Revisiting the hydrodynamic modulation of short surface waves by longer waves

Hydrodynamic modulation of short ocean surface waves by longer ambient waves significantly influences remote sensing, interpretation of in situ wave measurements, and numerical wave forecasting. This paper revisits the wave crest and action conservation laws and derives steady, nonlinear, analytical solutions for the change of short-wave wavenumber, action, and gravitational acceleration due to the presence of longer waves. We validate the analytical solutions with numerical simulations of the full conservation equations. The nonlinear analytical solutions of short-wave wavenumber, amplitude, and steepness modulation significantly deviate from the linear analytical solutions of Longuet-Higgins & Stewart (1960), and are similar to the nonlinear numerical solutions by Longuet-Higgins (1987) and Zhang & Melville (1990). The short-wave steepness modulation is attributed 5/8 to wavenumber, 1/4 due to wave action, and 1/8 due to effective gravity. Examining the homogeneity and stationarity requirements for the conservation of wave action reveals that stationarity is a stronger requirement and is generally not satisfied for very steep long waves. We examine the results of Peureux et al. (2021) who found through numerical simulations that the short-wave modulation grows unsteadily with each long-wave passage. We show that this unsteady growth only occurs for homogeneous initial conditions as a special case and not generally. The proposed steady solutions are a good approximation of the nonlinear crest-action conservation solutions in long-wave steepness $\lesssim$ 0.2. Except for a subset of initial conditions, the solutions to the non-linearised crest-action conservation equations are mostly steady in the reference frame of the long waves.