Periodic sandbars can scatter nearshore water waves, and wave nonlinearity can further induce complex hydrodynamic behaviours, either amplifying reflection or enhancing transmission, specifically through Class-III subharmonic or superharmonic Bragg resonance. While this phenomenon is crucial for understanding wave–seabed interactions, analytical quantification of key features, i.e. resonance detuning, cutoff frequencies and resonance bandwidth, remains limited. In this study, using the multiple-scale expansion method, we derive a new set of modified nonlinear Schrödinger (MNLS) equations that account for dispersion, wave nonlinearity and topographic effects up to third-order accuracy. By applying the frozen-coefficient method to the MNLS equations, we further formulate approximate closed-form solutions for the reflection and transmission coefficients, which remain bounded across all parameter regimes and can well capture resonance detuning. A theoretical formula is derived to quantify the detuning magnitude, which is validated against existing experimental and numerical results. Moreover, the closed-form nature of the solutions enables the first predictions of the cutoff frequencies and the resonance bandwidth for Class-III Bragg resonance, thereby clarifying the maximum capacity of the sandbars to scatter wave energy. Additionally, an asymptotic analysis in the infinite-sandbar limit reveals substantial differences between subharmonic and superharmonic resonance: the former exhibits a resonance bandwidth proportional to the product of the wave amplitude and the sandbar amplitude, whereas the latter presents a newly reported zero-bandwidth structure. Numerical simulations further support these findings and reveal two features near the superharmonic resonance: an asymmetry of the envelope, characterized by a sharp corner, and an additional upshift that is further evaluated analytically.
Fang et al. (Mon,) studied this question.