Two-dimensional surf beat: Long wave generation by a time-varying breakpoint

A two-dimensional parametrization of a surf zone with a time-dependent breakpoint due to groupiness in the incident wave field is developed. The breakpoint is defined in terms of a mean position X plus a modulation Δa. Free wave solutions are obtained on a plane beach at the group frequency and its harmonics. Shoreward of the breakpoint, standing waves are found, while seaward an outgoing progressive wave exists. The amplitude of the standing wave is relatively insensitive to the incident wave field. The amplitude of the outgoing wave depends on X, the group frequency σ, and the beach slope tan β. For certain values of X = (σ²X/g tan β) the amplitude of the outgoing wave goes to zero. It appears that any `resonant' response of the standing wave shoreward of the breakpoint is suppressed by seaward radiation of energy by the outgoing wave.