Bottom pressure distribution due to wave scattering near a submerged obstacle

2012 ◽  
Vol 702 ◽  
pp. 444-459 ◽  
Author(s):  
Julien Touboul ◽  
Vincent Rey
Keyword(s):  
Pressure Distribution ◽  
Deep Water ◽  
Water Waves ◽  
Dynamic Pressure ◽  
Finite Amplitude ◽  
Second Order ◽  
Local Modes ◽  
First Order ◽  
Pressure Zone ◽  
Water Conditions

AbstractThe dynamic pressure distribution on the bottom of a wave flume, due to the interaction of water waves with a submerged structure, is investigated experimentally and analytically, for both first- and second-order gravity waves of finite amplitude. The dynamic pressure excess is found to be very important, even for incoming waves propagating in deep water conditions. In this depth condition, a high pressure zone, thirty times larger than the dynamic pressure excess expected in the absence of the obstacle, is found in its vicinity. On the other hand, a low pressure zone is observed in the vicinity of the submerged obstacle for incoming waves propagating in smaller depth conditions. In any case, pressure gradients remain important. The second-order disturbance is found to be larger than first order in deep water conditions, for some specific conditions and locations. This result is interpreted in terms of nonlinear coupling of first-order components, including local modes.

Steep gravity waves in water of arbitrary uniform depth

1977 ◽  
Vol 286 (1335) ◽  
pp. 183-230 ◽  
Keyword(s):  
Gravity Waves ◽  
Deep Water ◽  
Water Waves ◽  
Perturbation Parameter ◽  
Intermediate Energy ◽  
Finite Amplitude ◽  
Wave Profile ◽  
Accurate Calculation ◽  
Theoretical Developments ◽  
Deep Water Waves

Modern applications of water-wave studies, as well as some recent theoretical developments, have shown the need for a systematic and accurate calculation of the characteristics of steady, progressive gravity waves of finite amplitude in water of arbitrary uniform depth. In this paper the speed, momentum, energy and other integral properties are calculated accurately by means of series expansions in terms of a perturbation parameter whose range is known precisely and encompasses waves from the lowest to the highest possible. The series are extended to high order and summed with Padé approximants. For any given wavelength and depth it is found that the highest wave is not the fastest. Moreover the energy, momentum and their fluxes are found to be greatest for waves lower than the highest. This confirms and extends the results found previously for solitary and deep-water waves. By calculating the profile of deep-water waves we show that the profile of the almost-steepest wave, which has a sharp curvature at the crest, intersects that of a slightly less-steep wave near the crest and hence is lower over most of the wavelength. An integration along the wave profile cross-checks the Padé-approximant results and confirms the intermediate energy maximum. Values of the speed, energy and other integral properties are tabulated in the appendix for the complete range of wave steepnesses and for various ratios of depth to wavelength, from deep to very shallow water.

Finite-amplitude deep-water waves on currents

1979 ◽  
Vol 292 (1392) ◽  
pp. 371-390 ◽  
Keyword(s):  
Wave Propagation ◽  
Deep Water ◽  
Water Waves ◽  
Large Scale ◽  
Finite Amplitude ◽  
The Other ◽  
Wave Trains ◽  
Deep Water Waves

Accurate integral properties of plane periodic deep-water waves of amplitudes up to the steepest are tabulated by Longuet-Higgins (1975). These are used to define an averaged Lagrangian which, following Whitham, is used to describe the properties of slowly varying wave trains. Two examples of waves on large-scale currents are examined in detail. One flow is that of a shearing current, V ( x ) j , which causes waves to be refracted. The other flow, U ( x ) i , varies in the direction of wave propagation and causes waves to either steepen or become more gentle. Some surprising features are found.

On some consequences of the canonical transformation in the Hamiltonian theory of water waves

2009 ◽  
Vol 637 ◽  
pp. 1-44 ◽  
Author(s):  
PETER A. E. M. JANSSEN
Keyword(s):  
Frequency Spectrum ◽  
Deep Water ◽  
Canonical Transformation ◽  
Water Waves ◽  
Second Order ◽  
Order Effects ◽  
Hamiltonian Theory ◽  
Wavenumber Spectrum ◽  
Second Order Effects ◽  
Change In Mean

We discuss some consequences of the canonical transformation in the Hamiltonian theory of water waves (Zakharov, J. Appl. Mech. Tech. Phys., vol. 9, 1968, pp. 190–194). Using Krasitskii's canonical transformation we derive general expressions for the second-order wavenumber and frequency spectrum and the skewness and the kurtosis of the sea surface. For deep-water waves, the second-order wavenumber spectrum and the skewness play an important role in understanding the so-called sea-state bias as seen by a radar altimeter. According to the present approach but in contrast with results obtained by Barrick & Weber (J. Phys. Oceanogr., vol. 7, 1977, pp. 11–21), in deep water second-order effects on the wavenumber spectrum are relatively small. However, in shallow water in which waves are more nonlinear, the second-order effects are relatively large and help to explain the formation of the observed second harmonics and infra-gravity waves in the coastal zone. The second-order effects on the directional-frequency spectrum are as a rule more important; in particular it is shown how the Stokes-frequency correction affects the shape of the frequency spectrum, and it is also discussed why in the context of the second-order theory the mean-square slope cannot be estimated from time series. The kurtosis of the wave field is a relevant parameter in the detection of extreme sea states. Here, it is argued that in contrast perhaps to one's intuition, the kurtosis decreases while the waves approach the coast. This is related to the generation of the wave-induced current and the associated change in mean sea level.

scholarly journals Three-dimensional surface gravity waves of a broad bandwidth on deep water

2021 ◽  
Vol 926 ◽  
Author(s):  
Yan Li
Keyword(s):  
Deep Water ◽  
Water Waves ◽  
Order Approximation ◽  
Three Dimensional ◽  
Finite Depth ◽  
Second Order ◽  
Leading Order ◽  
Broad Bandwidth ◽  
Stokes Waves ◽  
Deep Water Waves

A new nonlinear Schrödinger equation (NLSE) is presented for ocean surface waves. Earlier derivations of NLSEs that describe the evolution of deep-water waves have been limited to a narrow bandwidth, for which the bound waves at second order in wave steepness are described in leading-order approximations. This work generalizes these earlier works to allow for deep-water waves of a broad bandwidth with large directional spreading. The new NLSE permits simple numerical implementations and can be extended in a straightforward manner in order to account for waves on water of finite depth. For the description of second-order waves, this paper proposes a semianalytical approach that can provide accurate and computationally efficient predictions. With a leading-order approximation to the new NLSE, the instability region and energy growth rate of Stokes waves are investigated. Compared with the exact results based on McLean (J. Fluid Mech., vol. 511, 1982, p. 135), predictions by the new NLSE show better agreement than by Trulsen et al. (Phys. Fluids, vol. 12, 2000, pp. 2432–2437). With numerical implementations of the new NLSE, the effects of wave directionality are investigated by examining the evolution of a directionally spread focused wave group. A downward shift of the spectral peak is observed, owing to the asymmetry in the change rate of energy in a more complex manner than that for uniform Stokes waves. Rapid oblique energy transfers near the group at linear focus are observed, likely arising from the instability of uniform Stokes waves appearing in a narrow spectrum subject to oblique sideband disturbances.

Instability of periodic wavetrains in nonlinear dispersive systems

1967 ◽  
Vol 299 (1456) ◽  
pp. 59-76 ◽  
Keyword(s):  
Deep Water ◽  
Water Waves ◽  
Second Harmonic ◽  
Finite Amplitude ◽  
Dynamical Equations ◽  
Side Band ◽  
General Terms ◽  
Constant Form ◽  
Harmonic Components ◽  
State Of Rest

The defining property of the class of physical systems under consideration herein is that, by striking a balance between nonlinear and frequency-dispersive effects, they can transmit periodic waves of finite amplitude but constant form. For any such system, therefore, in respect of propagation in the x direction relative to a state of rest, the dynamical equations have exact periodic solutions of the form η ( x, t ) = H ( x - ct ), say, where c is a constant phase velocity depending on wave amplitude as well as on frequency or wavelength. This paper is concerned with the proposition that in many cases these uniform wavetrains are unstable to small disturbances of a certain kind, so that in practice they will disintegrate if the attempt is made to send them over great distances. The outstanding example only recently brought to light is that finite gravity waves on deep water are unstable: unmistakable experimental evidence of this property is now available, and it has also been demonstrated analytically. In §2 the essential factors leading to instability are explained in general terms. A disturbance capable of gaining energy from the primary wave motion consists of a pair of wave modes at side-band frequencies and wavenumbers fractionally different from the fundamental frequency and wavenumber. In consequence of a nonlinear effect on these modes counteracting the detuning effect of dispersion on them, they are forced into resonance with second-harmonic components of the primary motion and thereafter their amplitudes grow mutually at a rate that is exponential in time or distance. In §3 a detailed stability analysis is presented for wavetrains on water of arbitrary depth h , and it is shown that they are unstable if the fundamental wavenumber k satisfies kh > 1·363, but are otherwise stable. Finally, in §4, some experimental results regarding the instability of deep-water waves are discussed, and a few prospective applications to other specific systems are reviewed.

Integral properties of periodic gravity waves of finite amplitude

1975 ◽  
Vol 342 (1629) ◽  
pp. 157-174 ◽  
Keyword(s):  
Solitary Wave ◽  
Gravity Waves ◽  
Deep Water ◽  
Water Waves ◽  
Wave Theory ◽  
Finite Amplitude ◽  
Cnoidal Wave ◽  
The Mean ◽  
Exact Relations ◽  
Kinetic And Potential Energies

A number of exact relations are proved for periodic water waves of finite amplitude in water of uniform depth. Thus in deep water the mean fluxes of mass, momentum and energy are shown to be equal to 2T(4T—3F) and (3T—2V) crespectively, where T and V denote the kinetic and potential energies and c is the phase velocity. Some parametric properties of the solitary wave are here generalized, and some particularly simple relations are proved for variations of the Lagrangian The integral properties of the wave are related to the constants Q, R and S which occur in cnoidal wave theory. The speed, momentum and energy of deep-water waves are calculated numerically by a method employing a new expansion parameter. With the aid of Padé approximants, convergence is obtained for waves having amplitudes up to and including the highest. For the highest wave, the computed speed and amplitude are in agreement with independent calculations by Yamada and Schwartz. At the same time the computations suggest that the speed and energy, for waves of a given length, are greatest when the height is less than the maximum. In this respect the present results tend to confirm previous computations on solitary waves.

An experimental study of the pressure variations in standing water waves

1951 ◽  
Vol 206 (1086) ◽  
pp. 424-435 ◽  
Keyword(s):  
Experimental Study ◽  
Surface Waves ◽  
Standing Wave ◽  
Water Waves ◽  
Wave Length ◽  
Second Order ◽  
First Order ◽  
Standing Water ◽  
Plane Barrier ◽  
Progressive Waves

Although the first-order pressure variations in surface waves on water are known to decrease exponentially downwards, it has recently been shown theoretically that in a standing wave there should be some second-order terms which are unattenuated with depth. The present paper describes experiments which verify the existence of pressure variations of this type in waves of period 0·45 to 0·50 sec. When the motion consists of two progressive waves of equal wave-length travelling in opposite directions, the amplitude of the unattenuated pressure variations is found to be proportional to the product of the wave amplitudes. This property is used to determine the coefficient of reflexion from a sloping plane barrier.

scholarly journals Nonlinear Correction to Intensity of Dipole Radiation of Uncharged Drop Oscillating in External Electrostatic Field

2021 ◽  
Vol 57 (3) ◽  
pp. 50-61
Author(s):  
A. Grigoriev ◽  
◽  
S. Shiryaeva ◽  
N. Kolbneva ◽  
◽  
...  
Keyword(s):  
Electromagnetic Radiation ◽  
Electrostatic Field ◽  
Electromagnetic Waves ◽  
Finite Amplitude ◽  
Second Order ◽  
Nonlinear Correction ◽  
Dipole Radiation ◽  
First Order ◽  
Initial Drop

An asymptotic calculation of a dipole electromagnetic radiation of an uncharged conductive drop oscillating with a finite amplitude in an external electrostatic field to within the second order of smallness inclusive on the relation of the amplitude of oscillations to the radius of an initial drop is carried out. It is shown that the first mode of oscillations is excited in the calculation of the second order of smallness. As a result, the centers of the induced charges with different signs will synchronously oscillate, and the drop will radiate electromagnetic waves of dipolar type, forming the nonlinear correction to intensity of dipole radiation of the first order of smallness. An assessment of the intensity of electromagnetic radiation of this correction and of the width of a strip of frequencies depending on the size of a drop and the strength of an external electrostatic field is carried out.

Refraction of finite-amplitude water waves: deep-water waves approaching circular caustics

1981 ◽  
Vol 109 ◽  
pp. 63-74 ◽  
Author(s):  
D. H. Peregrine
Keyword(s):  
Wave Field ◽  
Deep Water ◽  
Water Waves ◽  
Finite Amplitude ◽  
Ray Theory ◽  
Wave Approximation ◽  
Comparison With Experiment ◽  
Straight Lines ◽  
The Waves ◽  
Deep Water Waves

The ‘numerically exact’ properties of plane periodic deep-water waves are used in a slowly-varying-wave approximation for a steady axisymmetric wave field. The linear ‘ray’ theory for such a wave field corresponds to waves approaching a circular caustic. A parameter, C, characterizes each solution. If C is smaller than 20 the wave behaviour is dominated by the convergence of wave energy and waves are expected to break. Comparison with experiment for C = 0 indicates that breaking may be accurately predicted. If C is greater than 50 then the waves propagate closer to the caustic and, since it is of Peregrine & Smith's (1979) type R, it is likely that the waves do not break. These solutions show that wave action does not flow along the straight lines of the linear rays.

scholarly journals Singularities in water waves and Rayleigh–Taylor instability

1991 ◽  
Vol 435 (1893) ◽  
pp. 137-158 ◽  
Keyword(s):  
Deep Water ◽  
Water Waves ◽  
Initial Conditions ◽  
Finite Depth ◽  
Finite Amplitude ◽  
Taylor Instability ◽  
Two Dimensional ◽  
Water Wave Problem ◽  
Unphysical Region ◽  
Rayleigh Taylor Instability

This paper is concerned with singularities in inviscid two-dimensional finite-amplitude water waves and inviscid Rayleigh–Taylor instability. For the deep water gravity waves of permanent form, through a combination of analytical and numerical methods, we present results describing the precise form, number and location of singularities in the unphysical domain as the wave height is increased. We then show how the information on the singularity can be used to calculate water waves numerically in a relatively efficient fashion. We also show that for two-dimensional water waves in a finite depth channel, the nearest singularity in the unphysical region has the form as for deep water waves. However, associated with such a singularity, there is a series of image singularities at increasing distances from the physical plane with possibly different behaviour. Further, for the Rayleigh–Taylor problem of motion of fluid over vacuum, and for the unsteady water wave problem, we derive integro-differential equations valid in the unphysical region and show how these equations can give information on the nature of singularities for arbitrary initial conditions. We give indications to suggest that a one-half point singularity on its approach to the physical domain corresponds to a spike observed in Rayleigh-Taylor experiment.

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