On the excitation of long nonlinear water waves by a moving pressure distribution

1984 ◽  
Vol 141 ◽  
pp. 455-466 ◽  
Author(s):  
T. R. Akylas
Keyword(s):  
Pressure Distribution ◽  
Water Waves ◽  
Vries Equation ◽  
Initial Conditions ◽  
Numerical Study ◽  
Equations Of Motion ◽  
Phase Speed ◽  
Finite Depth ◽  
Nonlinear Water Waves ◽  
De Vries

A study is made of the wave disturbance generated by a localized steady pressure distribution travelling at a speed close to the long-water-wave phase speed on water of finite depth. The linearized equations of motion are first used to obtain the large-time asymptotic behaviour of the disturbance in the far field; the linear response consists of long waves with temporally growing amplitude, so that the linear approximation eventually breaks down owing to finite-amplitude effects. A nonlinear theory is developed which shows that the generated waves are actually of bounded amplitude, and are governed by a forced Korteweg-de Vries equation subject to appropriate asymptotic initial conditions. A numerical study of the forced Korteweg-de Vries equation reveals that a series of solitons are generated in front of the pressure distribution.

scholarly journals On the equations of motion for mixtures of liquid and gas bubbles

1968 ◽  
Vol 33 (3) ◽  
pp. 465-474 ◽  
Author(s):  
L. Van Wijngaarden
Keyword(s):  
Vries Equation ◽  
Equations Of Motion ◽  
Bubble Diameter ◽  
Frequency Dispersion ◽  
Boussinesq Equations ◽  
Finite Depth ◽  
Finite Amplitude ◽  
Gas Content ◽  
Bubble Liquid ◽  
De Vries

On the basis of previous work by the author, equations are derived describing one-dimensional unsteady flow in bubble-fluid mixtures. Attention is subsequently focused on pressure waves of small and moderate amplitude propagating through the mixture. Four characteristic lengths occur, namely, wavelength, amplitude, bubble diameter and inter-bubble distance. The significance of their relative magnitudes for the theory is discussed. It appears that for high gas content the dispersion is weak and then the conservation of mass and momentum lead to equations similar to the Boussinesq equations, describing long dispersive waves of finite amplitude on a fluid of finite depth. For waves propagating in one direction only, the corresponding equation is similar to the Korteweg–de Vries equation.It is shown that for mixtures of low gas content the frequency dispersion is in most cases not small. Finally, solutions of the Korteweg–de Vries equation representing cnoidal and solitary waves in a bubble–liquid mixture are given explicitly.

The Korteweg-de Vries equation and water waves. Solutions of the equation. Part 1

1973 ◽  
Vol 59 (4) ◽  
pp. 721-736 ◽  
Author(s):  
Harvey Segur
Keyword(s):  
Water Waves ◽  
Large Time ◽  
Vries Equation ◽  
Asymptotic Properties ◽  
Series Representation ◽  
Convergent Series ◽  
Asymptotic Results ◽  
De Vries ◽  
Method Of Solution

The method of solution of the Korteweg–de Vries equation outlined by Gardneret al.(1967) is exploited to solve the equation. A convergent series representation of the solution is obtained, and previously known aspects of the solution are related to this general form. Asymptotic properties of the solution, valid for large time, are examined. Several simple methods of obtaining approximate asymptotic results are considered.

scholarly journals New Results on the Motions of Asteroids in Resonances

1992 ◽  
Vol 152 ◽  
pp. 145-152 ◽  
Author(s):  
R. Dvorak
Keyword(s):  
Initial Conditions ◽  
Numerical Study ◽  
Equations Of Motion ◽  
Close Approach ◽  
Integration Time ◽  
Time Interval ◽  
Time Intervals ◽  
3 Dimensional ◽  
Special Cases ◽  
Good Agreement

In this article we present a numerical study of the motion of asteroids in the 2:1 and 3:1 resonance with Jupiter. We integrated the equations of motion of the elliptic restricted 3-body problem for a great number of initial conditions within this 2 resonances for a time interval of 104 periods and for special cases even longer (which corresponds in the the Sun-Jupiter system to time intervals up to 106 years). We present our results in the form of 3-dimensional diagrams (initial a versus initial e, and in the z-axes the highest value of the eccentricity during the whole integration time). In the 3:1 resonance an eccentricity higher than 0.3 can lead to a close approach to Mars and hence to an escape from the resonance. Asteroids in the 2:1 resonance with Jupiter with eccentricities higher than 0.5 suffer from possible close approaches to Jupiter itself and then again this leads in general to an escape from the resonance. In both resonances we found possible regions of escape (chaotic regions), but only for initial eccentricities e > 0.15. The comparison with recent results show quite a good agreement for the structure of the 3:1 resonance. For motions in the 2:1 resonance our numeric results are in contradiction to others: high eccentric orbits are also found which may lead to escapes and consequently to a depletion of this resonant regions.

scholarly journals Soliton–Breather Interaction: The Modified Korteweg–de Vries Equation Framework

Symmetry ◽  
2020 ◽  
Vol 12 (9) ◽  
pp. 1445
Author(s):  
Ekaterina Didenkulova ◽  
Efim Pelinovsky
Keyword(s):  
Optical Fibers ◽  
Water Waves ◽  
Relative Phase ◽  
Vries Equation ◽  
Decisive Role ◽  
Pairwise Interactions ◽  
De Vries ◽  
Series Of Experiments ◽  
Stratified Ocean ◽  
The Moment

Pairwise interactions of particle-like waves (such as solitons and breathers) are important elementary processes that play a key role in the formation of the rarefied soliton gas statistics. Such waves appear in different physical systems such as deep water, shallow water waves, internal waves in the stratified ocean, and optical fibers. We study the features of different regimes of collisions between a soliton and a breather in the framework of the focusing modified Korteweg–de Vries equation, where cubic nonlinearity is essential. The relative phase of these structures is an important parameter determining the dynamics of soliton–breather collisions. Two series of experiments with different values of the breather’s and soliton’s relative phases were conducted. The waves’ amplitudes resulting from the interaction of coherent structures depending on their relative phase at the moment of collision were analyzed. Wave field moments, which play a decisive role in the statistics of soliton gases, were determined.

Irrotational water waves and the complex Korteweg-de Vries Equation

1996 ◽  
Vol 98 (2-4) ◽  
pp. 510-514 ◽  
Author(s):  
D. Levi ◽  
M. Sanielevici
Keyword(s):  
Water Waves ◽  
Vries Equation ◽  
De Vries

On the observability of finite-depth short-crested water waves

1996 ◽  
Vol 322 ◽  
pp. 1-19 ◽  
Author(s):  
M. Ioualalen ◽  
A. J. Roberts ◽  
C. Kharif
Keyword(s):  
Standing Wave ◽  
Water Waves ◽  
Numerical Study ◽  
Standing Waves ◽  
Finite Depth ◽  
Two Dimensional ◽  
Wave Fields ◽  
Progressive Waves ◽  
Wave Limit ◽  
Narrow Bands

A numerical study of the superharmonic instabilities of short-crested waves on water of finite depth is performed in order to measure their time scales. It is shown that these superharmonic instabilities can be significant-unlike the deep-water case-in parts of the parameter regime. New resonances associated with the standing wave limit are studied closely. These instabilities ‘contaminate’ most of the parameter space, excluding that near two-dimensional progressive waves; however, they are significant only near the standing wave limit. The main result is that very narrow bands of both short-crested waves ‘close’ to two-dimensional standing waves, and of well developed short-crested waves, perturbed by superharmonic instabilities, are unstable for depths shallower than approximately a non-dimensional depth d= 1; the study is performed down to depth d= 0.5 beyond which the computations do not converge sufficiently. As a corollary, the present study predicts that these very narrow sub-domains of short-crested wave fields will not be observable, although most of the short-crested wave fields will be.

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