Uniqueness in the water-wave problem for bodies intersecting the free surface at arbitrary angles

2004 ◽  
Vol 332 (1) ◽  
pp. 73-78 ◽  
Author(s):  
Nikolay Kuznetsov
Keyword(s):  
Free Surface ◽  
Water Wave ◽  
Wave Problem ◽  
Water Wave Problem

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.

scholarly journals An example of non-uniqueness in the two-dimensional linear water wave problem

1996 ◽  
Vol 315 ◽  
pp. 257-266 ◽  
Author(s):  
M. McIver
Keyword(s):  
Free Surface ◽  
Numerical Calculation ◽  
Water Wave ◽  
Two Dimensional ◽  
Wave Problem ◽  
Water Wave Problem ◽  
Source Point ◽  
The Waves

An example of non-uniqueness in the two-dimensional, linear water wave problem is obtained by constructing a potential which does not radiate any waves to infinity and whose streamline pattern represents the flow around two surface-piercing bodies. The potential is constructed from two wave sources which are positioned in the free surface in such a way that the waves radiated from each source cancel at infinity. A numerical calculation of the streamline pattern indicates that there are at least two streamlines which represent surface-piercing bodies, each of which encloses a source point. A proof of the existence of these lines is then given.

A variational principle for a fluid with a free surface

1967 ◽  
Vol 27 (2) ◽  
pp. 395-397 ◽  
Author(s):  
J. C. Luke
Keyword(s):  
Free Surface ◽  
Variational Principle ◽  
Potential Energy ◽  
Water Wave ◽  
Equations Of Motion ◽  
Lagrangian Function ◽  
Wave Problem ◽  
Water Wave Problem

The full set of equations of motion for the classical water wave problem in Eulerian co-ordinates is obtained from a Lagrangian function which equals the pressure. This Lagrangian is compared with the more usual expression formed from kinetic minus potential energy.

A note on uniqueness in the linearized water-wave problem

1999 ◽  
Vol 386 ◽  
pp. 5-14 ◽  
Author(s):  
N. G. KUZNETSOV ◽  
M. J. SIMON
Keyword(s):  
Free Surface ◽  
Finite Number ◽  
Water Wave ◽  
Axisymmetric Problem ◽  
Two Dimensional ◽  
Plane Region ◽  
Wave Problem ◽  
Water Wave Problem ◽  
Infinite Depth ◽  
The Uniqueness Theorem

The uniqueness theorem of Simon & Ursell (1984), concerning the linearized two-dimensional water-wave problem in a fluid of infinite depth, is extended in two directions. First, we consider a two-dimensional geometry involving two submerged symmetric bodies placed sufficiently far apart that they are not confined in the vertical right angle having its vertex on the free surface as the theorem of Simon & Ursell requires. A condition is obtained guaranteeing the uniqueness outside a finite number of bounded frequency intervals. Secondly, the method of Simon & Ursell is generalized to prove uniqueness in the axisymmetric problem for bodies violating John's condition provided the free surface is a connected plane region.

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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