At the ocean surface, short wind ripples ride on long waves and long waves ride on much longer swells or currents. Both can be classified as the short- and long-wave interaction. Short waves riding on long waves are modulated by and interact with, the long waves. They may break on the crest of, and transfer momentum to the long waves. Detailed knowledge of short- and long-wave interactions is essential to describe the kinematics and dynamics of steep irregular waves near the ocean surface and to understand the processes by which wind energy is transferred to the ocean surface.
The recent development of remote sensing from satellites has made it possible to measure the ocean wave spectrum and infer the wind velocity from microwave radar images of the ocean surface (Allan 1983, Stewart 1985). Accurate measurements, however, require more detailed quantitative knowledge of the modulation of short waves, of there stability, and of energy transfer from the wind to the waves. Innovative platform designs, such as tension leg platform (TLP) and compliant tower platform (CTP), have allowed the oil industry to push production to much deeper water. Under storm conditions, these deep-water offshore structures are subjected to severe wave impacts. Laboratory measurements have shown that the forces and pressures on structures due to steep-wave impacts may be two to three times and ten times larger than those predicted by the traditional wave theory, respectively (Ochi and Tsai 1984, Chan and Melville 1988, Zhou, Chan and Melville 1990). Differences between the predictions of wave kinematics near the steep-wave surface through two different approximations (Rodenbush and Forristal 1986) based on the same wave spectrum may result in a 50 percent difference in the computation of hydrodynamic loadings on a CTP (Chen 1990). The large discrepancies between the measurements and the prediction and between the predictions through different approximations can be resolved through the understanding of nonlinear wave interactions, including the short- and long-wave interaction. The developments in the microwave remote-sensing technique and in the deep-water offshore technology are typical examples requiring the knowledge of the short- and long-wave interaction, and this demand has already stimulated great interest in its study in recent years.
The evolution of short waves riding on the long waves is extremely complicated because it is influenced by the combination of the wave-wave interaction, wind-wave interaction, and wave breaking. Thus, heuristic models such as the wave breaking-relaxing model (Keller and Wright 1975 Valenzuela and Wright 1979, Phillips 1984), or assumptions, such as that of a steady profile of short-wave amplitude along with the long wave, are needed in computing the modulation of wind-generated short waves riding on long waves. In order to further understand these processes and establish better models, it is helpful to separate these coupled and complicated processes into simplified ones, and then thoroughly study each of them.
Here we concentrate on studying the steady solution and the stability of a weakly nonlinear short gravity wave train riding on a finite-amplitude periodic long wave, without considering the effect of wind, wave breaking and viscosity of water. Although short waves are limited to gravity waves and are collinear with the long wave in our study, it can be extended straightforwardly to include gravity-capillary waves and allow for three-dimensional short waves with more lengthy algebra.
Related Publications: Zhang, J. and Melville, W.K. “Evolution of Weakly Nonlinear Short Waves Riding on Long Gravity Waves,” J. Fluid Mech., Vol. 214, pp.321-346, 1990.
Zhang, J. “Fourth-Order Lagrangian of Short Waves Riding on Long Waves,” Phys. Fluids A 3 (12), December 1991.
Zhang, J. and Melville, W.K., “On the Stability of Weakly Nonlinear Short Waves on Finite-Amplitude Long Gravity Waves,” J. Fluid Mech., 243, pp.51-72, 1992.