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Modulation of short gravity‐capillary waves by longer‐scale periodic flows—A higher‐order theory
Author(s) -
Valenzuela G. R.,
Wright J. W.
Publication year - 1979
Publication title -
radio science
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.371
H-Index - 84
eISSN - 1944-799X
pISSN - 0048-6604
DOI - 10.1029/rs014i006p01099
Subject(s) - physics , gravity wave , mechanics , gravitational wave , mechanical wave , amplitude , computational physics , internal wave , dissipation , dissipative system , wave propagation , quantum electrodynamics , optics , classical mechanics , longitudinal wave , quantum mechanics
A higher‐order perturbation theory for the modulation of short gravity waves by longer‐scale periodic flows is developed, starting with Boltzmann's transport equation and including the input from the wind, energy transfer by nonlinear resonant interactions, and dissipation. Specific results are given for a model dealing with the straining of short gravity waves coupled to ‘parasitic’ capillary waves by second‐order resonant interactions. The initial value problem yields a relaxation rate for the short gravity waves equal to their growth rate. The predicted phase of the short gravity waves in relation to the crest of the longer gravity waves with this relaxation rate is in better agreement with measurements. Explicit expressions are obtained for the second‐order results in terms of U o / C (ratio of orbital to phase speed of the modulating wave). Numerical results are obtained for the reduction of the short gravity waves amplitude and for the modulation and phase of the backscattered power of microwave sensors probing this system of water waves. The second‐order results predict the reduction of the linear modulations, but the reduction is not large enough to match the measurements for U o / C ratios larger than 0.1. Accordingly, perturbation solutions do not converge fast enough for the larger U o / C ratios.