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.

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.

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.

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.

scholarly journals Regularity of rotational travelling water waves

2012 ◽  
Vol 370 (1964) ◽  
pp. 1602-1615 ◽  
Author(s):  
Joachim Escher
Keyword(s):  
Water Waves ◽  
Water Wave ◽  
Travelling Wave ◽  
Capillary Waves ◽  
Wave Profile ◽  
Wave Problem ◽  
Water Wave Problem ◽  
Common Assumption ◽  
Infinite Depth ◽  
Stagnation Points

Several recent results on the regularity of streamlines beneath a rotational travelling wave, along with the wave profile itself, will be discussed. The survey includes the classical water wave problem in both finite and infinite depth, capillary waves and solitary waves as well. A common assumption in all models to be discussed is the absence of stagnation points.

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