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, δ/a, and h0/b—govern wave reflection coefficients (KR). Key discoveries reveal that the a/b ratio controls resonance peak frequencies (inducing 12% shifts per 0.1 change), the radius parameter r=(h0−h1)/h0 triggers complete reflection (KR→1) at a critical value of 0.5, and optimal δ/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.