Revisiting the hydrodynamic modulation of short surface waves by longer waves

Hydrodynamic modulation of short ocean surface waves by longer ambient waves is a well-known ocean surface process that affects remote sensing, the interpretation of in situ wave measurements, and numerical wave forecasting. In this paper, we revisit the linear wave theory and derive higher-order steady solutions for the change of short-wave wavenumber, action density, and gravitational acceleration due to the presence of longer waves. We validate the analytical solutions with numerical simulations of the full wave crest and action 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 to be 5/8 due to the wavenumber, 1/4 due to the wave action, and 1/8 due to the effective gravity. We further examine the result 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 only for homogeneous initial conditions as a special case and does not generally occur, for example in more realistic long-wave groups. The proposed steady solutions are a good approximation of the fully nonlinear numerical solutions in long-wave steepness up to ~0.2. Except for a subset of initial conditions, the solutions to the fully nonlinear crest-action conservation equations are mostly steady in the reference frame of the long waves.