Dispersive Wave Focusing in Shoaling Water With Currents

A methodology creating dispersive focusing waves on constant depth, originally proposed by Rapp and Melville (1990), https://doi.org/10.1098/rsta.1990.0098 , has provided a wide range of applications studying nonlinear waves, wave breaking, and rogue waves in the open ocean. When the wave focusing is considered in coastal regions, generally having depth variations and coastal currents, effects of shoaling on beaches and interactions with currents must be included for predicting wave growth. Here we present the general form of a chirp frequency achieving complete wave‐energy focusing in arbitrary depth variations and currents. The proposed method has been validated through wave experiments in a wave flume with sloped beach and opposing currents. It was found that the current velocity significantly contributes to wave growth, achieving a maximum crest height 6.1 times higher than the incident wave in the strongest current considered. The dual effects of shoaling and radiation stress, intensified by the opposing current, contribute to the rapid growth of the wave energy density near the focusing point. Nonlinearity develops once waves are highly amplified, causing spectrum energy spread from the primary frequency band to wider frequencies depending on the current velocity. Whereas the nonlinear energy spread disappears after focusing when divergent waves travel in constant depth, the effects of shoaling and radiation stress maintain the nonlinear features of the waves behind the focus location on a beach. The proposed method achieving dispersive focusing will help in further understanding of nonlinear behaviors of breathers and wave breaking in arbitrary current and coastal geometry.