Abstract

The theoretical model for group-forced infragravity (IG) waves in shallow water is not well established for nonbreaking conditions. In the present study, analytical solutions of the group-forced IG waves at O(β ₁) (β ₁ = h _x /(Δkh), h _x = bottom slope, Δk = group wavenumber, h = depth) in intermediate water and at $O\left({\beta }_1^{-1}\right)$ in shallow water are derived separately. In case of off-resonance [β ₁ μ ⁻¹ = O(β ₁), where $\mu =1-c_g^2/\left(gh\right)$ is the resonant departure parameter, c _g = group speed] in intermediate water, additional IG waves in quadrature with the wave group forcing (hereinafter, the nonequilibrium response or component) are induced at O(β ₁) relative to the equilibrium bound IG wave solution of Longuet–Higgins and Stewart (1962) in phase with the wave group. The present theory indicates that the nonequilibrium response is mainly attributed to the spatial variation of the equilibrium bound IG wave amplitude instead of group-forcing. In case of near-resonance [β ₁ μ ⁻¹ = O(1)] in shallow water; however, both the equilibrium and nonequilibrium components are $~O\left({\beta }_1^{-1}\right)$ at the leading order. Based on the nearly-resonant solution, the shallow water limit of the local shoaling rate of bound IG waves over a plane sloping beach is derived to be ~h ⁻¹ for the first time. The theoretical predictions compare favorably with the laboratory experiment by Van Noorloos (2003) and the present numerical model results generated using SWASH. Based on the proposed solution, the group-forced IG waves over a symmetric shoal are investigated. In case of off-resonance, the solution predicts a roughly symmetric reversible spatial evolution of the IG wave amplitude, while in cases of near to full resonance the IG wave is significantly amplified over the shoal with asymmetric irreversible spatial evolution.