Wave jumps and caustics in the propagation of finite-amplitude water waves

Abstract

Nonlinear effects on the refraction of water waves are discussed. The existence of conjugate solutions for wave fields, one set of which corresponds to the ‘anomalous’ refraction solutions of Peregrine & Ryrie (1983) provides the stimulus for consideration of jumps in wave-field properties. Just such a jump is described by Yue & Mei (1980) for a case where near-linear waves are reflected with small deflection by a rigid wall.Wave jumps between conjugate solutions appear to be possible for finite-amplitude wavetrains. These are examined and the structure of wave jumps in the near-linear, small-deflection case is elucidated. They have a structure directly analogous to that of an undular bore on shallow water with surface tension (‘hydraulic analogy’). Numerical results of Yue & Mei (1980) provide valuable guidance and confirmation.Linear waves are reflected at a caustic, and can be described with Airy functions. Although equivalent weakly nonlinear solutions exist, the results from reflection by a wall and from use of the hydraulic analogy show that, unlike the caustics of linear theory, nonlinear caustics should not be considered in isolation. Caustic cusps, or wave focusing, must be considered, unless bed topography has discontinuities. A qualitative discussion of focusing based on the behaviour of unsteady waves in the hydraulic analogy shows that wave jumps can be expected. The relationship to linear theory is also put in perspective. Nonlinearity causes the linear rays to split into two sets of characteristics. The splitting of a ray focus leads to two wave jumps.Consideration of the case of a semi-infinite beach shows that anomalous refraction is most unlikely to occur because there is an offshore influence of the beach on the wave field which changes the incident wave conditions to prevent anomalous refraction.