Higher Order Resonant Interaction of Surface Waves by Undulatory Bottom Topography

Abstract

Higher order (quartet) Bragg resonance of water waves by bottom undulations and its effect on the evolution of ocean wave spectrum, particularly over continental shelves and littoral zones, are considered. Higher order Bragg resonance can provide a viable mechanism for distribution of (initially confined) energy across the spectrum. Contrary to classical Bragg resonances (Class I and II) where the resonant wave has to have the same frequency as the incident wave, Class III (quartet) Bragg resonance of three free waves and a bottom topography component allow participant waves to have different frequencies. Of particular interest here are higher-order resonances that lead to infragravity wave generation as a result of interaction of regular sea waves with bottom undulations (of the same order of wavelength as the primary waves), and, long/medium surface wave generation by nonlinear interaction between short surface waves and medium wavelength bottom undulations. These mechanisms can accelerate the rate by which energy is damped by the bottom friction. The second mechanism also provides a potential alternative mechanism for explaining microseismic noise observed in shallow waters. We further consider the oblique higher order Bragg resonance. Although for Class I, oblique resonance is less important than normal incidence, it is shown here via illustrative examples and direct simulations that there are strong oblique Class II and III Bragg cases. Inclusion of higher order interactions paves a path for the energy transfer to higher and lower frequencies of an initially narrow band spectrum. Ensuing multiple (exact/near) such resonant interactions can result in the generation of multiple new transmitted/reflected waves that fill a broad wavenumber band eventually leading to loss of order and chaotic motion of water surface.