Shoaling of bound infragravity waves on plane slopes for bichromatic wave conditions

In this paper, the cross-shore propagation and shoaling of bound infragravity (IG) waves have been investigated. By establishing a conceptual 1D model utilizing a constant slope in the shoaling zone, a semi-analytical solution for bound IG waves could be obtained. Through the use of this semi-analytical solution, a collection of bichromatic conditions have been simulated, thus confirming the dependence of the forced long wave shoaling rate on a normalized bed slope beta. beta was found to be the dominant factor controlling the shoaling of bound long waves, yet when the bichromatic wave steepness become more significant (in this study around 0.10), the mean primary wave frequencies come into play. A detailed analysis confirmed an over-estimation of the equilibrium solution by Longuet-Higgins and Stewart (1960) (LHS60), which was derived based on a flat bottom in the first place. This over-estimation was found to be the most significant for primary short waves with a small wave steepness in the offshore region. A correction to the equilibrium solution of LHS60 has been derived, and its validity has been verified through a comparison to a high-resolution laboratory experiment under bichromatic wave conditions. Furthermore, the phase shift between the short wave groups and the IG waves has been discussed. At the innermost of the shoaling zone, the bound IG waves phases are still found to be strongly governed by the local forcing of short wave groups, as has been pointed out by Baldock (2006). For the near-resonant case, the bound IG waves are anti-phase to the short wave groups, and a larger phase shift in the shoaling zone might be the result of superposition of incoming free IG waves.