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Showing papers on "Breaking wave published in 1970"


Journal ArticleDOI
TL;DR: In this paper, the authors used known results on the radiation stress associated with gravity waves, and showed that the total lateral thrust exerted by incoming waves on the beach and in the nearshore zone is rigorously shown to equal (E 0/4) sin 2θ 0 per unit distance parallel to the coastline, where E denotes the energy density of the waves in deep water and θ denotes the waves' angle of incidence.
Abstract: By using known results on the radiation stress associated with gravity waves, the total lateral thrust exerted by incoming waves on the beach and in the nearshore zone is rigorously shown to equal (E0/4) sin 2θ0 per unit distance parallel to the coastline, where E0 denotes the energy density of the waves in deep water and θ0 denotes the waves' angle of incidence. The local stress exerted on the surf zone in steady conditions is shown to be given by (D/c) sin θ per unit area, where D is the local rate of energy dissipation and c is the phase velocity. These relations are independent of the manner of the energy dissipation, but, because breaker height is related to local depth in shallow water, it is argued that ordinarily most of the dissipation is due to wave breaking, not to bottom friction. Under these conditions the local mean longshore stress in the surf zone will be given by (5/4)ρumax2 s sin θ, where ρ is the density, umax is the maximum orbital velocity in the waves, s is the local beach slope, and θ is the angle of incidence. It is further shown that, if the friction coefficient C on the bottom is assumed constant and if horizontal mixing is neglected, the mean longshore component of velocity is given by (5π/8)(s/C) umax sin θ. This value is proportional to the longshore component of the orbital velocity. When the horizontal mixing is taken into account, the longshore currents observed in field observations and laboratory experiments are consistent with a friction coefficient of about 0.010.

832 citations


Journal ArticleDOI
TL;DR: In this article, the authors used the time history of the center of gravity of sand tracer to predict the longshore transport rate of sand and found that the coherence of the models appeared to be based on the generation of longshore currents by the long-shore radiation stress.
Abstract: Simultaneous field measurements of wave and current parameters in the surf zone and the resulting longshore transport of sand have been made on two beaches under a variety of conditions. The direction and flux of wave energy was measured from an array of digital wave sensors placed in and near the surf zone. Quantitative measurements of the longshore sand transport rate were obtained from the time history of the center of gravity of sand tracer. The measurements have been used to test two models for the prediction of the longshore transport rate of sand. The first model gives the immersed weight longshore transport rate of sand, Il, as proportional to the longshore component of wave energy flux (power), Il = K(ECn)b sin αb cos αb, where E is the energy density, Cn is the wave group velocity, and αb is the breaker angle. The second model assumes that the waves provide the power to move and support the sand and that the superimposed longshore current 〈νl〉 provides a longshore component that results in the longshore transport of sand according to the relationship Il = K′ (ECn)b cos αb〈νl〉/um, where um is the magnitude of the maximum horizontal component of orbital velocity near the bottom under the breaking wave, assumed to be proportional to the rate of energy dissipation by friction on the beach bed. The measurements show that both models successfully predict the sand transport rate, with values of the dimensionless coefficients K = 0.77 and K′ = 0.28. The coherence of the models implies that they are interrelated, their common solution giving the relation as 〈νl〉 = K″ um sin αb, where K″ is a dimensionless constant equal to 2.7. This relation can be obtained directly by equating the longshore current and the longshore component of the momentum flux (radiation stress) of the breaking waves. Thus, the coherence of the models appears to be based on the generation of the longshore currents by the longshore radiation stress. The models will not be equivalent if 〈νl〉 owes its origin to some other generating mechanism such as tides or winds.

574 citations


Journal ArticleDOI
TL;DR: In this paper, numerical solutions to the linearized equation for wave propagation were obtained by assuming a realistic profile of zonal winds as the basic state and imposing observed monthly mean heights of the 500-mb surface as the lower boundary condition.
Abstract: Planetary-scale, stationary disturbances in the winter stratosphere are considered to be upward propagating internal Rossby waves forced from below. Numerical solutions to the linearized equation for wave propagation are obtained by assuming a realistic profile of zonal winds as the basic state and imposing observed monthly mean heights of the 500-mb surface as the lower boundary condition. The computed wave structures in the meridional section show good agreement with the observed state for the component of zonal wavenumber 1. For wavenumber 2, the computed amplitude is too small to compare with the observed. Wave energy density attains a maximum in the lower and middle stratosphere at high altitudes, where strong upward transfer of wave energy appears. A region of small latitudinal gradient of potential vorticity of the basic state is found above the tropospheric jet, which acts as a barrier for wave propagation and confines wave energy to the polar region. Above 40 km the wave tends to spread ...

559 citations


Journal ArticleDOI
TL;DR: In this article, the authors proposed a profile of the longshore current, as a function of distance from the swash line, using the concept of radiation stress (introduced in an earlier paper) together with a horizontal eddy viscosity μe of the form μe = ρNx(gh)1/2, where ρ is the density, x is the distance offshore, g is gravity, h is the local mean depth, and N is a numerical constant.
Abstract: The profile of the longshore current, as a function of distance from the swash line, is calculated by using the concept of radiation stress (introduced in an earlier paper) together with a horizontal eddy viscosity μe of the form μe = ρNx(gh)1/2, where ρ is the density, x is the distance offshore, g is gravity, h is the local mean depth, and N is a numerical constant. This assumption gives rise to a family of current profiles whose form depends only on the nondimensional parameter P = (Π/2)(sN/αC), where s denotes the bottom slope, α is a constant characteristic of breaking waves (α ≑ 0.41), and C is the drag coefficient on the bottom. The current profiles are of simple analytic form, having a maximum in the surf zone and tending to zero at the swash line. Comparison with the laboratory experiments of Galvin and Eagleson (1965) shows remarkably good agreement if the drag coefficient C is taken as 0.010. The theoretical profiles are insensitive to the exact value of P, but the experimental results suggest that P never exceeds a critical value of 2/5.

338 citations


Journal ArticleDOI
TL;DR: In this article, an asymptotic solution of these equations is obtained which describes a slowly varying solitary wave; also differential equations for the slow variations of the parameters describing the solitary wave are derived, and solved in the case when the solitary waves evolves from a region of uniform depth.
Abstract: Equations are derived for two-dimensional long waves of small, but finite, amplitude in water of variable depth, analogous to those derived by Boussinesq for water of constant depth. When the depth is slowly varying compared to the length of the wave, an asymptotic solution of these equations is obtained which describes a slowly varying solitary wave; also differential equations for the slow variations of the parameters describing the solitary wave are derived, and solved in the case when the solitary wave evolves from a region of uniform depth. For small amplitudes it is found that the wave amplitude varies inversely as the depth.

241 citations


Journal ArticleDOI
TL;DR: Turbulent boundary layer interaction with wavy wall in wind tunnel, discussing wall pressure drag and surface waves interaction as discussed by the authors, discussed wall pressure and surface wave interaction in the wind tunnel.
Abstract: Turbulent boundary layer interaction with wavy wall in wind tunnel, discussing wall pressure drag and surface waves interaction

167 citations


Journal ArticleDOI
TL;DR: In this article, the propagation of surface waves at the boundary between a piezoelectric crystal and an ideal fluid is investigated and the effect of the fluid on the character of the waves is studied and it is found that propagating waves take the form of leaky waves transfering energy to the fluid medium.
Abstract: The propagation of surface waves at the boundary between a piezoelectric crystal and an ideal fluid is investigated. The effect of the fluid on the character of the waves is studied and it is found that for the particular examples considered, propagating waves take the form of leaky waves transfering energy to the fluid medium. The numerical solution for the complex leaky wave velocity is obtained by two methods. One is a straightforward solution of the complex characteristic equation for the velocity; the other is a perturbation scheme that treats the leaky wave as a first-order perturbation on the nonleaky wave associated with surface-wave propagation when the mechanical properties of the fluid are neglected. Comparison of the two methods reveals that the perturbation procedure is very accurate over a broad range of fluid parameters. As an example, the leaky wave phase velocity and attenuation are obtained as functions of direction of propagation on X, Y, and Z cuts of lithium niobate.

137 citations


Journal ArticleDOI
29 Jan 1970
TL;DR: In this article, the authors investigated the impulse wave characteristics of landslides and found that the characteristics depend mainly on the slide volume and the Froude number of the slide upon impact with the water.
Abstract: A study programme has been initiated to investigate the impulse waves generated by landslides originating entirely above the water surface It may be seen that the characteristics of this wave depend mainly on the slide volume and the Froude number of the slide upon impact with the water The resulting wave goes through a transition period For the highest wave (usually the first), the wave height becomes stable relatively quickly and decays exponentially during the period of transition, the wave period continues to increase for a long time, the velocity of propagation may be approximated very closely by solitary wave theory.

131 citations


Journal ArticleDOI
W. D. Hayes1
TL;DR: In this article, multiple wave propagation in acoustic duct with winds, using perturbation Lagrangian for ideal fluid flow was performed using multiple wave propagators and perturbations.
Abstract: Multiple wave propagation in acoustic duct with winds, using perturbation Lagrangian for ideal fluid flow

115 citations


Journal ArticleDOI
TL;DR: In this article, a radar sounding system developed by Richter has a height resolution (about 1 meter) capable of resolving the detailed structure of small features in the atmosphere that were never before seen.
Abstract: A radar sounding system developed by Richter has a height resolution (about 1 meter) capable of resolving the detailed structure of small features in the atmosphere that were never before seen. Two distinctly different types of wave phenomena characterize many of the records. One type is a long-period internal wave. The mechanism of generation is discussed, and waves observed on the radar are compared with theory. They are shown to represent the fundamental mode of gravity waves for a stable lower troposphere. The other wave type is much shorter in period and typically shows a cusped structure like a breaking wave. Radar observations are compared with simultaneous wind and temperature soundings, and we conclude that wind shear is unquestionably the generation mechanism for the second type. We present evidence that untrapped waves above the region of instability indicate a flow of energy toward the shear zone, and we calculate the energy flux. The relation of these waves to atmospheric stability is described; their size is deduced from the observations, and their potential as a mechanism for generation of turbulence at the high-frequency end of the atmospheric turbulence spectrum is discussed.

112 citations


Journal ArticleDOI
TL;DR: In this paper, the authors present the results of an evaluation of the relative validity of eight wave theories, including seven analytical theories and the fifth-order stream function (numerical) wave theory, based on the fit provided by the various theories to the governing equations in the boundary value problem formulation.
Abstract: The engineer confronted with an offshore design problem can select one of a rather large number of wave theories; at present, there is an insufficient basis to indicate which of the available theories is best for a particular set of wave conditions. This paper presents the results of an evaluation of the relative validity of eight wave theories, including seven analytical theories and the fifth-order stream function (numerical) wave theory. Relative validity is based on the fit provided, by the various theories, to the governing equations in the boundary value problem formulation. The wave conditions included in the evaluation encompass a wide range of relative water depths, and wave steepnesses from one-quarter of breaking up to the conventional breaking limit. It is found that, of the theories tested, the best fit in deep water is provided by the fifth-order stream function wave theory; in intermediate and shallow water depths, the Airy and first-order Cnoidal wave theories provide the best fits to the governing equations.

Journal ArticleDOI
J. Ian Collins1
29 Jan 1970
TL;DR: In this article, a procedure was developed to transform an arbitrary probability density of wave characteristics in deep water into the corresponding breaking characteristics in shallow Water using hydrodynamic relationships for shoaling and refraction of waves approaching a shoreline over parallel bottom contours.
Abstract: Utilizing the hydrodynamic relationships for shoaling and refraction of waves approaching a shoreline over parallel bottom contours a procedure is developed to transform an arbitrary probability density of wave characteristics in deep water into the corresponding breaking characteristics in shallow Water A number of probability distributions for breaking wave characteristics are derived m terms of assumed deep water probability densities of wave heights wave lengths and angles of approach Some probability densities for wave heights at specific locations in the surf zone are computed for a Rayleigh distribution in deep water The probability computations are used to derive the expectation of energy flux and its distribution.

Journal ArticleDOI
TL;DR: In this article, a theoretical and experimental analysis of large amplitude non-breaking waves is presented, where the rate of energy dissipation as a function of bottom slope and convergence of wave orthogonals is determined theoretically and experimentally.
Abstract: Waves of large amplitude may form spilling breakers over a very gentle slope. A theoretical and experimental analysis of such waves is presented. In particular, the rate of energy dissipation as a function of bottom slope and convergence of wave orthogonals is determined theoretically and experimentally and related to the relative height of foam “white water” appearing in front of the breaker. Also the velocity field in gentle spilling breakers is measured. It was found that, except near the crest, the major features are essentially those of a limit height nonbreaking wave.

Journal ArticleDOI
TL;DR: In this article, the train of capillary wave which appears on the forward face of a steep gravity wave is discussed by considering the capillary waves as stationary waves on a slowly varying running stream, using the non-linear solution of Crapper (1957) and the recent method due to Whitham (1965a, b).
Abstract: The train of capillary waves which appears on the forward face of a steep gravity wave is discussed by considering the capillary waves as stationary waves on a slowly varying running stream, using the non-linear capillary wave solution of Crapper (1957) and the recent method due to Whitham (1965a, b). Then energy input and damping are introduced into an energy equation which becomes a non-linear ordinary differential equation for the capillary wave steepness. Numerical solutions for various gravity wavelengths and steepnesses are discussed, and some light is thrown on the problem of the breaking of gravity waves.

Journal ArticleDOI
TL;DR: Heat transfer from turbulent boundary layer interacting with shock and expansion waves in supersonic flow was studied in this article. But the results were limited to the case of a single wave.
Abstract: Heat transfer from turbulent boundary layer interacting with shock and expansion waves in supersonic flow

Journal ArticleDOI
TL;DR: In this paper, it was shown that if a solution has compact support then after a finite time the kinetic energy of the wave is constant and equals the potential energy, and the proof employs the Paley-Wiener theorem of Fourier analysis.

Journal ArticleDOI
29 Jan 1970
TL;DR: In this article, it was shown that for a plunging breaker the wave energy is dissipated on a very short way (less than on wave length), for a spilling breaker however, this way is of the order of some wave lengths.
Abstract: Even m shallow water, only a part of wave energy is lost by turbulent viscosity and bottom friction, most of wave energy transfer takes place m the narrow zone of surf at the shore. Till to the point of breaking, the theoretical conception of an one-phase flow may be applied to the problem. From beginning of breaking, however, the effect of aeration can not be neglected. Prom a simple physical consideration, the sudden reduction of wave height and wave energy inside the surf zone can be explained by the entrainment of air bubbles into the water. Except compression and surface tension effects, most of wave energy is stored at first by the static energy of the air bubbles which are driven into the water. Using idealized assumptions for calculation (uniform concentration of air bubbles a.s.o.), it can be shown that m a plunging breaker the wave energy is dissipated on a very short way (less than on wave length), for a spilling breaker however, this way is of the order of some wave lengths.

Journal ArticleDOI
TL;DR: In this paper, a mathematical model based on the wave-equation which governs the propagation of small amplitude pressure disturbances through a fluid is used to compute the temporal and spatial variation of impact or shock pressure against vertical wall subjected to breaking wave action.
Abstract: A mathematical model based on the wave-equation which governs the propagation of small amplitude pressure disturbances through a fluid is used to compute the temporal and spatial variation of impact or shock pressure against vertical wall subjected to breaking wave action. A separate development using conservation of momentum principles results in an equation for the maximum pressure in terms of both breaker characteristics and the time required for the pressure in terms of both breaker characteristics and the time required for the pressure to reach its maximum value. Experiments indicate the existence of two types of impact pressure. Significant impact pressures act over large areas of a structure simultaneously while ordinary impact pressures are more localized. The role of air in decreasing the magnitude of the maximum pressure and in retarding the propagation of the pressure disturbance appears important as the model and data compare favorably when c = 400 fps. This reduction in the sonic velocity through a mixture of air and water will occur with only a 1% volume of entrained air.

Journal ArticleDOI
TL;DR: In this article, a theory for evaluating the response of offshore structures to random wave action is described by the linear wave theory and the Gaussian processes, and the hydrodynamic forces are computed through the Morrision O'Brien-Britschneider formula.
Abstract: Response of offshore structures to random wave action is presented. A theory for evaluating the response of these structures in deep water is described by the linear wave theory and the Gaussian processes. Hydrodynamic forces are computed through the Morrision-O’Brien-Britschneider formula. Some results of this application to several towers are given.

Journal ArticleDOI
TL;DR: In this paper, the authors developed criteria for the design of a wave barrier to protect the proposed Honolulu International Airport Reef Runway from breaking waves using wave flume model tests, where tribar and quarrystone armor units placed in single and multiple layers on homogeneous and composite slopes were subjected to both overtopping and non-overtopping breaking waves.

Journal ArticleDOI
TL;DR: In this paper, the authors investigated the possible occurrence of periodic waves of finite-amplitude at the interface of two streams having different densities and velocities using a power-series expansion technique, where the solutions obtained are perturbations about the neutrally stable wave of linearized theory.
Abstract: In this paper, we investigate the possible occurrence of periodic waves of finite-amplitude at the interface of two streams having different densities and velocities. A power-series expansion technique is utilized wherein the solutions obtained are perturbations about the neutrally stable wave of linearized theory. It is found that periodic finite-amplitude motions are possible for wave numbers that lie in the region that is unstable on a linear basis. Although we are unable to find a unique expression for the equilibrium amplitude of such waves, a definite lower-bound is determined. The correction to the linear wave speed is computed and it is found that an O(e2) perturbation about the neutral wave number causes an O(e) correction to the wave speed, where e is an amplitude parameter. The latter result, however, is only true as the stability boundary is approached. For smaller (i.e. more stable) wave numbers, it is shown that the first-order wave speed correction is zero.


Journal ArticleDOI
TL;DR: In this article, a non-linear theory of periodic gravity waves in an incompressible inviscid liquid is presented and the problem which is found to be a singular perturbation problem is treated accordingly with considerable ease.
Abstract: A non-linear theory of periodic gravity waves in an incompressible inviscid liquid is presented. The problem which is found to be a singular perturbation problem is treated accordingly with considerable ease. Results for frequency, surface profile and potential function are calculated up to third order as series in powers of a parameter which is the ratio of the amplitude to wavelength. Unlike the linear theory, in this third-order theory, the frequency is found to depend on the amplitude also and decreases with increasing amplitude. The results agree with those obtained earlier by a complicated and labourious method and confirmed by experiments.

Journal ArticleDOI
TL;DR: In this article, a plate whose edge was hinged at the water surface was dropped into a water basin, and measured the resulting pressure on the plate by piezoelectric type cells.
Abstract: Shock pressures caused by impact between a solid and a liquid, such as the case of waves breaking against coastal structures, could better be described as an elastic wave wherein the elasticity of the solid and the compressibility of the liquid are taken into consideration. Tests were conducted to study the probability of occurrence of shock pressures as high as the theoretical pressure predicted by the elastic wave approach, for the case of waves breaking against coastal structures. The tests consisted of dropping a plate whose edge was hinged at the water surface, into a water basin, and measuring the resulting pressure on the plate by piezoelectric type cells. The probability distributions of the ratio between the magnitude of the recorded and the theoretical pressure, and of the duration of pressure were found to fit Poissons distribution well at the 5% level of significance. These distributions could serve as a guide for predicting the magnitude and possibly the duration of shock pressure to which coastal structures might be subjected.

Journal ArticleDOI
TL;DR: In this paper, the authors demonstrate that a finite-amplitude traveling cyclotron harmonic wave may decay into two similar waves by the process of resonant mode-mode coupling.
Abstract: Experimental results are presented which demonstrate that a finite‐amplitude traveling cyclotron harmonic wave may decay into two similar waves by the process of resonant mode‐mode coupling. The results presented here are restricted to cases where the group velocity of one of the perturbed waves is in the opposite direction from that of the other perturbed wave. The important feature of this process is that considerable growth, both in time and space, can occur for the perturbed waves. The spatial growth of these waves is observed, and the wave amplitudes are in qualitative agreement with theory. Furthermore, the experiments provide a quantitative measure of the resonant mode‐mode coupling coefficient for waves that propagate perpendicularly to the magnetic field in a warm plasma.

Journal ArticleDOI
TL;DR: In this paper, a traveling-wave solution to the hydrodynamic equations for an isothermal atmosphere stratified by a uniform gravitational field is obtained in two Cartesian spatial coordinates and one temporal coordinate and represent simple Riemannian waves of arbitrary amplitude and wave form.
Abstract: Exact traveling‐wave solutions to the hydrodynamic equations for an isothermal atmosphere stratified by a uniform gravitational field are obtained in two Cartesian spatial coordinates and one temporal coordinate and represent simple Riemannian waves of arbitrary amplitude and wave form which reduce directly into acoustic waves in the absence of gravity. While a number of characteristics are similar to the theory of small amplitude acoustic‐gravity waves, these solutions exhibit features such as shock formation and phase dispersion which cannot be treated in small‐amplitude theory. Solutions are given in terms of relations between physical wave characteristics and hydrodynamic variables. Comparison with traveling atmospheric disturbances are proposed.

Journal ArticleDOI
Li-San Hwang1, David Divoky1
29 Jan 1970
TL;DR: In this article, an analytical description based upon consideration of momentum flux has been developed which predicts this wave setup and the decay history of breaking wave height, which is particularly important near the shoreline, where setup dominates the vanishing mean depth.
Abstract: Waves of large amplitude on a gentle slope may form spilling breakers which propagate shoreward and are slowly transformed In addition, there occurs a modification of the mean water level termed wave setup An analytical description based upon consideration of momentum flux has been developed which predicts this wave setup and the decay history of breaking wave height The results have been compared with experiments and found quite satisfactory The effect of wave setup on breaking wave transformation is particularly important near the shoreline, where setup dominates the vanishing mean depth.

Journal ArticleDOI
29 Jan 1970
TL;DR: In this article, a study of the littoral regime of a section of the coastline of the city and county of San Francisco, California is presented, which includes a complete refraction analyses of all applicable deep water wave directions and periods, the determination of breaker locations, and a computation of alongshore energy and potential littral transport for seven stations located just offshore along the thirty foot depth contour.
Abstract: This work encompasses a study of the littoral regime of a section of the coastline of the city and county of San Francisco, California The study included a complete refraction analyses of all applicable deep water wave directions and periods, the determination of breaker locations, and a computation of alongshore energy and potential littoral transport for seven stations located just offshore along the thirty foot depth contour The waves are refracted from deep water locations to the shoreline using the Stanford Wave Refraction computer program Wave breaking is assumed to take place whenever the computed wave height exceeds 0 78 times the local water depth The effect of limiting the refraction coefficient is explored since the refraction theory, in analogy with its parent theory of geometric optics, fails along caustic curves and predicts unreasonably large values for the refraction coefficient.

DOI
01 Jan 1970
TL;DR: In this paper, a two-dimensional wave motion in a surf zone was numerically solved to obtain temporal and spatial characteristics of the velocity field and energy dissipation, and these results were compared with measurements by the particle image velocimetry.
Abstract: Two-dimensional wave motion in a surf zone was numerically solved to obtain temporal and spatial characteristics of the velocity field and energy dissipation, and these results were compared with measurements by the particle image velocimetry. This direct numerical analysis can express splash-up and propagation of a front bore with large vortexes after breaking. The characteristics of vorticity and energy dissipation for a plunging breaker, which vary in a complicated manner with space and time, were also investigated.

Journal ArticleDOI
TL;DR: In this paper, a new mode of ion wave with a slower phase velocity than that of the ion acoustic wave was observed in a turbulent plasma without magnetic field, where the phase velocity was different from the one of the acoustic wave.