Long waves are generated and transform when short-wave groups propagate into shallow water, but the generation and transformation processes are not fully understood. In this study we develop an analytical solution to the linearized shallow-water equations at the wave-group scale, which decomposes the long waves into a forced solution (a bound long wave) and free solutions (free long waves). The solution relies on the hypothesis that free long waves are continuously generated as short-wave groups propagate over a varying depth. We show that the superposition of free long waves and a bound long wave results in a shift of the phase between the short-wave group and the total long wave, as the depth decreases prior to short-wave breaking. While it is known that short-wave breaking leads to free-long-wave generation, through breakpoint forcing and bound-wave release mechanisms, we highlight the importance of an additional free-long-wave generation mechanism due to depth variations, in the absence of breaking. This mechanism is important because as free long waves of different origins combine, the total free-long-wave amplitude is dependent on their phase relationship. Our free and forced solutions are verified against a linear numerical model, and we show how our solution is consistent with prior theory that does not explicitly decouple free and forced motions. We also validate the results with data from a nonlinear phase-resolving numerical wave model and experimental measurements, demonstrating that our analytical model can explain trends observed in more complete representations of the hydrodynamics.

Low-frequency (infragravity) wave dynamics on a fringing coral reef were investigated using the numerical model XBeach (Roelvink et al., 2009). First, the skill of the model was evaluated in one- and two-dimensions based on its predictions of short waves (0.04-0.2. Hz), infragravity waves (0.004-0.04. Hz) and water level measurements (tidal and wave setup) obtained during a 2009 field study at Ningaloo Reef in Western Australia. The model calibration was sensitive to friction coefficients for short waves and current/infragravity bed friction, which were assumed independent in this model study. Although the one-dimensional cross-shore model captured the gradients in the dominant hydrodynamic processes at the site, a high current/IG bed friction coefficient was required. This resulted in an overestimation and a phase lag between the observed and predicted wave setup signal. In the two-dimensional model, a lower (more realistic) current/infragravity wave friction coefficient was required to achieve optimum performance due to the presence of significant reef and lagoon mean flows in the model, which led to reduced setup across the reef. The infragravity waves were found to propagate from the surf zone across the reef in a dominantly cross-shore direction towards the shore, but with substantial frictional damping. The infragravity waves were strongly modulated also over the reef by tidal depth variations, primarily due to the variability in frictional dissipation rates when the total water depth over the reef varied. Two mean wave-driven circulation cells were observed in the study area, with cross-shore flow becoming more alongshore-dominated before exiting the system via the two channels in the reef. The results reveal that short waves dominated bottom stresses on the forereef and near the reef crest; however, inside the lagoon, infragravity waves become increasingly dominant, accounting up to 50% of the combined bottom stresses.

Low-frequency (infragravity) wave dynamics on a fringing coral reef were investigated using the numerical model XBeach (Roelvink et al, 2009). First, the skill of the one-dimensional model was evaluated based on its predictions of short waves (0.04-0.2 Hz), infragravity waves (0.004-0.04 Hz) and water level measurements (tidal elevation and wave setup) obtained during a 2009 field study at Ningaloo Reef in Western Australia. The model calibration was sensitive to friction coefficients for short waves and current / infragravity bed friction, which were assumed independent in this model study. The infragravity waves were found to be generated primarily in the surf zone through the breakpoint generation mechanism rather than through offshore forcing. The infragravity waves were also strongly modulated over the reef by tidal depth variations, primarily due to the variability in frictional dissipation rates when the total water depth over the reef varied. The results revealed that short waves dominated bottom stresses on the forereef and near the reef crest; however, inside the lagoon, infragravity waves become increasingly dominant, accounting up to 50% of the combined bottom stresses.