scholarly journals Investigation of symmetry breaking in periodic gravity–capillary waves

2016 ◽  
Vol 811 ◽  
pp. 622-641 ◽  
Author(s):  
T. Gao ◽  
Z. Wang ◽  
J.-M. Vanden-Broeck
Keyword(s):  
Conformal Mapping ◽  
Symmetry Breaking ◽  
Incompressible Fluid ◽  
Numerical Method ◽  
Finite Depth ◽  
Capillary Waves ◽  
Fully Nonlinear ◽  
Infinite Depth ◽  
Breaking Points ◽  
Continuation Parameter

In this paper, fully nonlinear non-symmetric periodic gravity–capillary waves propagating at the surface of an inviscid and incompressible fluid are investigated. This problem was pioneered analytically by Zufiria (J. Fluid Mech., vol. 184, 1987c, pp. 183–206) and numerically by Shimizu & Shōji (Japan J. Ind. Appl. Maths, vol. 29 (2), 2012, pp. 331–353). We use a numerical method based on conformal mapping and series truncation to search for new solutions other than those shown in Zufiria (1987c) and Shimizu & Shōji (2012). It is found that, in the case of infinite-depth, non-symmetric waves with two to seven peaks within one wavelength exist and they all appear via symmetry-breaking bifurcations. Fully exploring these waves by changing the parameters yields the discovery of new types of non-symmetric solutions which form isolated branches without symmetry-breaking points. The existence of non-symmetric waves in water of finite depth is also confirmed, by using the value of the streamfunction at the bottom as the continuation parameter.

scholarly journals On the propagation of a disturbance in a fluid under gravity

1910 ◽  
Vol 83 (563) ◽  
pp. 347-356 ◽  
Keyword(s):  
Incompressible Fluid ◽  
Gravity Wave ◽  
Integral Form ◽  
Finite Depth ◽  
Infinite Depth ◽  
Poisson Problem

In a recent paper Lord Rayleigh has called attention to the fact of the instantaneous propagation of a limited disturbance over the surface of heavy incompressible fluid. This instantaneity occurs in spite of the fact that the velocity of any simple-harmonic gravity wave is finite, the depth being finite and constant. As the points thereby raised are of some delicacy, and are not completely settled in the paper quoted, some further remarks on the phenomenon may not be superfluous. 2. Solution of the Cauchy-Poisson Problem for Finite Depth . In his treatment of this case Lord Rayleigh used the integral form of solution. There are, however, great difficulties raised in this way on account of lack of convergence at the surface. I proceed to obtain a solution in the form of a series similar in type to the known serial solution of the problem for infinite depth.

Exact large amplitude capillary waves on sheets of fluid

1976 ◽  
Vol 77 (2) ◽  
pp. 229-241 ◽  
Author(s):  
William Kinnersley
Keyword(s):  
Exact Solution ◽  
Large Amplitude ◽  
Elliptic Functions ◽  
Finite Depth ◽  
Capillary Waves ◽  
Infinite Depth ◽  
The Waves

We generalize Crapper's exact solution for capillary waves on fluid of infinite depth. We find two finite-depth solutions involving elliptic functions. We show they can also be interpreted as large amplitude symmetrical and antisymmetrical waves on a fluid sheet. Particularly interesting are the waves obtained from our solution in the limit when the fluid sheet is extremely thin.

On the classical solutions for a Rosenau–Korteweg-deVries–Kawahara type equation

2021 ◽  
pp. 1-23
Author(s):  
Giuseppe Maria Coclite ◽  
Lorenzo di Ruvo
Keyword(s):  
Cauchy Problem ◽  
Small Amplitude ◽  
Type Equation ◽  
Discrete Systems ◽  
Finite Depth ◽  
Capillary Waves ◽  
Classical Solutions ◽  
Well Posedness ◽  
Kawahara Equation ◽  
The Cauchy Problem

The Rosenau–Korteweg-deVries–Kawahara equation describes the dynamics of dense discrete systems or small-amplitude gravity capillary waves on water of a finite depth. In this paper, we prove the well-posedness of the classical solutions for the Cauchy problem.

Reflexion of water waves by a permeable barrier

1979 ◽  
Vol 95 (1) ◽  
pp. 141-157 ◽  
Author(s):  
C. Macaskill
Keyword(s):  
Water Waves ◽  
Water Wave ◽  
Finite Depth ◽  
Infinite Depth ◽  
Permeable Barrier ◽  
Impermeable Barrier ◽  
Linearized Problem ◽  
Special Case ◽  
Thin Barrier ◽  
Standard Techniques

The linearized problem of water-wave reflexion by a thin barrier of arbitrary permeability is considered with the restriction that the flow be two-dimensional. The formulation includes the special case of transmission through one or more gaps in an otherwise impermeable barrier. The general problem is reduced to a set of integral equations using standard techniques. These equations are then solved using a special decomposition of the finite depth source potential which allows accurate solutions to be obtained economically. A representative range of solutions is obtained numerically for both finite and infinite depth problems.

scholarly journals Local uniqueness in boundary problems

1970 ◽  
Vol 17 (1) ◽  
pp. 23-36
Author(s):  
M. H. Martin
Keyword(s):  
Incompressible Fluid ◽  
Unit Circle ◽  
Finite Amplitude ◽  
Local Uniqueness ◽  
Boundary Problems ◽  
Infinite Depth ◽  
Image Position

The study of periodic, irrotational waves of finite amplitude in an incompressible fluid of infinite depth was reduced by Levi-Civita (1) to the determination of a functionregular analytic in the interior of the unit circle ρ = 1 and which satisfies the condition

A new family of capillary waves

1980 ◽  
Vol 98 (1) ◽  
pp. 161-169 ◽  
Author(s):  
Jean-Marc Vanden-Broeck ◽  
Joseph B. Keller
Keyword(s):  
Integral Equation ◽  
Nonlinear Integral Equation ◽  
Finite Amplitude ◽  
Capillary Waves ◽  
Wave Steepness ◽  
Algebraic Equations ◽  
Infinite Depth ◽  
New Family ◽  
Nonlinear Algebraic Equations ◽  
Finite Set

A new family of finite-amplitude periodic progressive capillary waves is presented. They occur on the surface of a fluid of infinite depth in the absence of gravity. Each pair of adjacent waves touch at one point and enclose a bubble at pressure P. P depends upon the wave steepness s, which is the vertical distance from trough to crest divided by the wavelength. Previously Crapper found a family of waves without bubbles for 0 [les ] s [les ] s* = 0·730. Our solutions occur for all s > s** = 0·663, with the trough taken to be the bottom of the bubble. As s → ∞, the bubbles become long and narrow, while the top surface tends to a periodic array of semicircles in contact with one another. The solutions were obtained by formulating the problem as a nonlinear integral equation for the free surface. By introducing a mesh and difference method, we converted this equation into a finite set of nonlinear algebraic equations. These equations were solved by Newton's method. Graphs and tables of the results are included. These waves enlarge the class of phenomena which can occur in an ideal fluid, but they do not seem to have been observed.

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