Formation Of Large-amplitude Waves Research Articles

Existence of a Highest Wave in a Fully Dispersive Two-Way Shallow Water Model

We consider the existence of periodic traveling waves in a bidirectional Whitham equation, combining the full two-way dispersion relation from the incompressible Euler equations with a canonical shallow water nonlinearity. Of particular interest is the existence of a highest, cusped, traveling wave solution, which we obtain as a limiting case at the end of the main bifurcation branch of $${2\pi}$$ -periodic traveling wave solutions continuing from the zero state. Unlike the unidirectional Whitham equation, containing only one branch of the full Euler dispersion relation, where such a highest wave behaves like |x|1/2 near its crest, the cusped waves obtained here behave like |x log |x||. Although the linear operator involved in this equation can be easily represented in terms of an integral operator, it maps continuous functions out of the Holder and Lipschitz scales of function spaces by introducing logarithmic singularities. Since the nonlinearity is also of higher order than that of the unidirectional Whitham equation, several parts of our proofs and results deviate from those of the corresponding unidirectional equation, with the analysis of the logarithmic singularity being the most subtle component. This paper is part of a longer research programme for understanding the interplay between nonlinearities and dispersion in the formation of large-amplitude waves and their singularities.

Statistical Properties of Directional Ocean Waves: The Role of the Modulational Instability in the Formation of Extreme Events

We discuss two independent, large scale experiments performed in two wave basins of different dimensions in which the statistics of the surface wave elevation are addressed. Both facilities are equipped with a wave maker capable of generating waves with prescribed frequency and directional properties. The experimental results show that the probability of the formation of large amplitude waves strongly depends on the directional properties of the waves. Sea states characterized by long-crested and steep waves are more likely to be populated by freak waves with respect to those characterized by a large directional spreading.

Modulational instability and non-Gaussian statistics in experimental random water-wave trains

We study random, long-crested surface gravity waves in the laboratory environment. Starting with wave spectra characterized by random phases we consider the development of the modulational instability and the consequent formation of large amplitude waves. We address both dynamical and statistical interpretations of the experimental data. While it is well known that the Stokes wave nonlinearity leads to non-Gaussian statistics, we also find that the presence of the modulational instability is responsible for the departure from a Gaussian behavior, indicating that, for particular parameters in the wave spectrum, coherent unstable modes are quite prevalent, leading to the occurrence of what might be called a “rogue sea.” Statistical results are also compared with ensemble numerical simulations of the Dysthe equation.

Large amplitude Kelvin-Helmholtz waves on thin liquid sheets

Experiments which show the formation of large amplitude waves on a thin liquid sheet when forced vibrations are applied to the nozzle are described in detail, after an initial look at the existing linearized theory of wave growth in this system. The experimental results show this theory to be inadequate and a preliminary theory for large amplitude waves is presented in a final section.

“Supersonic” flow in the animal world

The analogy between the formation of large amplitude waves (hydraulic jumps) on the surface of shallow flowing water, and shock waves in supersonic air flow is well-known. In addition to the correspondence between the wave patterns the variation in water depth can be related to the pressure distribution.This “hydraulic analogy” is usually demonstrated in the laboratory in a tank which produces a shallow flow of water over a horizontal surface, on which the appropriately shaped body is mounted. The resulting stationary wave patterns may then be examined. Fig. 1 shows a typical bow-wave on a double-wedge.