Abstract
To investigate the evolution of surface progressive waves propagating over a sloping wavy seabed, this study employes a perturbation expansion approach with three governing parameters, namely, the equivalent wave amplitude \({{a}_{0}}\) , the seabed slope \(\alpha \) , and the amplitude of wavy seabed structures \(b\) . These parameters are used to derive analytical solutions, including those under conditions involving the generation of reflected free waves, while avoiding singularities arising from resonant conditions in the wave field. Moreover, the reflected wave induced at the resonant point by a sloping wavy seabed is investigated. This resonance is amplified by the shoaling effect, which potentially contributes to the formation of large surges or freak waves. This amplification effect is stronger for longer-period waves near the coast and for steeper seabed slopes. When \(b = 0\) , the solution reduces to the classical sloping seabed case, capturing the effects of wave shoaling, wave set-down, and return flow generation (caused by variations in the water depth). Alternatively, when \(\alpha = 0\) , the solution corresponds to a wavy seabed in a scenario involving a constant average water depth. In both limiting cases, the analytical solutions satisfy the principle of wave energy conservation. The reflection coefficients, including those associated with resonance, are determined in each limiting case. Comparisons with experimental and analytical results from the literature confirm the accuracy and applicability of the developed formula, especially under resonant conditions. The findings of this study provide theoretical insights into wave amplification mechanisms relevant to the generation of extreme waves near coastal zones.
