Ray Theory of Ridge-Trapped Waves over a Triangular Profile

To elucidate the formation of ridge-trapped waves, this study employs ray theory to derive the ray trajectories and wave crest equations for waves propagating over a triangular ridge. The results indicate that the ray trajectories above such topography follow trochoidal curves. The envelope formed by the cycloidal arches constitutes the caustic, whose shape is influenced by the incident wave frequency, wavenumber, ridge slope, and water depth over the ridge crest. The condition for wave trapping requires that the trough line of the incident wave spatially coincides with the crest line of the reflected wave along the caustic. Based on this condition, the relationship between the crest line of the trapped wave and its wavelength is established, leading to the dispersion relation for trapped waves over a triangular ridge. Although the dispersion relation obtained from ray theory differs in form from that derived from the linear long-wave equation, the results are in close agreement for triangular ridges with gentle slopes. Furthermore, the spatial distribution of wave crest lines is used to explain the variation in wave height for ridge-trapped waves.