Singular solution of a water wave problem in an ocean of finite depth

1973 ◽  
Vol 11 (10) ◽  
pp. 1121-1130 ◽  
Author(s):  
A.S. Farooqui ◽  
D.L. Sharma
Keyword(s):  
Water Wave ◽  
Singular Solution ◽  
Finite Depth ◽  
Wave Problem ◽  
Water Wave Problem

Singular solution of a water wave problem

1973 ◽  
Vol 11 (2) ◽  
pp. 267-278 ◽  
Author(s):  
Darbari L. Sharma ◽  
A.S. Farooqui
Keyword(s):  
Water Wave ◽  
Singular Solution ◽  
Wave Problem ◽  
Water Wave Problem

scholarly journals Existence and amplitude bounds for irrotational water waves in finite depth

2017 ◽  
Vol 376 (2111) ◽  
pp. 20170094 ◽  
Author(s):  
Florian Kogelbauer
Keyword(s):  
Water Waves ◽  
Water Wave ◽  
Wave Speed ◽  
Finite Depth ◽  
Relative Mass ◽  
Wave Problem ◽  
Nonlinear Water Waves ◽  
Water Wave Problem ◽  
Pseudo Differential Equation ◽  
Nonlinear Solutions

We prove the existence of solutions to the irrotational water-wave problem in finite depth and derive an explicit upper bound on the amplitude of the nonlinear solutions in terms of the wavenumber, the total hydraulic head, the wave speed and the relative mass flux. Our approach relies upon a reformulation of the water-wave problem as a one-dimensional pseudo-differential equation and the Newton–Kantorovich iteration for Banach spaces. This article is part of the theme issue ‘Nonlinear water waves’.

Korteweg–de Vries type equations for waves propagating along the interface between air–water

2008 ◽  
Vol 86 (12) ◽  
pp. 1427-1435 ◽  
Author(s):  
A M Abourabia ◽  
M A Mahmoud ◽  
G M Khedr
Keyword(s):  
Water Waves ◽  
Water Wave ◽  
Fluid Layer ◽  
Wave Packets ◽  
Multiple Scale ◽  
Finite Depth ◽  
Wave Problem ◽  
Water Wave Problem ◽  
Kdv Equations ◽  
De Vries

We present solutions of the water wave problem for a fluid layer of finite depth in the presence of gravity and surface tension. The method of multiple scale expansion is employed to obtain the Korteweg–de Vries (KdV) equations for solitons, which describes the behavior of the system for the free surface between air and water in a nonlinear approach. The solutions of the water wave problem split up into two wave packets, one moving to the right and one to the left, where each of these wave packets evolves independently as the solutions of the KdV equations. The solutions of the KdV equations are obtained analytically by using the tanh-function method. The dispersion relations of the model KdV equations are studied. Finally, we observe that the elevation of the water waves are in the form of traveling solitary waves. The horizontal and vertical velocities, and the phase diagrams of the velocity components have a nonlinear characters.PACS No.: 47.11.St

scholarly journals An alternative approach to study irrotational periodic gravity water waves

2021 ◽  
Vol 72 (4) ◽  
Author(s):  
Biswajit Basu ◽  
Calin I. Martin
Keyword(s):  
Free Surface ◽  
Water Waves ◽  
Water Wave ◽  
Wave Problem ◽  
Water Wave Problem ◽  
Stagnation Points ◽  
Alternative Approach ◽  
New Formulation ◽  
Bounded Below ◽  
Surface Dynamic

AbstractWe are concerned here with an analysis of the nonlinear irrotational gravity water wave problem with a free surface over a water flow bounded below by a flat bed. We employ a new formulation involving an expression (called flow force) which contains pressure terms, thus having the potential to handle intricate surface dynamic boundary conditions. The proposed formulation neither requires the graph assumption of the free surface nor does require the absence of stagnation points. By way of this alternative approach we prove the existence of a local curve of solutions to the water wave problem with fixed flow force and more relaxed assumptions.

Unsteady non-linear waves in sloping channels

1958 ◽  
Vol 247 (1249) ◽  
pp. 186-198 ◽  
Keyword(s):  
Exact Solution ◽  
Water Wave ◽  
Legendre Function ◽  
Elliptic Integral ◽  
Vertical Wall ◽  
Auxiliary Function ◽  
Level Line ◽  
Wave Problem ◽  
Water Wave Problem ◽  
Non Linear

It is shown in general that the exact solution to every non-degenerate unsteady water-wave problem in a straight channel inclined at arbitrary slope, governed by the non-linear hydraulic equations, can be obtained in terms of the complete elliptic integral of the second kind, E . By means of a non-Newtonian reference frame, every such wave problem for a sloping channel can be replaced by an associated problem for a horizontal channel. For the latter, the partial differential equations become reducible and thus permit hodograph inversion. The Riemann integration method for the resulting Euler-Poisson equation yields an auxiliary function for these hydraulic problems which is transformable into a Legendre function and then into the elliptic integral. In particular, the procedure is applied to obtain the exact solution for the water wave in a sloping channel produced by sudden release of the triangular wedge of water (the reservoir) initially at rest behind a vertical wall. The behaviour of the solution is exhibited for convenience in two level-line charts, and representative wave profiles and velocity distributions are presented.

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