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Journal ArticleDOI

On cross-waves

01 May 1970-Journal of Fluid Mechanics (Cambridge University Press (CUP))-Vol. 41, Iss: 4, pp 837-849
TL;DR: In this paper, a second-order theory of the modes of oscillation of water in a tank with a free surface and wave-makers at each end leads to a form of Mathieu's equation for the amplitude of the cross-waves, which are thus an example of parametric resonance and may be excited at half the wave-maker frequency if this is within a narrow band.
Abstract: Cross-waves are standing waves with crests at right angles to a wave-maker. They generally have half the frequency of the wave-maker and reach a steady state at some finite amplitude. A second-order theory of the modes of oscillation of water in a tank with a free surface and wave-makers at each end leads to a form of Mathieu's equation for the amplitude of the cross-waves, which are thus an example of parametric resonance and may be excited at half the wave-maker frequency if this is within a narrow band. The excitation depends on the amplitude of the wave-maker at the surface and the integral over depth of its amplitude. Cross-waves may be excited even if the mean free surface is stationary. The effects of finite amplitude are that the cross-waves approach a steady state such that a given amplitude is achieved at a frequency greater than that for free waves by an amount proportional to the amplitude of the wave-maker. The theory agrees reasonably well with the experimental results of Lin & Howard (1960). The amplification of the cross-waves may be understood in terms of the rate of working of the wave-maker against transverse stresses associated with the cross-waves, one located at the surface and the other equal to Miche's (1944) depth-independent second-order pressure. The theory applies to the situation where the primary motion consists of standing waves and the cross-waves are constant in amplitude away from the wave-maker, but certain generalizations may be made to the situation where the primary waves are progressive and the cross-waves decay away from the wave-maker.
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19 May 2005
TL;DR: In this article, the authors present a detailed review of liquid sloshing dynamics in rigid containers, including linear forced and non-linear interaction under external and parametric excitations.
Abstract: Preface Introduction 1. Fluid field equations and modal analysis in rigid containers 2. Linear forced sloshing 3. Viscous damping and sloshing suppression devices 4. Weakly nonlinear lateral sloshing 5. Equivalent mechanical models 6. Parametric sloshing (Faraday's waves) 7. Dynamics of liquid sloshing impact 8. Linear interaction of liquid sloshing with elastic containers 9. Nonlinear interaction under external and parametric excitations 10. Interactions with support structures and tuned sloshing absorbers 11. Dynamics of rotating fluids 12. Microgravity sloshing dynamics Bibliography Index.

920 citations

Journal ArticleDOI

498 citations


Cites methods from "On cross-waves"

  • ...The corresponding free-surface displacement is given by WlJo = - ocP% t at z = o. 4 This problem was first treated by Garrett (1970), who linearized the boundary condition at thc wavcmaker and neglected the self-interaction of the cross wave in the free-surface conditions....

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Journal ArticleDOI
TL;DR: In this paper, the spacings of some cusps formed under reflective wave conditions both in the laboratory and in certain selected natural situations are shown to be consistent with models hypothesizing formation by either (1) subharmonic edge waves (period twice that of the incident waves) of zero mode number or (2) synchronous edge waves of low mode.
Abstract: Genetically, beach cusps are of at least two types: those linked with incident waves which are surging and mostly reflected (reflective systems) and those generated on beaches where wave breaking and nearshore circulation cells are important (dissipative systems). The spacings of some cusps formed under reflective wave conditions both in the laboratory and in certain selected natural situations are shown to be consistent with models hypothesizing formation by either (1) subharmonic edge waves (period twice that of the incident waves) of zero mode number or (2) synchronous (period equal to that of incident waves) edge waves of low mode. Experiments show that visible subharmonic edge wave generation occurs on nonerodable plane laboratory beaches only when the incident waves are strongly reflected at the beach, and this observation is quantified. Edge wave resonance theory and experiments suggest that synchronous potential edge wave generation can also occur on reflective beaches and is a higher-order, weaker resonance than the subharmonic type. In dissipative systems, modes of longshore periodic motion other than potential edge waves may be important in controlling the longshore scale of circulation cells and beach morphologies. On reflective plane laboratory beaches, initially large subharmonic edge waves rear-rage sand tracers into shapes which resemble natural beach cusps, but the edge wave amplitudes decrease as the cusps grow. Cusp growth is thus limited by negative feedback from the cusps to the edge wave excitation process. Small edge waves can form longshore periodic morphologies by providing destabilizing perturbations on a berm properly located in the swash zone. In this case the retreating incident wave surge is channelized into breeches in the berm caused by the edge waves, and there is an initially positive feedback from the topography to longshore periodic perturbations.

472 citations

References
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Journal ArticleDOI
TL;DR: In this article, the stability of the plane free surface is investigated theoretically when the vessel is a vertical cylinder with a horizontal base, and the liquid is an ideal frictionless fluid making a constant angle of contact of 90° with the walls of the vessel.
Abstract: A vessel containing a heavy liquid vibrates vertically with constant frequency and amplitude. It has been observed that for some combinations of frequency and amplitude standing waves are formed at the free surface of the liquid, while for other combinations the free surface remains plane. In this paper the stability of the plane free surface is investigated theoretically when the vessel is a vertical cylinder with a horizontal base, and the liquid is an ideal frictionless fluid making a constant angle of contact of 90° with the walls of the vessel. When the cross-section of the cylinder and the frequency and amplitude of vibration of the vessel are prescribed, the theory predicts that the m th mode will be excited when the corresponding pair of parameters (p m , q m ) lies in an unstable region of the stability chart; the surface is stable if none of the modes is excited. (The corresponding frequencies are also shown on the chart.) The theory explains the disagreement between the experiments of Faraday and Rayleigh on the one hand, and of Matthiessen on the other. An experiment was made to check the application of the theory to a real fluid (water). The agreement was satisfactory; the small discrepancy is ascribed to wetting effects for which no theoretical estimate could be given.

773 citations

01 Jan 1944
TL;DR: In this article, the authors confronterons dans l'ordre ci-dessus, les faits avec les theories en cours, nous en indiquerons.
Abstract: Dans cet expose preliminaire, nous confronterons dans l’ordre ci-dessus, les faits avec les theories en cours, nous en indiquerons. quelque points faible et le ameliorations ou complements que nous entendons y apporter comme constituant le but de cette etude. Nous preciserons et justifierons le choix de methodes de calcul et d'approximation adoptee . En effet, si les quelques solutions exactes deja trouvees ont un interet evident, ne serait-ce que pour asseoir sur un terrain plu solide les solutions approchees, elle conduisent, en general, a des calcul trop complexe pour etre utilisable couramment. Nous recapitulerons les solutions rigoureuses connues et les principales etudes parues anterieurement sur les sujets traites dans cette note.

285 citations

Journal ArticleDOI
TL;DR: In this paper, the nearshore circulation of water on a plane beach exposed to a uniform wave train, normally incident on the beach, was investigated experimentally in the laboratory and it was found that the interaction between these edge waves and the incident waves gave rise to steady flow patterns.
Abstract: The nearshore circulation of water on a plane beach exposed to a uniform wave train, normally incident on the beach, was investigated experimentally in the laboratory. The incident waves generated standing edge waves on the beach of the same frequency as the incoming waves. The interaction between these edge waves and the incident waves gave rise to steady flow patterns (nearshore circulation cells) consisting of an onshore flow toward the breakers, a longshore current in the surf zone, and an offshore flow in relatively strong, narrow rip currents. The rip currents were found to occur at alternate antinodes of the edge waves, and the spacing of the rip current was therefore equal to the longshore wavelength of the edge waves. Although the incoming wave may interact with all the possible edge wave modes of the same frequency, it was found that the interaction with one particular mode is often dominant. A useful estimate of the relative importance of the modes is given by the parameter w2xb/(g tan β), where ω is the radian frequency of the edge waves, tan β is the beach slope, and xb is the width of the surf zone. Field observations made in the Gulf of California strongly suggest that this mechanism is important on real beaches.

255 citations

Journal ArticleDOI
TL;DR: In this paper, a method of successive approximation to the solution of the hydrodynamical equations is formulated, and the solution is carried to the fifth order for the case of two-dimensional waves on a deep liquid.
Abstract: The possible existence, form and maximum height of strictly periodic finite stationary waves on the surface of a perfect liquid are discussed. A method of successive approximation to the solution of the hydrodynamical equations is formulated, and the solution is carried to the fifth order for the case of two-dimensional waves on a deep liquid. The convergence of the method has not been established, so that the existence of truly periodic stationary waves is not beyond doubt, but the calculations provide strong presumptive evidence for their existence, and for the existence of a finite stable wave of greatest height. The crest of this wave has a right-angled nodal form, in contrast with that of the greatest stable travelling wave for which the nodal angle is 120°. The maximum crest height is 0.141A, where A is the wave-length, and the maximum trough depth is 0.078 A. This means that the greatest stationary waves are greater than the maximum travelling waves, the ratio being 1.53. The motions of individual particles are studied and it is shown that particles in the surface, particularly those near the anti-nodes have large horizontal motions. For a given wave-length, the period increases with wave height. The wave pressure on a breakwater is examined, and the modification of the calculations to allow for the finite depth of water is considered. Doubly modulated oscillations in a deep rectangular tank are also briefly discussed.

178 citations

Journal ArticleDOI
TL;DR: In this article, the amplitude-frequency curve of a wave with a small amplitude and at frequencies where great amplification occurred, owing to resonance, is shown to have two non-intersecting branches, a result which can be explained theoretically.
Abstract: The experiments here described were designed to test experimentally some conclusions about free standing waves recently reached analytically by Penney & Price. A close approximation to free oscillations was produced in a tank by wave makers operating with small amplitude and at frequencies where great amplification occurred, owing to resonance. The amplitude-frequency curve proved to consist of two non-intersecting branches, a result which can be explained theoretically. A striking prediction made by Penney & Price was that when the height of the crests of standing waves reaches about 0·15 wave-length they will become pointed, in the form of a 90° ridge. Higher waves were expected to be unstable because the downward acceleration of the free surface near the crest would exceed that of gravity. The experimental conditions necessary for producing a crest in the form of an angled ridge were found and the wave photographed in this condition. Good agreement was found with the calculated form of the profile of the highest wave, which had an angle very near to 90°. The predicted instability for two-dimensional waves was found to begin at the moment the crest became a sharp ridge. It rapidly assumed a three-dimensional character which was revealed by two photographic techniques. Even when the amplitude of oscillation of the wave makers was only 0·85°, violent types of instability developed which produced effects that are here recorded.

168 citations