Exact large amplitude capillary waves on sheets of fluid

Kinnersley, William

doi:10.1017/s0022112076002085

Cited by 112 publications

(119 citation statements)

References 3 publications

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“…However, interestingly Crapper [16] showed the existence of explicit capillary solutions for irrotational water waves of infinite depth. Kinnersley [17] then showed a similar existence of waves in the case of irrotational flow over a flat bed. Both of these types of waves are symmetric regular waves, and in both cases, the streamlines are analytic.…”

Section: Introductionmentioning

confidence: 83%

Regularity for steady periodic capillary water waves with vorticity

Henry

2012

Phil. Trans. R. Soc. A.

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In the following, we prove new regularity results for two-dimensional steady periodic capillary water waves with vorticity, in the absence of stagnation points. Firstly, we prove that if the vorticity function has a Hölder-continuous first derivative, then the free surface is a smooth curve (C ∞ ) and the streamlines beneath the surface will be real analytic. Furthermore, once we assume that the vorticity function is real analytic, it will follow that the wave surface profile is itself also analytic. A particular case of this result includes irrotational fluid flow where the vorticity is zero. The property of the streamlines being analytic allows us to gain physical insight into small-amplitude waves by justifying a power-series approach.

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Section: Introductionmentioning

confidence: 83%

Regularity for steady periodic capillary water waves with vorticity

Henry

2012

Phil. Trans. R. Soc. A.

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“…It was followed by Crapper [14], who studied pure capillary waves in water of infinite depth, and Kinnersley [28], who generalized the theory to finite depth. For nonlinear water waves, the existence of the fully nonlinear equations was proven in [12,13,38,41], firstly for gravity water waves and then for waves incorporating surface tension.…”

Section: Introductionmentioning

confidence: 99%

A New Lagrangian Asymptotic Solution for Gravity–Capillary Waves in Water of Finite Depth

Hsu

Hwung

2010

J. Math. Fluid Mech.

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Abstract.A third-order Lagrangian asymptotic solution is derived for gravity-capillary waves in water of finite depth. The explicit parametric solution gives the trajectory of a water particle and the wave kinematics for Lagrangian points above the mean water level, and in a water column. The water particle orbits and mass transport velocity as functions of the surface tension are obtained. Some remarkable trajectories may contain one or multiple sub-loops for steep waves and large surface tension. Overall, an increase in surface tension tends to increase the motions of surface particles including the relative horizontal distance travelled by a particle as well as the time-averaged drift velocity.Mathematics Subject Classification (2010). Primary 76B15; Secondary 35Q35, 34C05, 34C25.

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“…Kinnersley extended the Crapper waves to the case of finite depth [4]. As in the infinite depth case that Crapper studied [1], the travelling waves are found as exact solutions, in this case involving elliptic functions.…”

Section: Introductionmentioning

confidence: 97%

Gravity perturbed Crapper waves

2014

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Crapper waves are a family of exact periodic travelling wave solutions of the free-surface irrotational incompressible Euler equations; these are pure capillary waves, meaning that surface tension is accounted for, but gravity is neglected. For certain parameter values, Crapper waves are known to have multi-valued height. Using the implicit function theorem, we prove that any of the Crapper waves can be perturbed by the effect of gravity, yielding the existence of gravity-capillary waves nearby to the Crapper waves. This result implies the existence of travelling gravity-capillary waves with multi-valued height. The solutions we prove to exist include waves with both positive and negative values of the gravity coefficient. We also compute these gravity perturbed Crapper waves by means of a quasi-Newton iterative scheme (again, using both positive and negative values of the gravity coefficient). A phase diagram is generated, which depicts the existence of singlevalued and multi-valued travelling waves in the gravity-amplitude plane. A new largest water wave is computed, which is composed of a string of bubbles at the interface.

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Exact large amplitude capillary waves on sheets of fluid

Cited by 112 publications

References 3 publications

Regularity for steady periodic capillary water waves with vorticity

Regularity for steady periodic capillary water waves with vorticity

A New Lagrangian Asymptotic Solution for Gravity–Capillary Waves in Water of Finite Depth

Gravity perturbed Crapper waves

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