Propagation Of Surface Gravity Waves Research Articles

Generation of Surface Gravity Waves by Two Moving Submerged Sources

Generation of Surface Gravity Waves by Two Moving Submerged Sources

A note on the diffraction of obliquely incident water waves by a stepwise obstacle

The second-order problem of the propagation of surface gravity waves over an asymmetric rectangular obstacle in an oblique sea is solved numerically using a Green's theorem integral-equation method.

Berkhoff approximation in a problem on surface gravity wave propagation in a basin with bottom irregularities

The problem of the propagation of small-amplitude surface gravity waves in a basin of constant mean depth with one- and two-dimensional bottom roughness is solved in the framework of the Berkhoff model by a mean-field method. In both cases the solutions obtained are compared with the solutions of sets of exact linearized equations of the hydrodynamics of an incompressible fluid. The comparison of the exact and approximate mean-field attenuation coefficients has shown that the Berkhoff approximation is appropriate for the solution of this problem in the case of shallow water for an arbitrary correlation length of bottom irregularities and in the case of arbitrary depth and large-scale irregularities. An explanation is given for the limits of applicability of the Berkhoff approximation which are connected with the weak variability of the vertical structure of the wave field in shallow water and in a basin with large-scale depth fluctuations. The mean-field attenuation coefficients reach their maximum values in the region k o h o≥1 (where k o is the wavenumber of the surface gravity wave in a basin of constant depth h o). The location of these maxima is practically independent of the correlation length of the bottom irregularities. For the case of one-dimensional irregularities the effect of bottom roughness on the surface gravity wave velocity is investigated. It is shown that the surface wave in a basin with an uneven bottom may propagate more slowly, as well as faster than the wave in a basin with an even bottom, depending on the relations between the wavelength, depth and correlation length of the bottom imperfections.

Surface Gravity Waves due to a Moving Body: Onset of Wave Resistance as a Critical Phenomenon

The onset of the wave resistance, $R$, via generation of surface gravity waves by a long body moving with a velocity $V$, is considered. We demonstrate that for a special shape of the disturbance moving uniformly with velocity $V$, the wave resistance $R(V)$ has a well-defined threshold (perfect bifurcation) at the critical value ${V}_{c}$, i.e., $R\ensuremath{\sim}\sqrt{{V\ensuremath{-}V}_{c}}$ for $Vg{V}_{c}$ and equals zero for $Vl{V}_{c}$. For arbitrary shape of a moving disturbance the bifurcation becomes imperfect. It results in a small nonzero $R$ usually observed at $0lVl{V}_{c}$. Using this idea, we have succeeded in describing quantitatively experimental data on the wave resistance of ship models obtained by D. W. Taylor in 1908.

Three-dimensional linear solution for wave propagation with sloping bottom

This paper presents a three-dimensional analytic linear wave solution for surface gravity wave propagation over a sloping bottom that is valid for small, but realistic, slopes. The sloping-bottom linear model is compared to published laboratory data and to predictions of two-dimensional, constant-bottom nonlinear theories. The model is shown to describe the measured wave-height growth in the wave transformation region up to a limiting local Ursell number U/sub r/ of 0.35-1.0, depending on the wave type, although, as a linear model, it does not predict the harmonics observed in that range. For U/sub r/<0.35, the harmonics can generally be neglected and the sloping-bottom linear theory agrees closely with both the published wave-height data and third-order Stokes nonlinear theory. As a three-dimensional linear model, superposition can be invoked to synthesize and relate wave structure in the transformation region to complex incident ocean spectra with both wind wave and swell components that arrive with a range of incidence angles. As such, the sloping-bottom linear model presented here should be a convenient useful tool for ocean modeling through a significant portion of the wave transformation region.

Ice‐coupled wave propagation across an abrupt change in ice rigidity, density, or thickness

The model of Fox and Squire [1990, 1991, 1994], which discusses the oblique propagation of surface gravity waves from the open sea into an ice sheet of constant thickness and properties, is augmented to include propagation across an abrupt transition of properties within a continuous ice sheet or across two dissimilar ice sheets that abut one another but are free to move independently. Rigidity, thickness, and/or density may change across the transition, allowing, for example, the modeling of ice‐coupled waves into, across, and out of refrozen leads and polynyas, across cracks, and through coherent pressure ridges. Reflection and transmission behavior is reported for various changes in properties under both types of transition conditions.

The influence of drift flow turbulence on surface gravity wave propagation

The theory of surface gravity waves scattering at vortex flows in the ocean is developed in this paper. A scattering amplitude is found in the Born approximation as a function of vorticity which appears very convenient for investigation of scattering at simple localized flows. It is shown that the wave scattering cross-section is determined by the vertical component of vorticity. For a random (turbulent) vortex field the scattering cross-section per unit voume is determined by a vorticity correlation function. The damping of the coherent wave component and the angular spectrum widening are calculated for multiple scattering by vortex turbulence of drift flows. The spectrum angular width evolution for waves scattered at self-similar vortices of the logarithmic boundary layer is determined only by its dynamical speed and the wave vector. The latter result may be used for a remote sensing of oceanic turbulent drift flows based on observations of surface waves.

Application of the ray technique to calculate the time of long wave propagation in the Black Sea

This paper provides calculations of the time of long surface gravity wave propagation in the Black Sea. The ray technique is used by the numerical model to calculate wave fronts with the seabed relief prescribed in tabular form on a rectangular grid. It is speculated how the derived estimates can be employed to predict tsunamis in the Black Sea.

A comparison of two spectral wave models in the Southern California Bight

Two models, a spectral refraction model (Longuet-Higgins) and a parabolic equation method (PEM) refraction-diffraction model (Kirby), are used to simulate the propagation of surface gravity waves across the Southern California Bight. The Bight contains numerous offshore islands and shoals and is significantly larger (≈ 300 km by 300 km) than regions typically studied with these models. The effects of complex bathymetry on the transformation of incident wave directional spectra, S 0( f, θ 0), which are very narrow in both frequency and direction are difficult to model accurately. As S 0( f, θ 0) becomes broader in both dimensions, agreement between the models improves and the spectra predicted at coastal sites become less sensitive to errors in the bathymetry grid, to tidal changes in the mean water depth, and to uncertainty in S 0( f, θ 0) itself. The smoothing associated with even relatively narrow (0.01 Hz-5° bandwidth) S 0( f, θ 0) is usually sufficient to bring the model predictions of shallow water energy into at least qualitative agreement. However, neither model is accurate at highly sheltered sites. The importance of diffraction degrades the predictions of the refraction model, and a positive bias [O (10%) of the deep ocean energy] in the refraction-diffraction model estimates, believed to stem from numerical “noise” (Kirby), may be comparable to the low wave energy. The best agreement between the predicted spectra generally occurs at moderately exposed locations in deeper waters within the Bight, away from shallow water diffractive effects and in the far-field of the islands. In these cases, the differences between the models are small, comparable to the errors caused by tidal fluctuations in water depth as waves propagate across the Bight. The accuracy of predicted energies at these sites is likely to be limited by the uncertainty in specifying S 0( f, θ 0).

Estimation of added-mass and damping coefficients of a tethered spherical float using potential flow theory

Added-mass (α) and damping coefficients (β) of a tethered spherical float, undergoing oscillatory motion in sinusoidal waves, have been derived from the motion generated velocity potential for one degree-of-freedom (surge) using potential flow theory. Variations of added-mass and damping coefficients with respect to water depth, wave period and float size have been obtained. Variations of added-mass and damping coefficients with wave period show that these hydrodynamic coefficients are frequently dependent, and this is because motion generated velocity potential is associated with the generation of surface gravity waves by the oscillating float. The added-mass and damping coefficients are unaffected by water depth, but increase with size of the float. Natural and resonant frequencies of the tethered float are computed for a wide range of wave periods using α and β values. Theoretical values of natural period of sscillation of the float agree closely with the experimental values.

An analytic model of the generation of surface gravity waves by turbulent air flow

Turbulent air flow over a surface gravity wave of small amplitude is studied on the basis of a family of first-order closure models, of which the eddy viscosity model and Prandtl's mixing-length model are members. Results are obtained by the method of matched asymptotic expansions in three layers. The problem is modelled by taking into account the combined effects of turbulence and molecular viscosity, which accommodates a proper imposition of the boundary conditions at the wave surface. The detailed structure of the various wave-induced field variables throughout the flow is then investigated. In addition, it is found that the growth rate of the waves by wind depends on the turbulence model. In particular, the more sensitively the mixing length depends on the shear in the mean air flow, the higher the growth rate. The validity of the results we obtain is restricted to small drag coefficient and small phase speed. Comparisons are made with other theoretical studies and with recent laboratory and field observations.

Propagation of surface gravity waves over a rectangular submerged bar

Experiments on the propagation of linear and weakly nonlinear gravity waves over a rectangular submerged bar were undertaken through very careful measurements in a wave tank. Effects arising from the finite amplitude of the surface wave and those coming from the generation of vortices around bar edges were examined. Experimental data are compared with results of two theoretical models. The first model was derived from Takano (1960) and Kirby & Dalrymple's (1983) work and the second model was developed by Devillard, Dunlop & Souillard (1988) using the renormalized transfer matrix introduced by Miles (1967).

Propagation of surface gravity waves in a layer of electrically conducting liquid

The wave motion of a weakly conducting incompressible liquid in a transverse magnetic field is investigated within the framework of the nonlinear theory of magnetohydrodynamics. The influence of MHD interaction effects on harmonic perturbations of infinitesimal amplitude is analyzed and a long-wave equation of the Kortewegde Vries-Burgers type describing the evolution of weakly nonlinear perturbations of the free surface is derived. It is shown that the influence of the electrical conductivity leads to a change in both the dissipative and the dispersive properties of the system.

On wave scattering by random inhomogeneities, with application to the theory of weak bores

This paper considers the propagation and scattering of waves in dispersive and non-dispersive media containing random inhomogeneities. A detailed discussion of the propagation of the coherent component of the wave field is presented, and a rather general method for obtaining the absorption coefficient and scattering cross-section per unit volume of medium is described. A major objective is to derive macroscopic equations for the propagation of the coherent field in the sense that the damping due to scattering appears in the wave equation as pseudo-viscosity. The paper is concluded by a detailed application of the theory to the problem of the propagation of surface gravity waves over a stretch of shallow water with a ‘rough’ bed. By including finite amplitude effects a balance is achieved between the dispersion of the pseudo-viscous term and the non-linear convective terms, which enables the steady profile of a weak bore to be calculated.

The Generation of Wind Waves on a Water Surface

The statistical methods for treating continuous media developed elsewhere are applied to the generation of surface gravity waves. The wind stress is assumed to be everywhere normal to the disturbed surface of the water and caused by a known ensemble of ``gusts.'' Each gust is considered to be a center of high or low pressure, which moves with the mean wind speed, and has a radius L, and duration T. The rms value of the variable wind pressure is assumed to vary systematically from point to point in a storm. The calculations are made for an idealized, long continuing storm; before applying the statistical equations, the effect of a single gust is calculated. It is shown that the single gust produces a V-shaped wake, which, after the gust has blown itself out (t&amp;gt;T) can be considered as a packet of free gravity waves, which moves and spreads under the influence of divergence and dispersion. The results of the statistical calculation are compared with known general facts, and with an actual storm. From the fact that the storm waves have a non-sinusoidal character, and arrive at any one point in groups of 5 to 10 crests, it is deduced that the duration of a gust is from 15 to 30 seconds. From the fact that the majority of the wave energy travels in directions that are within 30° of the wind, and with phase velocities that are nearly equal to the wind speed, the mean radius of the gusts is deduced to be about 40 meters in the case of a wind speed of 20 meters per second. The crest length of the waves is proportional to the distance r from the storm, and inversely to its diameter, D; when r/D=10, the dominant length of the crests is 2.2 wavelengths. The rms displacement, H, of the sea-surface from its horizontal mean is a function of the parameters already mentioned and of P, the rms value of the variable component of the wind pressure at the center of the storm; for the idealized storm considered, and for r/D≫1, this is approximated by H2=10P2D2/rL,all quantities being in centimeters, including P, which is measured in centimeters on a water barometer. Comparison with data for an actual storm indicates that this theory requires a value of P that is possibly 10 times greater than its actual value. While the theory explains many of the phenomena of storm waves, it is therefore incapable of accounting quantitatively for the wave height. The possible reasons for this inadequacy are discussed. The relation of the theory to Jeffreys' sheltering theory is also discussed, and it is shown that both postulate that the variable component of the wind stress is normal to the water surface. The sheltering theory also postulates that the wind pressure lags 90° behind the surface disturbance. A similar phase lag is a consequence of the assumptions of the present theory.