Phase velocity effects in tertiary wave interactions

1962 ◽  
Vol 12 (3) ◽  
pp. 333-336 ◽  
Author(s):  
M. S. Longuet-Higgins ◽  
O. M. Phillips
Keyword(s):  
Phase Velocity ◽  
Continuous Spectrum ◽  
Deep Water ◽  
Wave Train ◽  
The Other ◽  
Wave Number ◽  
Wave Interactions ◽  
Single Wave ◽  
Phase Velocities ◽  
Wave Trains

It is shown that, when two trains of waves in deep water interact, the phase velocity of each is modified by the presence of the other. The change in phase velocity is of second order and is distinct from the increase predicted by Stokes for a single wave train. When the wave trains are moving in the same direction, the increase in velocity Δc2 of the wave with amplitude a2, wave-number k2 and frequency α2 resulting from the interaction with the wave (a1, k1, σ1) is given by Δc2 = a21k1σ1, provided k1 < k2. If k1 > k2, then Δc2 is given by the same expression multiplied by k2/k1. If the directions of propagation are opposed, the phase velocities are decreased by the same amount. These expressions are extended to give the increase (or decrease) in velocity due to a continuous spectrum of waves all travelling in the same (or opposite) direction.

The disintegration of wave trains on deep water Part 1. Theory

1967 ◽  
Vol 27 (3) ◽  
pp. 417-430 ◽  
Author(s):  
T. Brooke Benjamin ◽  
J. E. Feir
Keyword(s):  
Deep Water ◽  
Wave Train ◽  
Low Frequency ◽  
Periodic Wave ◽  
Wave Number ◽  
Fundamental Wave ◽  
Side Band ◽  
Wave Trains ◽  
The Stability ◽  
Deep Water Part

The phenomenon in question arises when a periodic progressive wave train with fundamental frequency ω is formed on deep water—say by radiation from an oscillating paddle—and there are also present residual wave motions at adjacent side-band frequencies ω(1 ± δ), such as would be generated if the movement of the paddle suffered a slight modulation at low frequency. In consequence of coupling through the non-linear boundary conditions at the free surface, energy is then transferred from the primary motion to the side bands at a rate that, as will be shown herein, can increase exponentially as the interaction proceeds. The result is that the wave train becomes highly irregular far from its origin, even when the departures from periodicity are scarcely detectable at the start.In this paper a theoretical investigation is made into the stability of periodic wave trains to small disturbances in the form of a pair of side-band modes, and Part 2 which will follow is an account of some experimental observations in accord with the present predictions. The main conclusion of the theory is that infinitesimal disturbances of the type considered will undergo unbounded magnification if \[ 0 < \delta \leqslant (\sqrt{2})ka, \] where k and a are the fundamental wave-number and amplitude of the perturbed wave train. The asymptotic rate of growth is a maximum for δ = ka.

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.

scholarly journals Effect of Capillarity on Fourth Order Nonlinear Evolution Equation for Two Stokes Wave Trains in Deep Water in the Presence of Air Flowing Over Water

2015 ◽  
Vol 20 (2) ◽  
pp. 267-282
Author(s):  
A.K. Dhar ◽  
J. Mondal
Keyword(s):  
Fourth Order ◽  
Deep Water ◽  
Water Waves ◽  
Nonlinear Evolution ◽  
Wave Train ◽  
Nonlinear Evolution Equations ◽  
Wave Number ◽  
Stokes Wave ◽  
Instability Wave ◽  
Starting Point

Abstract Fourth order nonlinear evolution equations, which are a good starting point for the study of nonlinear water waves, are derived for deep water surface capillary gravity waves in the presence of second waves in which air is blowing over water. Here it is assumed that the space variation of the amplitude takes place only in a direction along which the group velocity projection of the two waves overlap. A stability analysis is made for a uniform wave train in the presence of a second wave train. Graphs are plotted for the maximum growth rate of instability wave number at marginal stability and wave number separation of fastest growing sideband component against wave steepness. Significant improvements are noticed from the results obtained from the two coupled third order nonlinear Schrödinger equations.

Single-wave run-up on sloping beaches

1976 ◽  
Vol 74 (4) ◽  
pp. 685-694 ◽  
Author(s):  
Lester Q. Spielvogel
Keyword(s):  
Deep Water ◽  
Single Wave ◽  
Initial Wave ◽  
Wave Trains ◽  
Run Up

Possibilities of high shoreline amplification and run-up are investigated. A shoreline amplification of magnitude 5·38 and a tsunamigenic (deep water) amplification of magnitude 5·71 are obtained from single waves without analytic or computational difficulties. It is not claimed that these are a maximum, but rather it is conjectured that arbitrarily high run-up and amplification can be obtained provided that the correct initial wave trains are chosen.

Non-linear dispersion of water waves

1967 ◽  
Vol 27 (2) ◽  
pp. 399-412 ◽  
Author(s):  
G. B. Whitham
Keyword(s):  
Water Waves ◽  
Wave Train ◽  
Periodic Wave ◽  
Wave Number ◽  
Amplitude Wave ◽  
Linear Dispersion ◽  
Elliptic Case ◽  
Non Linear ◽  
Wave Trains ◽  
Linear Water Waves

The slow dispersion of non-linear water waves is studied by the general theory developed in an earlier paper (Whitham 1965b). The average Lagrangian is calculated from the Stokes expansion for periodic wave trains in water of arbitrary depth. This Lagrangian can be used for the various applications described in the above reference. In this paper, the crucial question of the ‘type’ of the differential equations for the wave-train parameters (local amplitude, wave-number, etc.) is established. The equations are hyperbolic or elliptic according to whetherkh0is less than or greater than 1.36, wherekis the wave-number per 2π andh0is the undisturbed depth. In the hyperbolic case, changes in the wave train propagate and the characteristic velocities give generalizations of the linear group velocity. In the elliptic case, modulations in the wave train grow exponentially and a periodic wave train will be unstable in this sense; thus, periodic wave trains on water will be unstable ifkh0> 1·36, The instability of deep-water waves,kh0> 1·36, was discovered in a different way by Benjamin (1966). The relation between the two approaches is explained.

scholarly journals Emergence of Solitons from Irregular Waves in Deep Water

2021 ◽  
Vol 9 (12) ◽  
pp. 1369
Author(s):  
Weida Xia ◽  
Yuxiang Ma ◽  
Guohai Dong ◽  
Jie Zhang ◽  
Xiaozhou Ma
Keyword(s):  
Deep Water ◽  
Rogue Wave ◽  
Wave Train ◽  
Irregular Waves ◽  
Peak Wavelength ◽  
Long Distance ◽  
Irregular Wave ◽  
Resonant Interactions ◽  
Dispersion Effects ◽  
Wave Trains

Numerical simulations were performed to study the long-distance evolution of irregular waves in deep water. It was observed that some solitons, which are the theoretical solutions of the nonlinear Schrödinger equation, emerged spontaneously as irregular wave trains propagated in deep water. The solitons propagated approximately at a speed of the linear group velocity. All the solitons had a relatively large amplitude and one detected soliton’s height was two times larger than the significant wave height of the wave train, therefore satisfying the rogue wave definition. The numerical results showed that solitons can persist for a long distance, reaching about 65 times the peak wavelength. By analyzing the spatial variations of these solitons in both time and spectral domains, it is found that the third-and higher-order resonant interactions and dispersion effects played significant roles in the formation of solitons.

Effect of Finite Rotation on the Propagation of Elastic Waves in Constrained Mechanical Systems

1992 ◽  
Vol 114 (3) ◽  
pp. 384-393 ◽  
Author(s):  
Wei-Hsin Gau ◽  
A. A. Shabana
Keyword(s):  
Phase Velocity ◽  
Material Properties ◽  
Elastic Waves ◽  
Wave Motion ◽  
Coefficient Of Restitution ◽  
Wave Number ◽  
Harmonic Waves ◽  
Finite Rotation ◽  
Phase Velocities ◽  
The Impact

In structural systems, impact-induced longitudinal elastic waves travel with finite speeds that depend on the material properties. Using Fourier method of analysis, the exact wave motion can be described as the sum of infinite number of harmonic waves which have the same phase velocity. In this case the medium is said to be nondispersive, since the phase velocities of the harmonic waves are equal and equal to the group velocity of the resulting wave motion. In mechanism systems with intermittent motion, on the other hand, elastic members undergo finite rotations. In this investigation, the effect of the finite rotation, coefficient of restitution, and impact conditions on the propagation of the impact-induced waves in costrained elastic systems is examined. The system equations of motion are developed using the principle of virtual work in dynamics. The jump discontinuities in the system variables as the result of impact are predicted using the generalized impulse momentum equations that involve the coefficient of restitution. It is shown that the phase velocities of different harmonic waves are no longer equal, that is, dispersion occurs in perfectly elastic mechanism members as the result of the finite rotation. The analysis presented in this paper shows that the finite rotation has more significant effect on the phase velocity of the low frequency harmonics as compared to the high frequency harmonics. A rotation-wave number that depends on the material properties and the wave length is defined for each harmonic wave. It is shown that if the angular velocity of the elastic member becomes large such that the rotation-wave number of a mode exceeds one, the associated modal displacement is no longer oscillatory.

scholarly journals On the scattering of X- and γ- rays by rings of electrons.—The effect of damping of the incident radiation

1923 ◽  
Vol 104 (724) ◽  
pp. 153-164 ◽  
Keyword(s):  
High Frequency ◽  
Previous Investigation ◽  
Wave Train ◽  
Harmonic Wave ◽  
Incident Radiation ◽  
The Other ◽  
Other Hand ◽  
Γ Rays ◽  
Wave Trains

1. The object of the present investigation is to examine whether damping of the incident radiation can account for the abnormally small scattering of hard γ -rays by aluminium, iron and lead, which was observed by Ishino and was left unexplained in a previous investigation, based on the hypothesis that scattering is a phenomenon of the diffraction by electron rings in the atom of undamped simple harmonic wave-trains of high frequency. It may be stated at once that the result of the enquiry is negative: it is true that damping of an amount small enough to be consistent with the generation of moderately sharp lines in the X- and γ -ray spectrum diminishes the scattering of long waves, but it increases that of short waves, and in each case the change is far too small to explain Ishino’s result. On the other hand, very large damping probably diminishes the scattering of all waves, but it is practically certain that it does not diminish it below the amount required by the Simple Pulse theory, whilst Ishino’s values are only of the order of one-quarter of that amount. This result was to be expected, since an infinitely damped wave-train may be regarded as equivalent to a pulse.

scholarly journals On the Synthesis of Realistic Sea States

1980 ◽  
Vol 1 (17) ◽  
pp. 176 ◽  
Author(s):  
E.R. Funke ◽  
E.P.D. Mansard
Keyword(s):  
Spectral Density ◽  
Wave Train ◽  
Random Noise ◽  
Structural Response ◽  
The Other ◽  
Sea State ◽  
Floating Structures ◽  
Wave Heights ◽  
Wave Trains ◽  
The One

Recent investigations by some researchers (Johnson et al, 1978; Burchart, 1979; Gravesen and Sorensen, 1977) have indicated that it is no longer sufficient to match the variance spectral density of a simulated sea state to that of the prototype. When testing models of various fixed and floating structures, it appears to be most important to simulate the wave grouping phenomenon as veil. Some researchers also believe that the wave steepness, the particular sequencing of high and low waves (Burchart, 1979) and the ratio of the maximum to the significant wave height within a wave train are also of significance. Methods for the generation of 'random1 waves throughout the world vary greatly. One may, however, categorize these in terms of two substantially different approaches. These may be referred to as "probabilistic" on the one hand and "deterministic" on the other. In the former, a random or pseudo-random noise source is used which will never repeat or which has a very long repetition period. The assumption is then made that, in the course of the long testing period, all possible outcomes of wave heights, wave periods and wave groups will occur. The only constraint, which is usually placed on the synthesis, is the shape of the variance spectral density and its zeroth moment. The "deterministic" approach, on the other hand, attempts to create very specific and typically extreme conditions. Subsequent analysis of structural response to these conditions must, of course, be related to the likelihood of these conditions occurring in the prevailing climate. The old standby method of testing with monochromatic waves is a typical example of this category. However, other technigues such as Funke and Mansard (1979a) and the reproduction of prototype wave trains as favoured by several laboratories (Gravesen and Sorensen, 1977) may also be described as deterministic.

Effects of wavelength ratio on wave modelling

1993 ◽  
Vol 248 ◽  
pp. 107-127 ◽  
Author(s):  
Jun Zhang ◽  
Keyyong Hong ◽  
Dick K. P. Yue
Keyword(s):  
Wave Model ◽  
Short Wave ◽  
Wave Mode ◽  
The Other ◽  
Wave Steepness ◽  
Wave Interactions ◽  
Third Order ◽  
Long Wavelength ◽  
Modulated Phase ◽  
Wave Trains

The efficacy of perturbation approaches for short–long wave interactions is examined by considering a simple case of two interacting wave trains with different wavelengths. Frequency-domain solutions are derived up to third order in wave steepness using two different formulations: one employing conventional wave-mode functions only, and the other introducing a modulated wave-mode representation for the short-wavelength wave. For long-wavelength wave steepness and short-to-long wavelength ratio ε1 and ε3 respectively, the two results are shown to be identical for ε1 [Lt ] ε3 < 0.5. As ε1 approaches ε3, the conventional wave-mode approach converges slowly and eventually diverges for ε1 [Gt ] ε3. The loss of convergence is because the linear phase of conventional wave-mode functions is ineffective for modelling the modulated phase of the short wave. As expected, this difficulty can be removed by using a modulated wave-mode function for the short wave. On the other hand, for relatively large ε3 ∼O(1), the conventional wave-mode approach converges rapidly while the slowly varying interaction between the two waves cannot be accurately predicted by the present modulated wave-mode approach. These findings have important implications to (time-domain) numerical simulations of the nonlinear evolution of ocean wave fields, and suggest that a hybrid wave model employing both conventional (for large-ε3 interactions) and modulated (for small-ε3 interactions) wave-mode functions should be particularly effective.

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