Showing papers on "Breaking wave published in 1969"

The Transformation of a Solitary Wave over an Uneven Bottom

Abstract: Based on a set of approximate equations for long waves over an uneven bottom, numerical results show that as a solitary wave climbs a slope the rate of amplitude increase depends on the initial amplitude as well as on the slope. Results are also obtained for a solitary wave progressing over a slope onto a shelf. On the shelf a disintegration of the initial wave into a train of solitary waves of decreasing amplitude is found. Experimental evidence is also presented.

On wave breaking and the equilibrium spectrum of wind-generated waves

Abstract: A theoretical calculation is made of the loss of energy by wave breaking in a random sea state in terms of the spectral density function. In the special case of the equilibrium spectrum F(σ) = αg 2 σ -5 the proportion ɷ of energy lost per mean wave cycle is found to be given by ω ≑ e -1/8α irrespective of the low-frequency cut-off in the spectrum. Assuming that in the equilibrium state the loss of energy by breaking is comparable to that supplied by the wind, one can estimate the constant α in terms of the drag coefficient of the wind on the sea surface. It is found that α≑ -1/8/ln[1600C 3/2 ( ρ air/ ρ water)]. Taking a representative value of C one finds α ≑ 1.3 x 10 -2 , which falls within the range of observed values of α. The above equation for α is rather insensitive to the various assumptions made in the analysis. There is some evidence, derived from observation, that α may not in fact be quite constant, but may decrease slightly as the wave age ( gt/U ) or the non-dimensional fetch ( gx/U 2 ) is increased. It is suggested that the drag coefficient may behave similarly.

Eddy diffusion coefficients due to instabilities in internal gravity waves.

Abstract: Internal gravity waves in the upper atmosphere tend to grow in amplitude with increasing height to maintain continuous vertical transport of wave energy. The growth of amplitude of a wave is limited by convective instabilities that must form at the height where the temperature oscillation becomes great enough to include a superadiabatic region. This turbulence attenuates the wave by eddy transport of heat and viscous stress through the wave structure, essentially maintaining a constant amplitude of the wave at greater altitudes. Eddy diffusion coefficients of the order of 107 cm²/sec are readily produced by typical gravity waves near the mesopause. Since an average coefficient of only 106 cm²/sec is sufficient to explain the thermal structure of the mesopause region, sporadic turbulence produced by instabilities in gravity waves is probably an important process of the upper atmosphere.

A variational method for weak resonant wave interactions

Abstract: Whitham’s variational method is formulated so as to apply to weak second-order resonant interactions among waves whose amplitudes and phase angles vary slowly with position and time. The method is applied in detail to capillary-gravity wave interactions. An internal gravity waves problem is also discussed briefly. The method leads to new and substantial simplifications of the interaction equations. This makes possible the proof of local conservation of total mean wave energy and momentum laws. These, together with another integral of the motion, are found to be of central importance in classifying and characterizing the slow modulations of planewave-like form. Such a classification is given in detail for all initial values of phase angles and relative amplitudes. All progressive uniform waves in the capillary range are found to be unstable with perturbation growth rates which can be of first order in the wave slopes. In this formulation amplitude dependent first-order corrections of classical frequency and/or wave-number arise for all waves participating in a resonance. A few predictions which could be verified by simple experiments are made.

Breaker Travel and Choice of Design Wave Height

Abstract: Experiments on three-plane laboratory beaches show that plunging waves travel a horizontal distance of from four to eight times breaker height during the breaking process. This suggests that the potentially damaging effect of breaking waves may be spread over a significant horizontal distance. The experiments, as well as previously available data, show that breaker depth-to-height ratios for plunging waves decrease from above 1.3 to below 0.9 as beach slope increases, so that higher waves on steeper slopes may approach nearer to shore before breaking. The combined effect of breaker travel and breaker depth-to-height ratio is such that structures sited in shallow water on moderate or steep slopes can be subject to breaking wave heights significantly larger than the design heights computed according to accepted practice. The experimental results are consistent with a solitary wave description of oscillatory waves at breaking, if the breaker depths of oscillatory waves are appropriately defined, and they are consistent with the limiting heights of breaking waves measured on rubble-mound breakwaters.

Shoaling of Solitary Waves on Small Slopes

Abstract: It is proposed that solitary waves can be used as the initial wave shape to simulate the shoaling and breaking of large, long waves on a beach. Experiments were conducted with solitary waves to determine criteria for wave growth, celerity, and breaking height on various beach slopes. Various analytical means of representing shoaling waves are considered. Further considerations are given to the method of characteristics as a means of analyzing shoaling waves, and a qualitative analysis is presented of the effect of vertical acceleration terms on the characteristics solution. A comparison is made between experimental results and the analytical solutions.

Small amplitude waves in high β plasmas

Abstract: We present a brief survey of the phase velocities and polarizations of linear waves in an infinite, uniform, high β plasma using the two-fluid approximation. Questions of wave-particle damping are deferred. Several features of high β plasma waves differ from the familiar low β case.

Wave-generating apparatus

Abstract: APPARATUS FOR GENERATING WAVES IN RELATIVELY SMALL INLAND BODIES OF WATER, SUCH AS LAKES, OR LARGE POOLS, FOR PERMITTING SURFING AND THE LIKE, IS DESCRIBED THE WAVEGENERATING APPARATUS INCLUDES MEANS FOR DEFINING THE SURFING AREA SUCH AS EITHER ONE OR A PAIR OF VERTICAL WALLS DISPOSED IN THE WATER, WITH ONE OF THE WALLS OR DEFINING MEANS BEING SUBSTANTIALLY STRAIGHT AND THE SECOND WALL HAVING A FIRST PORTION ARRANGED PARALLEL TO A PORTION OF THE FIRST WALL AND A SECOND PORTION ARCUATELY DIVERGING FROM SAID FIRST WALL IN A MANNER TO ENHANCE THE WAVE CHARACTERISTICS, AND EXTENDING TOWARD A BEACH AREA THE CONFINED AREA IS PROVIDED WITH A SUBSTANTIALLY FLAT OR GRADUALLY INCLINING BOTTOM OR BASE, WHILE THE AREA DEFINED ADJACENT THE DIVERGING WALLS IS PROVIDED WITH AN INCLINED BOTTOM OR BASE, SO AS TO ENHANCE THE WAVE ACTIVITY A SURGE-GENERATING DEVICE IS LOCATED IN THE AREA WHERE THE FIRST AND SECOND WALLS ARE PARALLEL FOR IMPARTING MOTION TO THE WATER THEREIN, WITH BREAKING WAVES FORMING ALONG THE ARCUATE WALL SURFACE AND PROPAGATING OUTWARDLY TOWARD THE FIRST WALL, WHILE ADVANCING ALONG THE INCLINED BOTTOM AND TOWARD THE BATCHING AREA

Field occurrences of induced multiple gravity waves

Abstract: Field evidence is presented for the occurrence of additional waves arising from an interaction between incident shallow-water waves and submerged shoals. A given additional wave is induced by the passage of an incident wave crest over the crown and downwave flank of a bar. Induced waves were observed under conditions that commonly occur on beaches with wavebuilt bars. Such waves may significantly influence nearshore processes. Previous discussion or documentation of this interaction is unknown to the author.

Reflection of Acoustic Pulses from Stable and Instable Interfaces between Moving Fluids

Abstract: Cagniard's method is used to find the transient solution for the reflection of line‐source generated waves from an interface between two fluid half‐spaces in relative motion, but with the same density and sound speed. The resulting solution for the reflected wave in the (normally motionless) lower half‐space may be conveniently considered as the sum of a specularly reflected wave, a refracted arrival wave, a neutral stability wave comprising one or more resonance waves, and an instability wave. Critical angles limiting the appropriate domains of reception are derived for each wave type as a function of the Mach number M of the fluid motion in the upper half‐space. The instability wave appears in the solution when M < 2 2. This feature may possibly represent the prediction of the spatial extent of a turbulent region arising from an inherent instability of the ambient medium which has been triggered by the arival of the incident pulse at the interface.

Pressures of Standing Waves on Vertical Wall

Abstract: The extent to which theories of small amplitude and finite amplitude waves as well as Sainflou’s simplified method and the pressure equation derived from Gerstner’s trochoidal wave theory can be used to obtain the pressure-time curve, vertical distribution and resultant of pressures exerted by standing waves on a vertical wall, is shown by the comparison of the theories with experiments. Three alternative equations are presented for obtaining the maximum simultaneous pressures on a vertical wall of standing waves including shallow and deep water waves.

Probabilities of wave characteristics in the surf zone.

Abstract: : Utilizing the hydrodynamic relationships for shoaling and refraction of waves approaching a shoreline over parallel bottom contours, a number of probability distributions for breaking wave characteristics are derived in 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 at various locations in the surf zone A further application to determine longshore current expectation is proposed (Author)

A mathematical model for calculation of the run-up of tsunamis

Abstract: To understand the engineering implications of possible wave run-up resulting from tsunamis, a formulation of the run-up process capable of giving quantitative answers is required. In this thesis, a new mathematical run-up model suitable for computer evaluation is proposed and tested. The two-dimensional model uses a flow constrained so that the horizontal velocity is uniform in depth. However, unlike the usual shallow water theory, the terms representing the kinetic energy of the vertical motion are retained. It is shown that this formulation allows a solitary-like wave to propagate as well as giving a more accurate indication of wave breaking. An 'artificial viscosity' term is used to allow the formation of hydraulic shocks. The effects of bottom friction are also included. The model is derived for a linear beach slope, in Lagrangian coordinates. A finite element formulation of the problem is derived that is suitable for digital computer evaluation. Calculations with the model agree satisfactorily with experimental results for the fun-up of solitary waves and bores. The model is used to obtain run-up data on tsunami-like waves, which show the danger of large run-up from low initial steepness waves on shallow slopes. However, the data also show that bottom friction values can significantly attenuate run-up, especially on shallow slopes. Waves generated by a dipole-like displacement of the simulated ocean floor show that the run-up is usually larger when the upwards displacement is nearest the beach than when the downwards displacement is nearest the beach.

Dispersive long waves of finite amplitude over an uneven bottom.

Abstract: : Approximate equations for long waves are derived under assumptions similar to those of Boussinesg and Korteweg and deVries. Numerical studies are performed using the method of characteristics. Four cases are investigated (1) solitary wave on a beach, (2) solitary wave on a shelf, (3) periodic waves generated in a wave tank of constant depth, (4) periodic wave on a shelf. It is discovered that complicated disintegration and evolution appear due to combined effects of nonlinearity and dispersion. Experimental evidence is presented. (Author)

Dynamics of Fixed Towers in Deep-Water Random Waves

Abstract: This paper is concerned with the design of offshore, bottom-supported structures which are subject to deflections and stresses due to deep water wave forces. Both periodic and random waves are considered. It is shown that the wave force on the structure can be adequately expressed by the inertia force component and that this linearization of the wave force expression leads to a criterion for the optimum spacing of structural supports which is a function of the random wave spectrum. Equations for the dynamic response of the platform are developed. The theoretical considerations were verified experimentally and the results indicated that for structures designed with optimum support spacing, the magnitude of structural response is relatively insensitive to wave direction.

Spectra of Internal Waves and Turbulence in Stratified Fluids: 2. Experiments on the Breaking of Internal Waves in a Two-Fluid System

Abstract: The breaking of internal waves on a sloping surface in a two-fluid system is studied experimentally in a wave tank. The characteristics of breaking internal waves are examined through their frequency spectra for cases of different frequency, amplitude, and wavelength. Waveheight measurements are made with capacitance probes in conjunction with a recently developed capacitance bridge and are processed by computer utilizing the Cooley-Tukey fast Fourier transform algorithm. By using a direct, on-line computer system spectra can be computed with a high degree of accuracy and a very narrow bandwidth. Spectra measured in the breaking region of the tank reveal a series of sharp peaks in the low frequency range representing the different harmonics of the internal waves. These higher modes, which possess little or no energy in the incident wave, gain energy at the expense of the lowest mode as the wave train propagates over the sloping surface. The result is a transfer of energy to higher frequencies through growth of the harmonics, each mode gaining energy at the expense of lower modes and transferring it to higher modes. At higher frequencies the spectra no longer possess these distinct harmonics. The high frequency end of the spectrum in the breaking region has a slope of approximately −3 for all cases considered.

The transport of energy by internal waves

Abstract: Two different concepts of energy transport by waves are developed and compared. The “wave energy flux” is related to the transport of a quantity known as the wave energy, and has often been used in analyzing the characteristics of the waves themselves. The “correlation wave energy flux” is related to the transport of energy by a phenomenon called a wave, is analogous to turbulence energy transports, and is of use in total system energy budgets. In dissipative fluids and fluids with spatially varying mean flows, the two are not the same. In the course of the analysis, a comparison is made between the Stokes expansion commonly used in linear wave theory and the mean and deviation approach of turbulence. DOI: 10.1111/j.2153-3490.1969.tb00429.x

Weak Reflection of Water Waves by Bottom Obstacles

Abstract: For a gradually varying bottom, an approximate theory is developed for the calculation of scattering properties of two-dimensional water waves. An integral representation is first formed with the help of a Green's function. An iterative solution is then obtained by using Rayleigh's wave shoaling equation as the first approximation. The reflection coefficient is found to depend on the smoothness of the obstacle; more abrupt depth changes at the ends cause stronger wave interference. For obstacles with a flat top it is found that under certain conditions only the sloping ends are responsible for wave scattering.

Modulated simple waves: an approach to attenuated finite amplitude waves

Abstract: : The paper describes some of the techniques which are currently being used to investigate finite amplitude waves in elastic and viscoelastic materials In particular we show how the simple wave solutions, which describe finite amplitude plane progressing waves in elastic materials, may be modified to describe the effects of reflection from boundaries, deformations behind curved wave fronts, and the effects of locally small damping mechanisms

The propagation of sound in imperfect ocean surface ducts

Abstract: : In the report, the author attempts to organize a field of knowledge. He reviews the field of the imperfect ocean surface sound duct and makes original contributions of detail where he believes that the greatest and most important needs exist. One of these contributions is the introduction of the concept of statistical reciprocity to explain the observed depth dependence of acoustic intensity within the duct at the higher frequencies. An expression is obtained for surface-coupled attenuation for acoustic propagation in surface ducts, and it is shown that bubble layers, established by wind and breaking waves, can be important in causing absorption losses. The author uses these facts, together with a model of strong acoustic scattering depending on the mean-square slope of the surface irregularities, to account for the acoustic intensity in the shadow zone below the surface duct. The importance of the mean-square slope parameter is also demonstrated by its use to account for wind-speed and acoustic frequency effects observed in acoustic backscattering measurements, especially in the presence of a surface duct. In addition, the author evaluates the importance of wavy thermoclines, drift currents, and the biomass in relation to surface duct propagation. (Author)

Wave forces on the Eider evacuation sluices

Abstract: For a safe and efficient dimensioning of the Eider Evacuation Sluices it was necessary to know the magnitude and probability of the occurring wave forces To determine those data a model investigation has been carried out in one of the wind flumes of the Delft Hydraulics Laboratory in charge of and in co-operation with the Bundesanstalt fur Wasserbau in Karlsruhe For this purpose it was necessary to consider all combinations of wave conditions and waterlevels in nature which can lead to important wave forces, taking into account their probability of occurrence It was on these grounds that the conditions for the model tests were chosen The results of the model investigation had to be translated into probabilities of exceedance of the wave forces per year Taking into account the results of the model tests as well as the general knowledge about the distribution of the wave forces, suitable load figures have been determined especially for the dimensioning of the structure Only this extensive investigation could provide the guarantee of a safe and efficient dimensioning of the structure against the impacts of breaking waves

The structure of internal gravity waves in the atmosphere from high resolution radar measurements

Abstract: A radar sounding system developed by one of us (Richter) has a height resolution (about 1 meter) capable of resolving the detailed structure of small features in the atmosphere 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 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 it is concluded that wind shear is unquestionably the mechanism of generation of the second type. Evidence is presented that untrapped waves above the region of instability can cause a flow of energy away from the shear zone and the energy flux is calculated. Their relation 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.