Waves in shallow water due to arbitrary surface disturbances

Summary The velocity potential and the surface elevation are calculated for the three-dimensional motion of surface waves excited by any local disturbance of the surface of a sea with constant depth. Approximate values of the resulting wave integrals are given for large values of time and distance. The results are illustrated using physically plausible distributions of the initial disturbance. Some features of the waves are discussed.     

Instability of periodic capillary-gravity waves on deep water

Pure gravity waves of finite amplitude in infinitely deep water are unstable to small disturbances in the form of modulated side-bands (Benjamin-Feir instability). These disturbances will undergo unbounded magnification if the parameter Ω associated with the frequency of the side-bands lies in the range $\text{0<Ω<√2 ka}$, where k and a are the wavenumber and amplitude of the basic wavetrain. The present paper is devoted to studying the situation for capillary-gravity waves. It is shown that the corresponding range of Ω, for which instability can arise, is smaller, see eq. 4.17, than for the pure gravity wave situation. Even pure capillary waves admit instabilities.     

Surface waves induced by external periodic pressure in a fluid with an uneven bottom

The behavior of waves generated by periodic pressure on the free surface is considered within the linear shallow-water theory. The fluid depth is a piecewise-constant function, which implies the presence of a finite-size bottom trench or elevation. For an arbitrary shape of bottom unevenness, the solution of the problem reduces to a system of integral boundary equations. Manifestation of wave-guiding properties of bottom unevenness is illustrated by an example of an extended rectangular elevation.Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 1, pp. 70–77, January–February, 2005.     

Parametric instability of gravity waves on the surface of deep water

We propose to investigate the characteristics of the parametric generation of gravity waves on the surface of a body of deep water. The threshold conditions for the onset of generation are determined, and the results are compared with the experimental data. The singularities of the excitation of parametric oscillations in a resonator are noted.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 182–185, May–June, 1972.The authors are grateful to V. N. Pshenichnikov for assisting with the experiments.     

Stability of periodic waves of finite amplitude on the surface of a deep fluid

We study the stability of steady nonlinear waves on the surface of an infinitely deep fluid [1, 2]. In section 1, the equations of hydrodynamics for an ideal fluid with a free surface are transformed to canonical variables: the shape of the surface (r, t) and the hydrodynamic potential (r, t) at the surface are expressed in terms of these variables. By introducing canonical variables, we can consider the problem of the stability of surface waves as part of the more general problem of nonlinear waves in media with dispersion [3,4]. The resuits of the rest of the paper are also easily applicable to the general case.In section 2, using a method similar to van der Pohl's method, we obtain simplified equations describing nonlinear waves in the small amplitude approximation. These equations are particularly simple if we assume that the wave packet is narrow. The equations have an exact solution which approximates a periodic wave of finite amplitude.In section 3 we investigate the instability of periodic waves of finite amplitude. Instabilities of two types are found. The first type of instability is destructive instability, similar to the destructive instability of waves in a plasma [5, 6], In this type of instability, a pair of waves is simultaneously excited, the sum of the frequencies of which is a multiple of the frequency of the original wave. The most rapid destructive instability occurs for capillary waves and the slowest for gravitational waves. The second type of instability is the negative-pressure type, which arises because of the dependence of the nonlinear wave velocity on the amplitude; this results in an unbounded increase in the percentage modulation of the wave. This type of instability occurs for nonlinear waves through any media in which the sign of the second derivative in the dispersion law with respect to the wave number (d²/dk²) is different from the sign of the frequency shift due to the nonlinearity.As announced by A. N. Litvak and V. I. Talanov [7], this type of instability was independently observed for nonlinear electromagnetic waves.The author wishes to thank L. V. Ovsyannikov and R. Z. Sagdeev for fruitful discussions.     

Standing vortex waves in deep water

Standing vibrations of the free surface of an infinitely deep fluid exist in the presence of a weak (of the order of the wave amplitude) shear flow. As a result of the interaction with the flow the waves acquire a vorticity proportional to the cube of their amplitude.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 158–161, May–June, 1996.     

Deformation to breaking of deep water gravity waves

The results of laboratory observations of the deformation of deep water gravity waves leading to wave breaking are reported. The specially developed visualization technique which was used is described. A preliminary analysis of the results has led to similar conclusions than recently developed theories. As a main fact, the observed wave breaking appears as the result of, first, a modulational instability which causes the local wave steepness to approach a maximum and, second, a rapidly growing instability leading directly to the breaking.List of symbols L total wave length - H total wave height - crest elevation above still water level - trough depression below still water level - wave steepness =H/L - crest steepness =/L - trough steepness =/L - F ₁ forward horizontal length from zero-upcross point (A) to wave crest - F ₂ backward horizontal length from wave crest to zero-downcross point (B) - crest front steepness =/F ₁ - crest rear steepness =/F ₂ - vertical asymmetry factor=F ₂/F ₁ (describing the wave asymmetry with respect to a vertical axis through the wave crest) - µ horizontal asymmetry factor=/H (describing the wave asymmetry with respect to a horizontal axis: SWL) - T ₀ wavemaker period - L ₀ theoretical wave length of a small amplitude sinusoïdal wave generated at T _inf0 ^sup–1 frequency - ₀ average wave height     

Exact similarity and traveling wave solutions to an integrable evolution equation for surface waves in deep water

In this paper, the Lie symmetry analysis and the dynamical system method are performed on an integrable evolution equation for surface waves in deep water

$$\begin{aligned} 2\sqrt{\frac{k}{g}}u_{xxt}=k^2u_x-\frac{3}{2}k(uu_x)_{xx}. \end{aligned}$$

All of the geometric vector fields of the equation are presented, as well as some exact similarity solutions with an arbitrary function of t are obtained by using a special symmetry reduction and the dynamical system method. Different kinds of traveling wave solutions also be found by selecting the function appropriately.

     

Action of an adverse current on the shape of deep water surface waves

It is well established that the shape of surface waves changes when waves meet a current. The effect of an adverse current is particularly interesting from a practical point of view. A visualization method is used to study the shape change when waves meet such a current, in a large scale laboratory flume. The corresponding observations and measurements are presented in this paper.     

Instability of capillary-gravity waves in water of arbitrary uniform depth

The Benjamin-Feir instability of periodic capillary-gravity waves on a liquid layer of arbitrary uniform depth is investigated. When surface tension is present, there is always instability for some wavenumber and liquid depth and bounds on the sideband frequencies for unbounded amplification are derived. The results are compared with the slow modulation theory using an averaged Lagrangian.     

Surface and interfacial waves of arbitrary form in isotropic elastic media

A method due to Friedlander of accommodating disturbances of arbitrary form into the theory of surface waves in a semi-infinite isotropic elastic body is extended and shown to yield a simple closed form solution for the displacement field. An analogous treatment of interfacial waves of arbitrary form at a plane contact discontinuity separating different isotropic elastic materials is also given.

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Résumé On développe une méthode, conçue par Friedlander, qui fait entrer les perturbations de forme arbitraire dans la théorie des ondes de surface dans un corps élastique isotropique semi-infini, et on montre qu'elle permet d'obtenir une solution simple et exacte pour le champ de déplacement. Les ondes de forme arbitraire qui existent dans le plan à la frontière de materiaux élastiques isotropiques differents sont traitées de façon analogue.

     

Stability of bichromatic gravity waves on deep water

The stability of bichromatic gravity waves with small but finite amplitudes propagating in two directions on deep water is considered. Starting from the Zakharov equation, elementary quartet interactions are isolated and stability criteria are formulated. Results are illustrated for various combinations of bichromatic wave trains, from long-crested to standing waves. Two generic mechanisms operate: the first one is a modulational instability of one of the two components of the bichromatic wave train; the second mechanism is a modulation which couples both components of the wave train. However a third mechanism eventually comes into play: the resonant interaction of Phillips and Longuet-Higgins which leads initially to the linear growth of a third wave. When this latter is active, in particular for wave trains with wave vectors close together, it is shown by numerical integration that the long-time recurrence is destroyed.     

Magnetohydrodynamic flow through a rectangular duct due to a periodic pressure gradient-II

An analysis is made of the flow of an incompressible, viscous, electrically conducting fluid in a long channel of rectangular cross section due to a periodic pressure gradient, in the presence of a uniform transverse magnetic field. Exact solutions are obtained and asymptotic forms valid for large Hartmann numbers in the boundary layers parallel to the field are discussed.     

Shallow-water waves on a rotating ocean due to oscillatory surface stresses

The initial-value problem of shallow-water waves due to an oscillatory surface stress distribution on a homogeneous rotating ocean is solved by the method of integral transforms. For the wave integral, an asymptotic analysis is given which is uniform across the line produced by the coalescing of the pole and the stationary point of the wave spectrum; the result, unlike previous findings, is non-singular when the circular frequency of oscillation equals the Coriolis parameter. Some limiting cases of interest are deduced and the asymptotic envelope of the progressive waves at the surface is illustrated graphically.     

Shear waves in an elastic-plastic material due to a supersonic surface step load

A step shear load moves steadily on the surface of an elastic-plastic half space at a speed exceeding the elastic shear wave speed of the material. The orientation of the shear traction is such that the deformation is two-dimensional antiplane strain. Two different representations of the rate independent elastic-plastic material response are considered. The first material model is based on the associated flow rule and the Mises yield condition with isotropic hardening, whereas the second model is based on a particular flow theory of plasticity which represents incremental behavior at a corner of the instantaneous yield surface. Both models predict the same response under the same proportional loading. The stress history experienced by a typical material particle during passage of the load step is determined, and the variation of final strain with the magnitude of the load step is calculated. One conclusion resulting from comparison of results for the two material models for this problem is that the influence of yield surface vertex formation is not significant.     

Solitary waves with surface tension I: Trajectories homoclinic to periodic orbits in four dimensions

Communicated by the Editor     

Surface waves on the periodic boundary of an elastic half space

The existence of free surface waves on the periodic boundary of an elastic half space is established. These waves are a generalization of Rayleigh waves, and they can propagate both along and — at low frequencies and small profile heights — normal to the ridges of the periodic surface (periodic in one direction and constant in the other). It is shown how the wave number depends on the height and shape of the periodic surface, the frequency, and the direction of propagation. To give a further insight into behaviour of the surface waves some computations of surface displacements are given.     

On steady three-dimensional deep water weakly nonlinear gravity waves

Various kinds of steady weakly nonlinear gravity waves are examined. Corrections to the linear phase speed and the direction of modulation are calculated.     

Water waves generated due to initial axisymmetric disturbances in water with an ice-cover

This paper is concerned with the generation of water waves due to prescribed initial axisymmetric disturbances in a deep ocean with an ice-cover modelled as a thin elastic plate. The initial disturbances are either in the form of an impulsive pressure distributed over a certain region of the ice-cover or an initial displacement of the ice-cover. Assuming linear theory, the problem is formulated as an initial-value problem in the velocity potential describing the ensuing motion in the fluid. In the mathematical analysis, the Laplace and Hankel transform techniques have been utilised to obtain the deformation of the ice-covered surface as an infinite integral in each case. The method of stationary phase is used to evaluate the integral for large values of time and distance. Figures are drawn to show the effect of the presence of ice-cover on the wave motion.