The resonance of modes in the problem of two-dimensional capillary-gravity waves

Okamoto, Hisashi; Shoji, Masataka

doi:10.1016/0167-2789(96)00071-1

“…The curvature κ can be expressed by f as in [12]. Using |v| 2 = |W | 2 and iΓ ζ = w (ζ) = W (f (ζ))f (ζ) (8) we obtain a boundary condition for f…”

Section: Equationsmentioning

confidence: 99%

“…The situation is much more complicated, when secondary bifurcation and mode interaction are considered [1]. Resonance has been studied by Christodoulides and Dias [2] and also by Okamoto and Shoji [8].…”

Section: Introductionmentioning

confidence: 97%

Numerical calculation of capillary-gravity waves

Wegmann

¹

2002

Numerische Mathematik

0

View full text Add to dashboard Cite

show abstract

“…[15,25,30] for analytic results and, e.g. [26,28,29] and the references contained therein for some numerical results.) For general vorticity, this scenario was analyzed by Wahlén [32].…”

Section: Introductionmentioning

confidence: 99%

Steady periodic gravity waves with surface tension

Walsh

2009

Preprint

View full text Add to dashboard Cite

In this paper we consider two-dimensional, stratified, steady water waves propagating over an impermeable flat bed and with a free surface. The motion is assumed to be driven by capillarity (that is, surface tension) on the surface and a gravitational force acting on the body of the fluid. We prove the existence of global continua of classical solutions that are periodic and traveling. This is accomplished by first constructing a 1-parameter family of laminar flow solutions, T , then applying bifurcation theory methods to obtain local curves of small amplitude solutions branching from T at an eigenvalue of the linearized problem. Each solution curve is then continued globally by means of a degree theoretic theorem in the spirit of Rabinowitz. Finally, we complement the degree theoretic picture by proving an alternate global bifurcation theorem via the analytic continuation method of Dancer.

show abstract

“…[14,21,25] for analytic results and, e.g. [22,23,24] and the references contained therein for some numerical results.) For general vorticity, this scenario was analyzed by Wahlén [27].…”

mentioning

confidence: 99%

Steady stratified periodic gravity waves with surface tension I: Local bifurcation

Walsh¹

2014

Discrete &Amp; Continuous Dynamical Systems - A

View full text Add to dashboard Cite

In this paper we consider two-dimensional, stratified, steady water waves propagating over an impermeable flat bed and with a free surface. The motion is assumed to be driven by capillarity (that is, surface tension) on the surface and a gravitational force acting on the body of the fluid. We prove the existence of small-amplitude solutions. This is accomplished by first constructing a 1-parameter family of laminar flow solutions, T , then applying bifurcation theory methods to obtain local curves of small amplitude solutions branching from T at an eigenvalue of the linearized problem.

show abstract

The resonance of modes in the problem of two-dimensional capillary-gravity waves

Cited by 6 publications

References 13 publications

Numerical calculation of capillary-gravity waves

Numerical calculation of capillary-gravity waves

Steady periodic gravity waves with surface tension

Steady stratified periodic gravity waves with surface tension I: Local bifurcation

Contact Info

Product

Resources

About