Freak Waves in the Linear Regime: A Microwave Study

Microwave transport experiments have been performed in a quasi-two-dimensional resonator with randomly distributed conical scatterers. At high frequencies, the flow shows branching structures similar to those observed in stationary imaging of electron flow. Semiclassical simulations confirm that caustics in the ray dynamics are responsible for these structures. At lower frequencies, large deviations from Rayleigh’s law for the wave height distribution are observed, which can only partially be described by existing multiple-scattering theories. In particular, there are “hot spots” with intensities far beyond those expected in a random wave field. The results are analogous to flow patterns observed in the ocean in the presence of spatially varying currents or depth variations in the sea floor, where branches and hot spots lead to an enhanced frequency of freak or rogue wave formation.

Figure 1

Photograph of one of the two scattering arrangements used. The platform has width 260 mm and length 360 mm. Each cone has diameter 25 mm and height 15 mm. The probe antenna is fixed in a horizontally movable top plate located 20 mm above the bottom (not shown).

Figure 2

Comparison of an experimental wave pattern with a classical ray simulation. Left: A wave function at frequency $f=30.95\mathrm{GHz}$. Right: The corresponding semiclassical simulation, with modes 1 through 4 added together.

Figure 3

Probability distribution of intensities. The dark (black) histogram includes all data, while the light (yellow) histogram excludes frequencies associated with the hot spots. The dotted line is the Rayleigh distribution, while the dashed (blue) line is a best fit using the theoretical distribution given by Eq. (2) ($\gamma \approx 23.5$).

Figure 4

A “hot spot,” observed at a frequency of 8.85 GHz. The experimental probability density for observing such a hot spot is 1 to 2 orders of magnitude larger than that expected from multiple-scattering theory.

Figure 5

Intensity distribution for the time-dependent wave patterns generated by Eq. (3) for all measured points [dark (black) crosses], for the hot spot region only [light (orange) solid circles], and for all points excluding the hot spot [light (blue) open squares]. The dotted line is the random wave expectation, while the solid (red) line is given by Eq. (6). The arrow indicates the extreme event studied more closely in Fig. 7. The inset shows a sketch of the setup: the dark crosses mark the different exciting antenna positions, the (red) shaded region corresponds to the measured field, and the empty rectangle inside the measured field indicates the hot spot region. The cross inside the hot spot region is the position of the maximal measured intensity.

Figure 6

Distribution of the time-averaged intensities $s$ found for the 780 pixels of our measurement. The inset shows the same data using a semilogarithmic scale. The solid curve is a ${\chi }^2$ distribution with $\nu =32$ degrees of freedom.

Figure 7

A freak wave event. The time evolution of wave intensity at the center of one of the hot spots is shown for the most extreme event observed. The inset shows the region surrounding the hot spot at the moment of the freak event. A movie of the time evolution in the entire field surrounding the hot spot is available [13].