scholarly journals Third-order stokes wave solutions of the free surface capillary-gravity wave and the interfacial internal wave

2017 ◽  
Vol 31 (6) ◽  
pp. 781-787
Author(s):  
Rui-jun Meng ◽  
Ji-feng Cui ◽  
Xiao-gang Chen ◽  
Bao-le Zhang ◽  
Hong-bo Zhang
Keyword(s):  
Free Surface ◽  
Internal Wave ◽  
Gravity Wave ◽  
Stokes Wave ◽  
Third Order ◽  
Wave Solutions

The limiting form of internal waves

1984 ◽  
Vol 394 (1807) ◽  
pp. 329-344 ◽  
Keyword(s):  
Free Surface ◽  
Internal Wave ◽  
Internal Waves ◽  
Solitary Waves ◽  
The Other ◽  
Stokes Wave ◽  
Internal Solitary Waves ◽  
Two Dimensional ◽  
Extreme Form ◽  
Eddy Formation

The bifurcation of two-dimensional internal solitary waves in a perfect density stratified fluid between horizontal walls under gravity is studied near to a point of incipient eddy formation. It is shown that eddies do not attach to the walls. Moreover, along the bifurcating branch there is always a flow with a singular cusped streamline before the formation of eddies. This flow with the cusped streamline is an example of what we call an internal wave of limiting form, by analogy with the Stokes wave of extreme form in the free surface problem. Two examples are given where the primary density stratification ensures the existence of a limiting wave of depression in one case, and of elevation in the other.

Third-order Stokes wave solutions for interfacial internal waves in a three-layer density-stratified fluid

2009 ◽  
Vol 18 (5) ◽  
pp. 1906-1916 ◽  
Author(s):  
Chen Xiao-Gang ◽  
Guo Zhi-Ping ◽  
Song Jin-Bao ◽  
He Xiao-Dong ◽  
Guo Jun-Ming ◽  
...  
Keyword(s):  
Internal Waves ◽  
Stratified Fluid ◽  
Stokes Wave ◽  
Third Order ◽  
Layer Density ◽  
Wave Solutions

Abundant rogue wave solutions for the (2 + 1)-dimensional generalized Korteweg–de Vries equation

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Huanhuan Lu ◽  
Yufeng Zhang
Keyword(s):  
Vries Equation ◽  
Rogue Wave ◽  
Rogue Waves ◽  
Third Order ◽  
First Order ◽  
Wave Solutions ◽  
De Vries ◽  
Rogue Wave Solutions ◽  
Bilinear Formulation ◽  
Hirota Bilinear

AbstractIn this paper, we analyse two types of rogue wave solutions generated from two improved ansatzs, to the (2 + 1)-dimensional generalized Korteweg–de Vries equation. With symbolic computation, the first-order rogue waves, second-order rogue waves, third-order rogue waves are generated directly from the first ansatz. Based on the Hirota bilinear formulation, another type of one-rogue waves and two-rogue waves can be obtained from the second ansatz. In addition, the dynamic behaviours of obtained rogue wave solutions are illustrated graphically.

scholarly journals Surface and internal waves in a liquid of variable depth

1969 ◽  
Vol 1 (1) ◽  
pp. 29-46 ◽  
Author(s):  
D. G. Hurley ◽  
J. Imberger
Keyword(s):  
Free Surface ◽  
Gravity Wave ◽  
Internal Waves ◽  
Infinite Number ◽  
Variable Depth ◽  
Constant Depth ◽  
Stratified Liquid

Consider a stably stratified liquid, whose density varies exponentially with the vertical co-ordinate, that is bounded above by a free surface and below by a bed whose height depends on only one of the horizontal co-ordinates. Suppose that a gravity wave, that may be either a surface or an internal one, is travelling in a direction normal to the lines of constant depth. It is shown that if the frequency is below a certain value an infinite number of waves, all of the same frequency but having differing wave lengths, are generated and expressions for their amplitude are given in terms of the changes in depth which are assumed to be small.

Lax pair, Darboux transformation and rogue-periodic waves of a nonlinear Schrödinger–Hirota equation with the spatio-temporal dispersion and Kerr law nonlinearity in nonlinear optics

2021 ◽  
pp. 2150451 ◽  
Author(s):  
Cheng-Cheng Wei ◽  
Bo Tian ◽  
Qi-Xing Qu ◽  
Su-Su Chen ◽  
Dan-Yu Yang
Keyword(s):  
Darboux Transformation ◽  
Periodic Wave ◽  
Lax Pair ◽  
Third Order ◽  
Periodic Wave Solutions ◽  
The Third ◽  
Third Order Dispersion ◽  
Wave Solutions ◽  
Minimum Amplitude ◽  
Order Dispersion

For a nonlinear Schrödinger–Hirota equation with the spatio-temporal dispersion and Kerr law nonlinearity in nonlinear optics, we derive a Lax pair, a Darboux transformation and two families of the periodic-wave solutions via the Jacobian elliptic functions dn and cn. We construct the linearly-independent and non-periodic solutions of that Lax pair, and substitute those solutions into the Darboux transformation to get the rogue-periodic-wave solutions. When the third-order dispersion or group velocity dispersion (GVD) or inter-modal dispersion (IMD) increases, the maximum amplitude of the rogue-periodic wave remains unchanged. From the rogue-dn-periodic-wave solutions, when the GVD decreases, the minimum amplitude of the rogue-dn-periodic wave decreases. When the third-order dispersion decreases, the minimum amplitude of the rogue-dn-periodic wave rises. Decrease of the IMD causes the period of the rogue-dn-periodic wave to decrease. From the rogue-cn-periodic-wave solutions, when the GVD increases, the minimum amplitude of the rogue-cn-periodic wave decreases. Increase of the third-order dispersion or IMD leads to the decrease of the period.

A Two-Dimensional Potential Flow Model of the Wave Field Generated by a Semisubmerged Body in Heaving Motion

1988 ◽  
Vol 32 (02) ◽  
pp. 83-91
Author(s):  
X. M. Wang ◽  
M. L. Spaulding
Keyword(s):  
Free Surface ◽  
Boundary Conditions ◽  
Potential Flow ◽  
Flow Model ◽  
Second Order ◽  
Two Dimensional ◽  
Third Order ◽  
The Third ◽  
Potential Flow Model ◽  
Good Agreement

A two-dimensional potential flow model is formulated to predict the wave field and forces generated by a sere!submerged body in forced heaving motion. The potential flow problem is solved on a boundary fitted coordinate system that deforms in response to the motion of the free surface and the heaving body. The full nonlinear kinematic and dynamic boundary conditions are used at the free surface. The governing equations and associated boundary conditions are solved by a second-order finite-difference technique based on the modified Euler method for the time domain and a successive overrelaxation (SOR) procedure for the spatial domain. A series of sensitivity studies of grid size and resolution, time step, free surface and body grid redistribution schemes, convergence criteria, and free surface body boundary condition specification was performed to investigate the computational characteristics of the model. The model was applied to predict the forces generated by the forced oscillation of a U-shaped cylinder. Numerical model predictions are generally in good agreement with the available second-order theories for the first-order pressure and force coefficients, but clearly show that the third-order terms are larger than the second-order terms when nonlinearity becomes important in the dimensionless frequency range 1≤ Fr≤ 2. The model results are in good agreement with the available experimental data and confirm the importance of the third order terms.

Nonlinear Aspects of Subcritical Shallow-Water Flow Past Two-Dimensional Obstructions

1978 ◽  
Vol 22 (04) ◽  
pp. 203-211
Author(s):  
Nils Salvesen ◽  
C. von Kerczek
Keyword(s):  
Free Surface ◽  
Shallow Water ◽  
Flow Problem ◽  
Free Stream ◽  
The Body ◽  
Two Dimensional ◽  
Third Order ◽  
Submerged Body ◽  
The Mean ◽  
The Waves

Some nonlinear aspects of the two-dimensional problem of a submerged body moving with constant speed in otherwise undisturbed water of uniform depth are considered. It is shown that a theory of Benjamin which predicts a uniform rise of the free surface ahead of the body and the lowering of the mean level of the waves behind it agrees well with experimental data. The local steady-flow problem is solved by a numerical method which satisfies the exact free-surface conditions. Third-order perturbation formulas for the downstream free waves are also presented. It is found that in sufficiently shallow water, the wavelength increases with increasing disturbance strength for fixed values of the free-stream-Froude number. This is opposite to the deepwater case where the wavelength decreases with increasing disturbance strength.

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