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

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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 O(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 O(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.
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Document Type: Research Article

Publication date: January 1, 1990

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