Abstract

Whitham modulation theory (WMT) has had an impact on the theory of water waves since the beginning of the subject in the early 1960s. This talk will focus on three topics of current interest. The first is ocean wave forecasting, where the “fundamental law for wave prediction” is the conservation of wave action. This theory is reviewed and generalisations to finite amplitude discussed, including how coalescing characteristics in WMT arise in wave forecasting. The second problem is the transition point of the Benjamin-Feir instability which occurs in relatively shallow water. The Whitham (1967) modulation equations for this problem are reviewed, new properties discussed, and re-modulation is introduced at the transition point to find a dispersive higher-order modulation equation due to Ratliff (2017). This latter equation contains wave jumps and frequency shifting. The third topic is how WMT illuminates defects in shallow water hydrodynamics, where Stokes waves of differing amplitude and wavenumber are joined by a shock. Sprenger & Hoefer (2020) use WMT to construct these shocks in the fifth-order KdV equation. This latter theory is reviewed, and implications and generalisations discussed.