Nonlinear Fourier analysis of deep-water, random surface waves: theoretical formulation and experimental observations of rogue waves

Unidirectional deep-water waves are studied theoretically and experimentally. Theoretically we apply the theory of the nonlinear Schroedinger equation (NLS) using the inverse scattering transform, a kind of generalized, nonlinear Fourier analysis. We discover from the theoretical study that there are essentially four kinds of physical effects that can lead to extreme waves: (1) the superposition of sine waves, (2) the Stokes wave nonlinearity, (3) the BenjaminFeir instability and (4) the wave/current interaction. In the context of nonlinear random wave trains, item (3) is of prime interest here and we discuss how the theory predicts the existence of unstable wave packets which constitute a second population of nonlinear waves, the first population being the weakly nonlinear superposition of sine waves (item (1)) together with the Stokes-wave correction (item (2)). The second population, unstable wave packets, which can rise up to more than twice the significant wave height, are a new nonlinear spectral component in nonlinear random waves. Do second population waves actually exist in water waves? To answer this question we conducted a number of random wave experiments at Marintek, Trondheim, and assessed the results in terms of the NLS spectral theory. We find that the second population waves not only exist, but they can also, under the circumstances discussed herein, dominate the energetics of the wave train leading to a condition we call a “rogue sea.” Indeed, we also find in a companion paper (Onorato, Osborne, and Serio, 2005) that the extreme events associated with the Benjamin-Feir instability can also lead to a large enhancement in the tail of the Rayleigh distribution for crest and wave heights.