On resonant interactions of gravity-capillary waves without energy exchange

Document Type

Conference Proceeding

Publication Date

5-1-2019

Abstract

We consider resonant triad interactions of gravity-capillary waves and investigate in detail special resonant triads that exchange no energy during their interactions so that the wave amplitudes remain constant in time. After writing the resonance conditions in terms of two parameters (or two angles of wave propagation), we first identify a region in the two-dimensional parameter space, where resonant triads can be always found, and then describe the variations of resonant wavenumbers and wave frequencies over the resonance region. Using the amplitude equations recovered from a Hamiltonian formulation for water waves, it is shown that any resonant triad inside the resonance region can interact without energy exchange if the initial wave amplitudes and relative phase satisfy the two conditions for fixed point solutions of the amplitude equations. Furthermore, it is shown that the symmetric resonant triad exchanging no energy forms a transversely modulated traveling wave field, which can be considered a two-dimensional generalization of Wilton ripples.

Identifier

85058860304 (Scopus)

Publication Title

Studies in Applied Mathematics

External Full Text Location

https://doi.org/10.1111/sapm.12249

e-ISSN

14679590

ISSN

00222526

First Page

528

Last Page

550

Issue

4

Volume

142

Grant

1634939

Fund Ref

National Science Foundation

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