The Third‐Harmonic Resonance for Capillary‐Gravity Waves with O(2) Spatial Symmetry

Abstract

It is well known that the addition of surface-tension effects to the classic Stokes model for water waves results in a countable infinity of values of the surface tension coefficient at which two traveling waves of differing wavelength travel at the same speed. In this paper the third-harmonic resonance (interaction of a one-crested wave with a three-crested wave) with 0(2) spatial symmetry is considered. Nayfeh analyzed the third-harmonic resonance for traveling waves and found two classes of solutions. It is shown that there are in fact six classes of periodic solutions when the 0(2) symmetry is acknowledged. The additional solutions are standing waves, mixed waves and secondary branches of "Z-waves." The normal form and symmetry group for each of the solution classes are developed, and the coefficients in the normal form are formally computed using a perturbation method. The physical aspects of the most unusual class of waves (three-mode mixed waves) are illustrated by plotting the wave height as a function of x for discrete values of t.

Soliton solutions to coupled nonlinear evolution equations modelling a third harmonic resonance in the theory of capillary–gravity waves

“…All the reports above were devoted to travelling wave solutions of the third harmonic resonance. However the problem was re-considered by Dias & Bridges in [6] who showed by acknowledging the presence of O(2) symmetry that additional classes of solutions may exist. These new classes consist of standing waves, mixed waves and 'Z-waves'.…”

O(2)‐lnvariant Hamiltonians on ℂ4

“…We do not compute values of the coefficients, appearing in the normal forms, that are associated with the water-wave problem. However, in a companion paper [7] the values of the coefficients for the capillary-gravity (1,3) mode interaction are computed and the physical wave profiles for each of the classes of waves are plotted.…”

“…The coefficients ai' f3, and yare defined in Proposition 2.3. The stationary solutions of the normal form in (3.7) are given for the special case m = 1, n = 3 in [7], The values of Zl' Z3' and Z4 obtained from (3.7) are used in [7] to plot the wave profiles for the Z6 (m = 1, n = 3) class of waves associated with the water-wave problem.…”

Finite-amplitude steady-state resonant waves in a circular basin

“…Higher-order resonant waves also exist (Hammack & Henderson 1993; Annenkov & Shrira 2006; Liu & Liao 2014). Resonant triads can occur in capillary-gravity waves (Wilton 1915; McGoldrick 1965; Vanden-Broeck 1984; Jones & Toland 1985; Bridges 1990; Dias & Bridges 1990; Chabane & Choi 2019), in interfacial waves (Christodoulides & Dias 2019; Chossat & Dias 1995), or in acoustic-gravity waves (Kadri & Stiassnie 2013; Kadri & Akylas 2016). In addition, the second-harmonic resonance associated with a resonant triad can occur in a circular basin (Mack 1962; Miles 1984).…”