurring at the minimum phase speed cmin = (4gγ/ρ) 1/4 depends on the size of the pressure distribution (ρ is the liquid density, γ is the water-air surface tension, and g is the acceleration due to gravity). We also show how for pressure distributions broader than a few capillary lengths, the result obtained by Havelock for the wave resistance in the particular case of pure gravity waves (i.e., γ = 0) is progressively recovered.
NONLINEAR GRAVITY AND CAPILLARY-GRAVITY WAVES
Annual Review of Fluid Mechanics, 1999
This review deals primarily with the bifurcation, stability, and evolution of gravity and capillary-gravity waves. Recent results on the bifurcation of various types of capillary-gravity waves, including two-dimensional solitary waves at the minimum of the dispersion curve, are reviewed. A survey of various mechanisms (including the most recent ones) to explain the frequency downshift phenomenon is provided. Recent significant results are given on "horseshoe" patterns, which are three-dimensional structures observable on the sea surface under the action of wind or in wave tank experiments. The so-called short-crested waves are then discussed. Finally, the importance of surface tension effects on steep waves is studied.
On the stability of gravity waves on deep water
Journal of Fluid Mechanics, 1990
This note presents numerical results on the stability of large-amplitude gravity waves on deep water. The results are then used to predict new two-dimensional superharmonic instabilities. They are due to collisions of eigenvalues of opposite signatures, confirming the recent condition for instability of .
Bounds for Arbitrary Steady Gravity Waves on Water of Finite Depth
Journal of Mathematical Fluid Mechanics, 2009
Necessary conditions for the existence of arbitrary bounded steady waves are proved (earlier, these conditions, that have the form of bounds on the Bernoulli constant and other wave characteristics, were established only for Stokes waves). It is also shown that there exists an exact upper bound such that if the free-surface profile is less than this bound at infinity (positive, negative, or both), then the profile asymptotes the constant level corresponding to a unform stream (supercritical or subcritical). Finally, an integral property of arbitrary steady waves is obtained. A new technique is proposed for proving these results; it is based on modified Bernoulli’s equation that along with the free surface profile involves the difference between the potential and its vertical average.
An intermediate wavelength, weakly nonlinear theory for the evolution of capillary gravity waves
2011
A new nonlinear evolution equation is derived for surface solitary waves propagating on a liquid-air interface where the wave motion is induced by a harmonic forcing. Instead of the traditional approach involving a base state of the long wave limit, a base state of harmonic waves is assumed for the perturbation analysis. This approach is considered to be more appropriate for channels of length just a few multiples of the depth. The dispersion relation found approaches the classical long wave limit. The weakly nonlinear dispersive waves satisfy a KdV-like nonlinear evolution equation with steeper nonlinearity.
On dispersion of directional surface gravity waves
Journal of Fluid Mechanics, 2017
Using a nonlinear evolution equation we examine the dependence of the dispersion of directional surface gravity waves on the Benjamin–Feir index (BFI) and crest length. A parameter for describing the deviation between the dispersion of simulated waves and the theoretical linear dispersion relation is proposed. We find that for short crests the magnitude of the deviation parameter is low while for long crests the magnitude is high and depends on the BFI. In the present paper we also consider laboratory data of directional waves from the Marine Research Institute of the Netherlands (MARIN). The MARIN data confirm the simulations for three cases of BFI and crest length.
Nonlinear Effects in Gravity Waves Propagating in Shallow Water
Coastal Engineering Journal, 2012
Nonlinear energy transfers due to triad interactions change the characteristics of the wave-field in the shoaling region. The degree of nonlinear coupling is examined using numerical simulations based on an accurate set of deterministic evolution equations for the propagation of fully dispersive weakly nonlinear waves. The model validation, using existing experimental measurements for wave transformation over a shoal, showed that it accurately predicts nonlinear energy transfer for irregular waves with large wave-numbers. The bound higher harmonics and nonlinear statistical measures, i.e. the wave skewness and asymmetry, are well simulated by the model in both the shoaling and deshoaling regions. Numerical simulation of steep waves in shallow water with the Ursell number O(1), showed that nonlinear dispersion and phase locking lead to triad interactions even on a horizontal bottom. Nonlinear energy transfers in monochromatic waves lead to rapid spatial recurrence of the primary wave amplitudes. This is in contrast to the case of irregular waves where the Fourier coefficients of the wave-field do not recur due to the presence of innumerable interactions, which are expected to cancel resulting in no spatial evolution of the wave spectrum.
On short-crested waves: experimental and analytical investigations
European Journal of Mechanics B-fluids, 1999
Analytical and experimental investigations were conducted on short-crested wave fields generated by a sea-wall reflection of an incident plane wave. A perturbation method was used to compute analytically the solution of the basic equations up to the sixth order for capillary-gravity waves in finite depth, and up to the ninth order for gravity waves in deep water. For the experiments, we developed a new video-optical tool to measure the full three dimensional wave field η(x, y, t). A good agreement was found between theory and experiments. The spatio-temporal bi-orthogonal decomposition technique was used to exhibit the periodic and progressive properties of the short-crested wave field. © 1999 Éditions scientifiques et médicales Elsevier SAS O. Kimmoun et al.
Comptes Rendus Mécanique, 2004
The method developed by Debiane and Kharif for the calculation of symmetric gravity-capillary waves on infinite depth is extended to the general case of non-symmetric solutions. We have calculated non-symmetric steady periodic gravity-capillary waves on deep water. It is found that they appear via bifurcations from a family of symmetric waves. On the other hand we found that the symmetry-breaking bifurcation of periodic steady class 1 gravity wave on deep water is possible when it approaches the limiting profile, if it is very weakly influenced by surface tension effects. To cite this article: R. Aider, M. Debiane, C. R. Mecanique 332 (2004). 2004 Académie des sciences. Published by Elsevier SAS. All rights reserved. Résumé Brisure de symétrie dans les vagues de gravité avec une faible tension de surface et les vagues de gravité-capillarité périodiques en eau profonde. La méthode développée par Debiane et Kharif pour le calcul des ondes de gravité-capillarité symétriques en profondeur infinie a été étendue aux cas de vagues à profils non-symétriques. Nous avons calculé des ondes de gravité-capillarité non-symétriques périodiques et de formes permanentes. Elles apparaissent via des bifurcations à partir d'une solution symétrique. D'autre part, nous avons trouvé qu'en présence d'une très faible tension de surface, la brisure de symétrie d'une onde de gravité périodique de classe 1 en profondeur infinie est possible à l'approche de sa forme limite. Pour citer cet article : R. Aider, M. Debiane, C. R. Mecanique 332 (2004).
Model Equations for Gravity-Capillary Waves in Deep Water
Studies in Applied Mathematics, 2008
The Euler equations for water waves in any depth have been shown to have solitary wave solutions when the effect of surface tension is included. This paper proposes three quadratic model equations for these types of waves in infinite depth with a two-dimensional fluid domain. One model ...
Comptes Rendus Mécanique, 2006
In this Note the method developed by for the calculation of nonsymmetric water waves on infinite depth is extended to finite depth. The water-wave problem is reduced to a system of nonlinear algebraic equations which is solved by using Newton's method. Solutions are computed up to their limiting forms by decrementing the depth from the infinity to a value of the depth-wavelength ratio h/λ less than 0.025. It is found that the waves become symmetric when the depth becomes very small. Relations giving some integral properties are derived. To cite this article: R. Aider, M. Debiane, C. R. Mecanique 334 (2006).
Evolution of short gravity waves on long gravity waves
Physics of Fluids A: Fluid Dynamics, 1993
The evolution of short gravity waves on long gravity waves on the surface of deep water is studied. Both wave trains are assumed to be irrotational, mild in slope, and slowly modulated in space and time, but their scales are so different that the short wavelength is very much less than the long-wave amplitude. Here, it is shown that the use of Lagrangian instead of the usual Eulerian coordinates is advantageous for yielding analytical results. Linear and nonlinear evolutions of short waves over intermediate and very long distances are discussed.
The run-up on a cylinder in progressive surface gravity waves: harmonic components
Applied Ocean Research, 2004
Wave run-up is the vertical uprush of water that occurs when an incident wave impinges on a partially immersed body. In this present work, wave run-up is studied on a fixed vertical cylinder in plane progressive waves. These progressive waves are of a form suitable for description by Stokes' wave theory whereby the typical energy content of a wave train consists of one fundamental harmonic and corresponding phase locked Fourier components. The limitations of canonical wave diffraction theory-whereby the free-surface boundary condition is treated by a Stokes expansion-in predicting the harmonic components of the wave run-up are discussed. An experimental campaign is described and the choice of monochromatic waves is indicative of the diffraction regime for large volume structures where the assumption of potential flow theory is applicable, or more formally A=a! Oð1Þ (A and a being the wave amplitude and cylinder radius, respectively). The wave environment is represented by a parametric variation of the scattering parameter ka and wave steepness kA (where k denotes the wave number). The zeroth-, first-, second-and third-harmonics of the wave run-up are examined to determine the importance of each with regard to local wave diffraction and incident wave nonlinearities. It is shown that the complete wave run-up is not well accounted for by second-order diffraction theory. This is, however, dependent upon the coupling of ka and kA. In particular, whilst the modulus and phase of the second-harmonic are moderately predicted, the mean set-up is not well predicted by a second-order diffraction theory. Experimental evidence suggests this to be caused by higher than second-order nonlinear diffraction. Moreover, these effects most noticeably operate at the first-harmonic in waves of moderate to large steepness when ka/1.
Gravity waves with effect of surface tension and fluid viscosity
Journal of Hydrodynamics, Ser. B, 2006
The potential flow in a viscous fluid due to a point impulsive source is considered within the framework of linear Stokes equations. The combined effect of fluid viscosity and surface tension on the potential function below the water surface is studied. Dependent on the wavenumbers associated with the level of the effect due to surface tension, the oscillations can be grouped as gravity-dominant waves and capillary-dominant waves. It is shown that the wave form of gravity-dominant oscillations is largely modified by the surface tension while the wave amplitude of capillary-dominant oscillations is mostly reduced by the fluid viscosity.
Non-symmetric gravity waves on water of infinite depth
Journal of Fluid Mechanics, 1987
Two different numerical methods are used to demonstrate the existence of and calculate non-symmetric gravity waves on deep water. It is found that they appear via spontaneous symmetry-breaking bifurcations from symmetric waves. The structure of the bifurcation tree is the same as the one found by Zufiria (1987) for waves on water of finite depth using a weakly nonlinear Hamiltonian model. One of the methods is based on the quadratic relations between the Stokes coefficients discovered by Longuet-Higgins (1978a). The other method is a new one based on the Hamiltonian structure of the water-wave problem.
On the structure of steady parasitic gravity-capillary waves in the small surface tension limit
Journal of Fluid Mechanics
When surface tension is included in the classical formulation of a steadily-travelling gravity wave (a Stokes wave), it is possible to obtain solutions that exhibit parasitic ripples: small capillary waves riding on the surface of steep gravity waves. However, it is not clear whether the singular small-surface-tension limit is well-posed. That is, is it possible for an appropriate travelling gravity-capillary wave to be continuously deformed to the classic Stokes wave in the limit of vanishing surface tension? The work of Chen & Saffman [Stud. Appl. Math. 1980, 62 (1) 1-21] had suggested smooth continuation was not possible, while the work of Schwartz & Vanden-Broeck [J. Fluid Mech. 1979, 95 (1) 119-139] presented an incomplete bifurcation diagram of the nonlinear numerical solutions. In this paper, we numerically explore the low surface tension limit of the steep gravity-capillary travellingwave problem. Our results allow for a classification of the bifurcation structure that arises, and serve to unify a number of previous numerical studies. Crucially, we demonstrate that different choices of solution amplitude can lead to subtle restrictions on the continuation procedure. When wave energy is used as a continuation parameter, solution branches can be continuously deformed to the zero surface tension limit of a travelling Stokes wave.
Effects of viscosity on linear gravity waves due to surface disturbances in water of finite depth
ZAMM, 2003
The three-dimensional problem of waves due to an arbitrary initial time-dependent surface pressure together with an elevation of the surface in a viscous fluid of constant finite depth h is examined. It is shown that the multiple-integral expression for the surface displacement ζ is reducible to one which is correct to terms of order O(ν) , ν = ν/(4gh 3) 1/2 , for small coefficient of viscosity ν , under certain conditions. When the Laplace inversion is completed in ζ we arrive at new results which differ significantly from those obtained in an earlier analysis of the problem by Nikitin and Potetyunko [1].
Effects of high-order nonlinear wave-wave interactions on gravity waves
2000
Numerical simulations of gravity waves with high-order nonlinearities in two and three dimensional domain are performed by using pseudo spectral method. High-order nonlinearities more than thirdorder excite apparently chaotic evolutions of the Fourier energy in deepwater random waves. The high-order nonlinearities increase kurtosis, wave height distribution and Hmax/H 1/3 in deep-water and decrease these wave statistics in shallow water. They, moreover, can generate a single extreme high wave with an outstanding crest height in deep-water. The high-order nonlinearities more than third-order can be regarded as one of a reason of a cause of a freak wave in deep-water.
Linear theory of gravity waves on a maxwell fluid
Journal of Non Newtonian Fluid Mechanics, 1990
Surface gravity waves on a semi-infinite incompressible Maxwell fluid are studied by means of linear theory. A dimensionless (memory) time-number (O), different from the Deborah number, is introduced, together with a dimensionless wave-number and a dimensionless surface tension. A characteristic equation describing the waves is derived. This is an S-degree complex polynomial which is solved to give the complex dispersion relation. Two critical time-numbers are found, 0, = 0.321 and 0, = 0.906. These are important for the number of pure decay solutions and propagating waves. The dispersion relation is shown for 0 in the range from 0 to 2. A similarity to gravity waves on inviscid fluids is seen for small wave numbers. For large wave numbers there are solutions of the Rayleigh wave type.