scholarly journals A nonlinear Schrödinger equation for gravity–capillary water waves on arbitrary depth with constant vorticity. Part 1

2018 ◽  
Vol 854 ◽  
pp. 146-163 ◽  
Author(s):  
H. C. Hsu ◽  
C. Kharif ◽  
M. Abid ◽  
Y. Y. Chen
Keyword(s):  
Surface Tension ◽  
Growth Rate ◽  
Gravity Waves ◽  
Capillary Wave ◽  
Modulational Instability ◽  
Capillary Waves ◽  
Rate Of Growth ◽  
Positive Vorticity ◽  
Constant Vorticity ◽  
Wave Trains

A nonlinear Schrödinger equation for the envelope of two-dimensional gravity–capillary waves propagating at the free surface of a vertically sheared current of constant vorticity is derived. In this paper we extend to gravity–capillary wave trains the results of Thomas et al. (Phys. Fluids, 2012, 127102) and complete the stability analysis and stability diagram of Djordjevic & Redekopp (J. Fluid Mech., vol. 79, 1977, pp. 703–714) in the presence of vorticity. The vorticity effect on the modulational instability of weakly nonlinear gravity–capillary wave packets is investigated. It is shown that the vorticity modifies significantly the modulational instability of gravity–capillary wave trains, namely the growth rate and instability bandwidth. It is found that the rate of growth of modulational instability of short gravity waves influenced by surface tension behaves like pure gravity waves: (i) in infinite depth, the growth rate is reduced in the presence of positive vorticity and amplified in the presence of negative vorticity; (ii) in finite depth, it is reduced when the vorticity is positive and amplified and finally reduced when the vorticity is negative. The combined effect of vorticity and surface tension is to increase the rate of growth of modulational instability of short gravity waves influenced by surface tension, namely when the vorticity is negative. The rate of growth of modulational instability of capillary waves is amplified by negative vorticity and attenuated by positive vorticity. Stability diagrams are plotted and it is shown that they are significantly modified by the introduction of the vorticity.

Measurement of high frequency capillary waves on steep gravity waves

1978 ◽  
Vol 86 (3) ◽  
pp. 401-413 ◽  
Author(s):  
John H. Chang ◽  
Richard N. Wagner ◽  
Henry C. Yuen
Keyword(s):  
Surface Tension ◽  
High Resolution ◽  
Gravity Wave ◽  
Gravity Waves ◽  
Deep Water ◽  
High Frequency ◽  
Qualitative Agreement ◽  
Capillary Wave ◽  
Capillary Waves ◽  
Good Qualitative Agreement

The properties of high frequency capillary waves generated by steep gravity waves on deep water have been measured with a high resolution laser optical slope gauge. The results have been compared with the steady theory of Longuet-Higgins (1963). Good qualitative agreement is obtained. However, the quantitative predictions of the capillary wave slopes cannot be verified by the data because the theory requires knowledge of an idealized quantity - the crest curvature of the gravity wave in the absence of surface tension - which cannot be measured experimentally.

scholarly journals Transverse instability of gravity–capillary solitary waves on deep water in the presence of constant vorticity

2019 ◽  
Vol 871 ◽  
pp. 1028-1043
Author(s):  
M. Abid ◽  
C. Kharif ◽  
H.-C. Hsu ◽  
Y.-Y. Chen
Keyword(s):  
Growth Rate ◽  
Solitary Wave ◽  
Solitary Waves ◽  
Deep Water ◽  
Capillary Waves ◽  
Two Dimensional ◽  
Transverse Instability ◽  
Constant Vorticity ◽  
Dimensional Gravity ◽  
Wave Properties

The bifurcation of two-dimensional gravity–capillary waves into solitary waves when the phase velocity and group velocity are nearly equal is investigated in the presence of constant vorticity. We found that gravity–capillary solitary waves with decaying oscillatory tails exist in deep water in the presence of vorticity. Furthermore we found that the presence of vorticity influences strongly (i) the solitary wave properties and (ii) the growth rate of unstable transverse perturbations. The growth rate and bandwidth instability are given numerically and analytically as a function of the vorticity.

scholarly journals Nonlinear Evolution Equations for Broader Bandwidth Wave Packets in Crossing Sea States

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
S. Debsarma ◽  
S. Senapati ◽  
K. P. Das
Keyword(s):  
Growth Rate ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Modulational Instability ◽  
Wave Packets ◽  
Nonlinear Evolution Equations ◽  
Wave Spectra ◽  
Surface Gravity Wave ◽  
Freak Waves ◽  
Wave Trains

Two coupled nonlinear equations are derived describing the evolution of two broader bandwidth surface gravity wave packets propagating in two different directions in deep water. The equations, being derived for broader bandwidth wave packets, are applicable to more realistic ocean wave spectra in crossing sea states. The two coupled evolution equations derived here have been used to investigate the instability of two uniform wave trains propagating in two different directions. We have shown in figures the behaviour of the growth rate of instability of these uniform wave trains for unidirectional as well as for bidirectional perturbations. The figures drawn here confirm the fact that modulational instability in crossing sea states with broader bandwidth wave packets can lead to the formation of freak waves.

scholarly journals Two-dimensional stability of finite-amplitude gravity waves on water of finite depth with constant vorticity

2017 ◽  
Vol 830 ◽  
pp. 631-659 ◽  
Author(s):  
M. Francius ◽  
C. Kharif
Keyword(s):  
Growth Rate ◽  
Modulational Instability ◽  
Finite Depth ◽  
Finite Amplitude ◽  
Numerical Results ◽  
Two Dimensional ◽  
Weakly Nonlinear ◽  
Constant Vorticity ◽  
Shear Current ◽  
Modulational Instabilities

A numerical investigation of normal-mode perturbations of a two-dimensional periodic finite-amplitude gravity wave propagating on a vertically sheared current of constant vorticity is considered. For this purpose, an extension of the method developed by Rienecker & Fenton (J. Fluid Mech., vol. 104, 1981, pp. 119–137) is used for the numerical computations of the finite-amplitude waves on a linear shear current. This method enables to compute accurately waves with or without critical layers and pressure anomalies. The numerical results of the linear stability analysis extend the weakly nonlinear analytical results of Thomas et al. (Phys. Fluids, vol. 24, 2012, 127102) to fully nonlinear waves. In particular, the restabilization of the Benjamin–Feir modulational instability, whatever the depth, for an opposite shear current is confirmed. For these sideband instabilities, the numerical results show some deviations with the weakly nonlinear theory as the wave steepness of the basic wave and vorticity are increased. Besides the modulational instabilities, new instability bands corresponding to quartet and quintet instabilities, which are not sideband disturbances, are discovered. The present numerical results show that with opposite shear currents, increasing the shear reduces the growth rate of the most unstable sideband instabilities but enhances the growth rate of these quartet instabilities, which eventually dominate the Benjamin–Feir modulational instabilities.

Numerical calculations of steady gravity-capillary waves using an integro-differential formulation

1979 ◽  
Vol 94 (4) ◽  
pp. 777-793 ◽  
Author(s):  
James W. Rottman ◽  
D. B. Olfe
Keyword(s):  
Gravity Waves ◽  
Small Amplitude ◽  
Numerical Solutions ◽  
Capillary Wave ◽  
Capillary Waves ◽  
The Other ◽  
Wave Number ◽  
Infinite Depth ◽  
Free Surface Waves ◽  
Differential Formulation

A new integro-differential equation is derived for steady free-surface waves. Numerical solutions of this equation for periodic gravity-capillary waves on a fluid of infinite depth are presented. For the two limiting cases of gravity waves and capillary waves, our results are in excellent agreement with previous calculations. For gravity-capillary waves, detailed calculations are performed near the wave-number at which the classical second-order perturbation solution breaks down. Our calculations yield two solutions in this region, which in the limit of small amplitudes agree with the results obtained by Wilton in 1915; one solution has the small amplitude behaviour of a gravity wave and the other that of a capillary wave, but the numerical results show that at large amplitudes both waves have the characteristics of capillary waves. The calculations also show that the wavenumber range in which two solutions exist increases with increasing wave height.

scholarly journals Experimental study of dispersion and modulational instability of surface gravity waves on constant vorticity currents

2019 ◽  
Vol 884 ◽  
Author(s):  
James N. Steer ◽  
Alistair G. L. Borthwick ◽  
Dimitris Stagonas ◽  
Eugeny Buldakov ◽  
Ton S. van den Bremer
Keyword(s):  
Experimental Study ◽  
Gravity Waves ◽  
Modulational Instability ◽  
Surface Gravity ◽  
Surface Gravity Waves ◽  
Constant Vorticity ◽  
Image Position

scholarly journals Registration of high-frequency waves on the surface by the interference methods

2019 ◽  
Vol 213 ◽  
pp. 02075
Author(s):  
Anastasia Shmyrova ◽  
Andrey Shmyrov ◽  
Irina Mizeva ◽  
Alexey Mizev
Keyword(s):  
Surface Tension ◽  
Capillary Wave ◽  
Surface Profile ◽  
Wave Generation ◽  
Cylindrical Wave ◽  
Capillary Waves ◽  
Practical Implementation ◽  
Excitation Techniques ◽  
High Frequency Waves ◽  
Profile Reconstruction

Capillary waves are frequently used to measure the surface tension of liquids. However, this approach has not found wide application in the manufacture of modern commercial tensiometers because of the limitations imposed by capillary wave excitation techniques and the labor input associated with its practical implementation. In this paper we introduce a modified version of the capillary wave method which allows one to avoid the existing limitations and disadvantages. The distinguishing features of the proposed technique are as follows: acoustic wave generation and application of an interferometry technique for 3D surface profile reconstruction. A dynamic speaker with controlled vibration frequency and amplitude is used to produce acoustic vibrations. Application of a conventional Fizeau interferometer and the spatial phase shifting method makes it possible to perform surface form measurements with a high accuracy. For calculating wavelengths and the damping co-efficient, the surface profile is fitted with a decaying cylindrical wave equation. The accuracy of surface tension measurement by the modified capillary wave technique is 0.3 %. Owing to the non-contact way of wave generation and the small amounts of the examined fluid, the proposed method can be used in different studies.

Particle trajectories in linear periodic capillary and capillary–gravity water waves

2007 ◽  
Vol 365 (1858) ◽  
pp. 2241-2251 ◽  
Author(s):  
David Henry
Keyword(s):  
Surface Tension ◽  
Linear Theory ◽  
Significant Role ◽  
Water Waves ◽  
Small Amplitude ◽  
Capillary Wave ◽  
Capillary Waves ◽  
Particle Trajectories ◽  
Particle Paths

Surface tension plays a significant role as a restoration force in the setting of small-amplitude waves, leading to pure capillary and gravity-capillary waves. We show that within the framework of linear theory, the particle paths in a periodic gravity–capillary or pure capillary wave propagating at the surface of water over a flat bed are not closed.

Efficient Simulation of Short and Long-Wave Interactions With Applications to Capillary Waves

1994 ◽  
Vol 116 (1) ◽  
pp. 77-82 ◽  
Author(s):  
D. G. Dommermuth
Keyword(s):  
Gravity Waves ◽  
Capillary Wave ◽  
Perturbation Expansion ◽  
Capillary Waves ◽  
Front Face ◽  
Wave Interactions ◽  
Efficient Simulation ◽  
Ripple Formation ◽  
Long Wave ◽  
Unsteady Interaction

A perturbation expansion and a multigrid technique are developed for simulating the fully-nonlinear unsteady-interaction of short waves riding on long gravity waves. Both numerical techniques are capable of simulating wave slopes near breaking and wavelength ratios greater than thirty, but the multigrid technique converges more rapidly and it is more efficient. The results of numerical simulations agree qualitatively with experimental measurements of ripple formation on the front face of a gravity-capillary wave.

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