Multiscale wave resonance in composite sinusoidal-elliptical topographies: Critical transitions and analytical control

This study presents the first analytical solution for wave propagation over composite seabeds integrating sinusoidal sandbars with truncated semi-elliptical topographies, overcoming limitations of conventional mild-slope equations in handling elliptical curvature effects, coupled Bragg scattering, and singularities at truncated boundaries. Utilizing Frobenius series expansion and multi-region field matching, we systematically quantify how geometric parameters—$a/b$ ratio, $\delta /a$, and $h_0/b$—govern wave reflection coefficients ($K_R$). Key discoveries reveal that the $a/b$ ratio controls resonance peak frequencies (inducing 12% shifts per 0.1 change), the radius parameter $r=(h_0-h_1)/h_0$ triggers complete reflection ($K_R\to 1$) at a critical value of 0.5, and optimal $\delta /a$ expands reflection bandwidth by up to 22%. This work transcends classical studies on singular seabed types, establishes a theoretical foundation for designing wave-control metamaterials via multiscale resonances, and bridges classical potential flow theory with modern coastal engineering applications in wave energy harvesting, coastal protection, and offshore structure design.