Statistics of Extreme Waves in Random Media

Waves traveling through random media exhibit random focusing that leads to extremely high wave intensities even in the absence of nonlinearities. Although such extreme events are present in a wide variety of physical systems and the statistics of the highest waves is important for their analysis and forecast, it remains poorly understood, in particular, in the regime where the waves are highest. We suggest a new approach that greatly simplifies the mathematical analysis and calculate the scaling and the distribution of the highest waves valid for a wide range of parameters.

Figure 1

The formation of extreme waves in a random medium. (Center panel) Illustration of a wave flow (gray scale, darker shades correspond to higher wave intensities) through a weak random medium (color), starting from an initially plane wave. The standard deviation of the random potential is 10% of the energy of the flow, and the wavelength is 5% of the correlation length of the potential ${\ell }_c$. The image shows an extract of a larger simulation with periodic boundary conditions on top and bottom, and absorbing boundary conditions on the left and right. (Top panel) The variance of the wave intensities averaged over 100 realizations of the random medium. A high variance corresponds to the occurrence of extreme events. (Bottom panel) Two sections of the wave amplitudes at the locations indicated by arrows in the central panel. The amplitude is normalized to the standard deviation of the fluctuations in the saturated regime far away from the source (thick black dashed lines). Wave amplitudes of up to 10 times of this standard deviation can be observed. The probability to observe such an event in random waves according to Rayleigh’s law would be smaller than ${10}^{-20}$.

Figure 2

Mapping the complex dynamics of the propagation in the random medium to the simpler problem of an initially curved wave front propagating through free space. (a) An initially plane ray bundle propagating through a weak random medium (colored background) develops a fold caustic, with typical transverse velocity $v_{\text{typ}}$. (b) In a situation mimicking the random ray propagation, a ray bundle emanating perpendicularly to an initially curved wave front described by a deviation $f(y_0)$ from the plane initial condition along the initial $y_0$ axis propagates through free space. The path length corresponding to $f(y_0)$ is given by $\varphi (\xi ,y_0)$. By matching the velocities $v_{\text{typ}}$ of the ray bundles in both cases the caustic parameter $\alpha$ can be related to the parameters of the stochastic medium.

Figure 3

Distribution of extreme waves. (a) Unscaled distributions for different wavelengths $\lambda$ and different parameters of the random medium, $\epsilon$ (standard deviation), and ${\ell }_c$ (correlation length). The insets show curves for fixed medium parameters and fixed wavelength, respectively. (b) The same curves, but rescaled according to Eq. (5), together with the distribution, Eq. (5) (dashed line). The inset shows the same curves in a double-logarithmic plot.