A method for the calculation of nonsymmetric steady periodic capillary–gravity waves on water of arbitrary uniform depth (original) (raw)

Investigation of symmetry breaking in periodic gravity–capillary waves

Journal of Fluid Mechanics, 2016

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 stre...

Model Equations for Gravity-Capillary Waves in Deep Water

Studies in Applied Mathematics, 2008

The Euler equations for water waves in any depth have been shown to have solitary wave solutions when the effect of surface tension is included. This paper proposes three quadratic model equations for these types of waves in infinite depth with a two-dimensional fluid domain. One model ...

A new limiting form for steady periodic gravity waves with surface tension on deep water

Physics of Fluids, 1996

The method developed by Longuet-Higgins [J. Inst. Math. Appl. 22, 261 (1978)] for the computation of pure gravity waves is extended to capillary-gravity waves in deep water. Surface tension provides an additional term in the identities between the Fourier coefficients in Stokes' expansion. This term is then reduced to a simple function of the slope of the local tangent to the profile of the free surface. A set of nonlinear algebraic equations is derived and solved by using the Newton's method. A new family of limiting profiles of steady gravity waves with surface tension is found.

Transversally periodic solitary gravity-capillary waves

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2013

When both gravity and surface tension effects are present, surface solitary water waves are known to exist in both two- and three-dimensional infinitely deep fluids. We describe here solutions bridging these two cases: travelling waves which are localized in the propagation direction and periodic in the transverse direction. These transversally periodic gravity–capillary solitary waves are found to be of either elevation or depression type, tend to plane waves below a critical transverse period and tend to solitary lumps as the transverse period tends to infinity. The waves are found numerically in a Hamiltonian system for water waves simplified by a cubic truncation of the Dirichlet-to-Neumann operator. This approximation has been proved to be very accurate for both two- and three-dimensional computations of fully localized gravity–capillary solitary waves. The stability properties of these waves are then investigated via the time evolution of perturbed wave profiles.

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

Journal of Mathematical Fluid Mechanics, 2010

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