Long waves forced by symmetric and asymmetric wave groups

Two solutions have been proposed for forced waves by wave groups. Two interpretations are proposed by Longuet-Higgins and Stewart (1962), LHS1962, and Mei (1989) for steady groups. These indicate a forced wave oscillating either around or solely below the mean water level for the short waves, respectively. The existence of forced waves as either pure depressions or as oscillations about the still water level has significant implications in terms of shoaling behaviour and the generation of free long waves. With a bound wave with both positive and negative components then shoaling (an increase in amplitude in shallower water) does not necessarily require radiation of free waves to ensure continuity of volume, but with a bound wave of pure depression, then an increase in amplitude of that depression requires radiation of free waves to ensure continuity (Nagase and Mizuguchi, 1996, Nielsen and Baldock, 2010). Radiation of those free waves then changes the shape of the total infragravity wave during shoaling and may also influence the commonly observed lag between incident bound waves and the incoming short-wave envelope. Here, we show that pure long wave depression can be generated by Gaussian wave groups, and that asymmetric forced long waves, with a leading crest and following depression, are generated by triangular / Gaussian groups.