Date of Award

8-31-2020

Document Type

Dissertation

Degree Name

Doctor of Philosophy in Mathematical Sciences - (Ph.D.)

Department

Mathematical Sciences

First Advisor

Wooyoung Choi

Second Advisor

Diane M. Henderson

Third Advisor

Roy Goodman

Fourth Advisor

Richard O. Moore

Fifth Advisor

David Shirokoff

Abstract

This dissertation is a study of the weakly nonlinear resonant interactions of a triad of gravity-capillary waves in systems of one and two fluid layers of arbitrary depth, in one and two-dimentions. For one-layer systems, resonant triad interactions of gravity-capillary waves are considered and a region where resonant triads can be always found is identified, in the two-dimensional wavevector angles-space. Then a description of the variations of resonant wavenumbers and wave frequencies over the resonance region is given. The amplitude equations correct to second order in wave slope are used to investigate special resonant triads that, providing their initial amplitude and relative phase satisfy appropriate conditions, exchange no energy during their interactions, which implies that the wave amplitudes remain constant in time. From the fact that the steadiness of the wave amplitudes is a necessary condition for resonant triads to form traveling waves, a transversely modulated two-dimensional wave field of permanent form is found and can be considered as a generalization of Wilton ripples. For two-layer systems, resonant triad interactions between surface and interfacial gravity waves propagating in two horizontal dimensions are considered. As the system supports both surface and internal wave modes, two different types of resonant triad interactions are possible: one with two surface and one internal wave modes and the other with one surface and two internal wave modes. Presented are the spectral domains, where, for given physical parameters, the two resonance scenarios can be found. It is shown that one-dimensional triads occur on the boundary of the spectral domain of resonance. Using a set of amplitude equations recently derived by Choi et al, [10], the necessary and sufficient conditions to form traveling waves are

found when the three wave trains travel in the same directions (class-III and class-IV resonance). In addition, a set of physical parameters for one-dimensional triads, for which these traveling waves are possible, is presented.

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