Radiation stress and mass transport in gravity waves, with application to `surf-beats'
TL;DR: In this article, the second-order currents and changes in mean surface level which are caused by gravity waves of non-uniform amplitude are investigated, and the effects are interpreted in terms of the radiation stresses in the waves.
Abstract: This paper studies the second-order currents and changes in mean surface level which are caused by gravity waves of non-uniform amplitude. The effects are interpreted in terms of the radiation stresses in the waves.The first example is of wave groups propagated in water of uniform mean depth. The problem is solved first by a perturbation analysis. In two special cases the second-order currents are found to be proportional simply to the square of the local wave amplitude: (a) when the lengths of the groups are large compared to the mean depth, and (b) when the groups are all of equal length. Then the surface is found to be depressed under a high group of waves and the mass-transport is relatively negative there. In case (a) the result can be simply related to the radiation stresses, which tend to expel fluid from beneath the higher waves.The second example considered is the propagation of waves of steady amplitude in water of gradually varying depth. Assuming no loss of energy by friction or reflexion, the wave amplitude must vary horizontally, to maintain the flux of energy constant; it is shown that this produces a negative tilt in the mean surface level as the depth diminishes. However, if the wave height is limited by breaking, the tilt is positive. The results are in agreement with some observations by Fairchild.Lastly, the propagation of groups of waves from deep to shallow water is studied. As the mean depth decreases, so the response of the fluid to the radiation stresses tends to increase. The long waves thus generated may be reflected as free waves, and so account for the 'surf beats’ observed by Munk and Tucker.Generalle speaking, the changes in mean sea level produced by ocean waves are comparable with those due to horizontal wind stress. It may be necessary to allow for the wave stresses in calculating wind stress coefficients.
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TL;DR: The radiation stresses in water waves play an important role in a variety of oceanographic phenomena, for example in the change in mean sea level due to storm waves (wave set-up), the generation of "surf-beats", the interaction of waves with steady currents, and the steepening of short gravity waves on the crests of longer waves as discussed by the authors.
1,567 citations
29 Jan 1978
TL;DR: In this article, a model was developed for the prediction of the dissipation of energy in random waves breaking on a beach and the probability of occurrence of breaking waves was estimated on the basis of a wave height distribution with an upper cut-off which in shallow water is determined mainly by the local depth.
Abstract: A description is given of a model developed for the prediction of the dissipation of energy in random waves breaking on a beach The dissipation rate per breaking wave is estimated from that in a bore of corresponding height, while the probability of occurrence of breaking waves is estimated on the basis of a wave height distribution with an upper cut-off which in shallow water is determined mainly by the local depth A comparison with measurements of wave height decay and set-up, on a plane beach and on a beach with a bar-trough profile, indicates that the model is capable of predicting qualitatively and quantitatively all the main features of the data
1,463 citations
TL;DR: In this paper, the Boussinesq equations for long waves in water of varying depth are derived for small amplitude waves, but do include non-linear terms, and solutions have been calculated numerically for a solitary wave on a beach of uniform slope, which is also derived analytically by using the linearized long-wave equations.
Abstract: Equations of motion are derived for long waves in water of varying depth. The equations are for small amplitude waves, but do include non-linear terms. They correspond to the Boussinesq equations for water of constant depth. Solutions have been calculated numerically for a solitary wave on a beach of uniform slope. These solutions include a reflected wave, which is also derived analytically by using the linearized long-wave equations.
1,352 citations
TL;DR: In this paper, a new form of the Boussinesq equations is derived using the velocity at an arbitrary distance from the still water level as the velocity variable instead of the commonly used depth-averaged velocity.
Abstract: Boussinesq‐type equations can be used to model the nonlinear transformation of surface waves in shallow water due to the effects of shoaling, refraction, diffraction, and reflection. Different linear dispersion relations can be obtained by expressing the equations in different velocity variables. In this paper, a new form of the Boussinesq equations is derived using the velocity at an arbitrary distance from the still water level as the velocity variable instead of the commonly used depth‐averaged velocity. This significantly improves the linear dispersion properties of the Boussinesq equations, making them applicable to a wider range of water depths. A finite difference method is used to solve the equations. Numerical and experimental results are compared for the propagation of regular and irregular waves on a constant slope beach. The results demonstrate that the new form of the equations can reasonably simulate several nonlinear effects that occur in the shoaling of surface waves from deep to shallow w...
1,112 citations
TL;DR: In this paper, the authors introduce the concept of rogue waves, which is the name given by oceanographers to isolated large amplitude waves, that occur more frequently than expected for normal, Gaussian distributed, statistical events.
851 citations
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TL;DR: In this article, a general theory of mass transport is developed, which takes account of the viscosity, and leads to results in agreement with observation, and is shown that the nature of the motion in the interior depends upon the ratio of the wave amplitude a to the thickness δ of the boundary layer.
Abstract: It was shown by Stokes that in a water wave the particles of fluid possess, apart from their orbital motion, a steady second-order drift velocity (usually called the mass-transport velocity). Recent experiments, however, have indicated that the mass-transport velocity can be very different from that predicted by Stokes on the assumption of a perfect, non-viscous fluid. In this paper a general theory of mass transport is developed, which takes account of the viscosity, and leads to results in agreement with observation. Part I deals especially with the interior of the fluid. It is shown that the nature of the motion in the interior depends upon the ratio of the wave amplitude a to the thickness $\delta $ of the boundary layer: when a$^{2}$/$\delta ^{2}$ is small the diffusion of vorticity takes place by viscous 'conduction'; when a$^{2}$/$\delta ^{2}$ is large, by convection with the mass-transport velocity. Appropriate field equations for the stream function of the mass transport are derived. The boundary layers, however, require separate consideration. In part II special attention is given to the boundary layers, and a general theory is developed for two types of oscillating boundary: when the velocities are prescribed at the boundary, and when the stresses are prescribed. Whenever the motion is simple-harmonic the equations of motion can be integrated exactly. A general method is described for determining the mass transport throughout the fluid in the presence of an oscillating body, or with an oscillating stress at the boundary. In part III, the general method of solution described in parts I and II is applied to the cases of a progressive and a standing wave in water of uniform depth. The solutions are markedly different from the perfect-fluid solutions with irrotational motion. The chief characteristic of the progressive-wave solution is a strong forward velocity near the bottom. The predicted maximum velocity near the bottom agrees well with that observed by Bagnold.
1,186 citations
TL;DR: In this article, the changes in wavelength and amplitude of the shorter wave train are rigorously calculated by taking into account the non-linear interactions between the two wave trains, and the results differ in some essentials from previous estimates by Unna.
Abstract: Short gravity waves, when superposed on much longer waves of the same type, have a tendency to become both shorter and steeper at the crests of the longer waves, and correspondingly longer and lower in the troughs. In the present paper, by taking into account the non-linear interactions between the two wave trains, the changes in wavelength and amplitude of the shorter wave train are rigorously calculated. The results differ in some essentials from previous estimates by Unna. The variation in energy of the short waves is shown to correspond to work done by the longer waves against the radiation stress of the short waves, which has previously been overlooked. The concept of the radiation stress is likely to be valuable in other problems.
597 citations
TL;DR: In this article, it was shown that the wave amplitude is also affected by the horizontal advection of wave energy from the sides of a non-uniform current, and the change in wave amplitude was shown to be such as would be found by adding the radiation stress term.
Abstract: The common assumption that the energy of waves on a non-uniform current U is propagated with a velocity (U + c) where cg is the group-velocity, and that no further interaction takes place, is shown in this paper to be incorrect. In fact the current does additional work on the waves at a rate γijSij where γij is the symmetric rate-of-strain tensor associated with the current, and Sij is the radiation stress tensor introduced earlier (Longuet-Higgins & Stewart 1960).In the present paper we first obtain an asymptotic solution for the combined velocity potential in the simple case (1) when the non-uniform current U is in the direction of wave propagation and the horizontal variation of U is compensated by a vertical upwelling from below. The change in wave amplitude is shown to be such as would be found by inclusion of the radiation stress term.In a second example (2) the current on the x-axis is assumed to be as in (1), but the horizontal variation in U is compensated by a small horizontal inflow from the sides. It is found that in that case the wave amplitude is also affected by the horizontal advection of wave energy from the sides.From cases (1) and (2) the general law of interaction between short waves and non-uniform currents is inferred. This is then applied to a third example (3) when waves encounter a current with vertical axis of shear, at an oblique angle. The change in wave amplitude is shown to differ somewhat from the previously accepted value.The conclusion that non-linear interactions affect the amplification of the waves has some bearing on the theoretical efficiency of hydraulic and pneumatic breakwaters.
428 citations
TL;DR: In this paper, the authors give a summary of useful relationships derived by means of the solitary wave theory, and plot these relations using dimensionless parameters for the purpose of making the theory accessible to numerical examples.
Abstract: The purposes of this paper are: (a) to give a summary of useful relationships derived by means of the solitary wave theory, and to plot these relations using dimensionless parameters for the purpose of making the theory accessible to numerical examples;? (b) to review various studies a t the Scripps Institution dealing with the application of this theory to surf problems; and (c) to discuss the problem of sand transport in or near the surf zone, in the light of the solitary wave theory. This investigation represents part of a general project undertaken during the war for the purpose of providing useful wave forecasts for the amphibious forces. By 1943, methods for forecasting sea and swell had been dwelopedlJ and a study of the transformation of waves in shallow water was initiated for the purpose of extending the wave forecasts right into the surf zone. It should be noted that the outer edge of the surf zone (the greatest depth where waves break) is usually the most critical from the point of view of bringing landing craft ashore. The problem was attacked in-three ways: (a) by field observations along the East Coast by the Woods Hole Oceanographic Institution and along the West Coast by the Scripps Institution of Oceanography; (b) by laboratory observations at the Beach Erosion Board wave tank, in Washington, D. C., and later at the Department of Engineering of the University of California in Berkeley, California; (c) by theoretical studies. A theoretical investigation by Burnside: based on the assumptions of constancy of wave periods, conservation of energy, and the linear shallow water (4iry) wave theory, reveals that the waves decrease somewhat in height after entering shallow water, reach a minimum height and then increase!, The initial decrease in wave height had been noticed by O’Brien in laboratory investigations. A comparison between the subsequent increase in heigh& as derived from Burnside’s equations with that obtained from field and laboratory observations mentioned above, showed the computed increase to be considerably smaller than the observed increase. This discrepancy became increasingly large the nearer one came to the breaking zone, the zone most important for practical forecasts. One reason for this discrepancy is contained in an assumption underlying the linear Airy theory, namely that the wave height be small compared to
301 citations