Role of detuned frequency on Bragg scattering of surface gravity waves over an array of sinusoidal bottom patches: An analytic study

Abstract

The study uses linearized water wave theory to examine the role of detuned frequency on Bragg scattering of surface gravity waves over an array of bottom-standing submerged sinusoidal patches. Explicit formulae for reflection and transmission coefficients are derived using the matrix transfer method in the case of an array of patches, with each patch having a finite number of ripples. Bragg resonance occurs in the case of more than two patches beyond a certain cutoff frequency corresponding to supercritical detuning, while a monotonic increasing trend is observed below the cut-off frequency which is referred to as subcritical detuning. The number of sub-harmonic peaks between two consecutive harmonic peaks is one less than the number of patches. As the number of patches grows, so does the number of zero reflections, while the number of sub-harmonic peaks is invariant with the number of ripples within a patch. The corrugated length of the submerged sinusoidal patches and the resonator length determine the highly resonating/wave trapping features of wave reflection within the resonator and the corrugated patches. The time-domain simulation of surface displacement reveals the scattering and splitting of wave pulses over the submerged patches.