Analytical study of Bragg resonance by singly periodic sinusoidal ripples based on the modified mild-slope equation

Abstract

The wave propagation over singly periodic sinusoidal ripples is studied analytically based on the modified mild-slope equation (MMSE). Firstly, each ripple region is divided into four monotonic subintervals such that there is only one regular singular point of the MMSE within each subinterval which is located at one of the two endpoints. Secondly, in each subinterval, due to the monotonicity, the implicit MMSE can be transformed into an explicit MMSE (EMMSE). Thirdly, in each subinterval, due to the unique regular singularity, a general solution in terms of Frobenius series to the EMMSE can be constructed and the convergence condition of the series solution is analyzed and graphically demonstrated. Finally, by using the mass-conserving matching conditions at each common boundary between any two adjacent subintervals, an analytical formula of the reflection coefficient is established. The present solution is validated against various existing solutions, especially, it is identical to a numerical solution to the MMSE. It is shown that, as the number of ripples increases, the peak amplitudes of both the primary and subharmonic Bragg resonances increase while the resonance bandwidths decrease, and the zero reflection occurs more frequently. When the number becomes very large, the peak amplitudes of the primary and subharmonic Bragg resonances achieve unity and the resonance bandwidths approach to their limit resonance bands. As the height of ripples increases, the peak amplitudes of both the primary and subharmonic Bragg resonances and the bandwidth of the primary resonance increase accordingly, and the positions of both the primary and subharmonic Bragg resonances downward shift to low frequency more significantly, but the number of the zero reflection keeps the same.