Bragg scattering of flexural-gravity waves induced by an array of periodic sinusoidal bars in the purview of blocking dynamics and triad resonance

A mathematical model is developed using a multiple-scale method to explore the impact of detuned frequency on the triadic resonant interactions of flexural-gravity waves over sinusoidal beds of finite length (patches). Explicit expressions for scattering coefficients linked to multiple sinusoidal beds are derived using the transfer matrix method. With the introduction of detuned frequency, the reflection coefficient increases monotonically under subcritical detuning frequencies, reaches a cutoff value, and then oscillates under supercritical conditions. In contrast, the reflection coefficient as a function of the detuned frequency reveals a Fabry–Pérot resonance pattern in the case of multiple patches, with the number of subharmonics between consecutive harmonic peaks being two less than the number of sandbars. With increased plate rigidity, the reflected wave amplitude transitions from monotonically increasing to an oscillatory pattern. Additionally, the reflection coefficient increases for higher values of compressive force, patch amplitude, and number of patches. Compressive stress in the flexible plate causes a non-monotonic reflection trend due to structural bending and wave blocking for higher compressive stress before structural buckling. The reflection coefficient oscillates before primary blocking (PB) and decays beyond secondary blocking (SB), with the occurrence of removable discontinuities at saddle points, PB, and SB points in frequency space. The plate deflection modulates with a rise and fall in amplitude within the submerged sandbar regions and the gaps, depending on the constructive or destructive interference of the flexural-gravity waves.