[4] The rogue holes were mainly analysed in two‐dimensional models of deep water waves [ Osborne et al. , 2000 ; Porubov , 2007 ]. The holes were also observed in the open ocean within the frame of MaxWave project [ Monbaliu , 2003 ; Moes , 2003 ]. However, simple observations of holes in one‐dimensional case are still absent. Meanwhile, the Peregrine soliton [ Peregrine , 1983 ; Shrira and Geogjaev , 2010 ], which is a doubly localized solution of the nonlinear Schrödinger equation can explain both, the elevated rogue waves as well as deep holes on the water surface. In this work, we present first observations of rogue holes on the water surface and give a theoretical explanation in terms of a rational solution of the nonlinear Schrödinger equation (NLS).

[3] Presently, there are no known observations of hole rogue waves in a laboratory. Revealing their existence in a water wave tank is an important step in further understanding of rogue wave in the ocean. One of the ways to see the problem is the introduction of carrier‐envelope phase. As other types of waves, water waves are defined by the carrier and the envelope. When the envelope has dimensions comparable with the wavelength of the carrier, the carrier‐envelope phase is becoming an important parameter in the description of rogue waves. Namely, this parameter defines the difference between the “bright” and hole rogue waves.

[2] Rogue waves in the ocean are presently under intense studies [ Kharif et al. , 2009 ; Garret and Gemmrich , 2009 ; Müller et al. , 2005 ; Ruban et al. , 2010 ]. Inverse formations, rogue wave holes have been mentioned by Osborne et al. [2000] as deep troughs occurring before/after large crests. Thus, they usually accompany the elevated rogue waves. On the other hand, they can also be observed as truly “hole” rogue waves that are surrounded by the crests of equal size before and after the hole itself. When the depth of such hole from trough to crest is much higher than the average height of the surrounding waves, it can equally be dangerous for navigation as the elevated rogue wave.

2. Theory and Experiments

[5] One of the most direct approaches modeling the evolution of gravity water waves is the use of the NLS [ Zakharov, 1968 Osborne, 2010 Yuen and Lake, 1975 Lake et al., 1977 t and x are time and longitudinal coordinates, respectively, while k 0 and ω 0 = ω(k 0 ) denote the wave number and the frequency of the carrier wave, respectively. ω 0 and k 0 are linked by the dispersion relation of linear deep water wave theory, ω 0 = , where g denotes the gravitational acceleration. Accordingly, the group velocity is c g := = . The surface elevation η(x, t) of the seawater is then given by ϕ is a carrier phase. When the envelope function is much wider than the wavelength, the phase of the carrier does not influence much the wave dynamics. However, for highly localized functions a(x, t) this parameter strongly influences the measured wave profile. Then we can talk about ϕ as the “carrier‐envelope phase.” This parameter relates the phase of the carrier to the characteristic points (say the maximum) of the envelope. One of the most direct approaches modeling the evolution of gravity water waves is the use of the NLS []. It is known to be good for capturing weakly nonlinear evolution of narrow‐band processes. In dimensional form, NLS can be written aswhereandare time and longitudinal coordinates, respectively, whileand) denote the wave number and the frequency of the carrier wave, respectively.andare linked by the dispersion relation of linear deep water wave theory,, wheredenotes the gravitational acceleration. Accordingly, the group velocity is:=. The surface elevation) of the seawater is then given bywhereis a carrier phase. When the envelope function is much wider than the wavelength, the phase of the carrier does not influence much the wave dynamics. However, for highly localized functions) this parameter strongly influences the measured wave profile. Then we can talk aboutas the “carrier‐envelope phase.” This parameter relates the phase of the carrier to the characteristic points (say the maximum) of the envelope.

[6] A dimensionless form of the NLS [ Zakharov and Shabat, 1972 X is the normalized coordinate in the frame moving with the wave group velocity and T is the normalized time. The scaling parameter a 0 in A dimensionless form of the NLS [is obtained from (1) using the rescaled variablesHere,is the normalized coordinate in the frame moving with the wave group velocity andis the normalized time. The scaling parameterin (4) is the characteristic water elevation in this problem which allows us to adjust the wavelength to the length and the depth of the tank.

[7] The Peregrine breather [ Peregrine, 1983 Shrira and Geogjaev, 2010 Akhmediev et al., 2009 X = 0, T = 0 of The Peregrine breather [] is the lowest‐order rational solution of the NLS that is localized in both space and timeAs such, it describes unique wave events, i.e., waves that appear from nowhere and disappear without a trace []. The solution is complex and depending on the its location relative to the carrier wave, it may describe both elevated wave or a hole. When the position= 0,= 0 of (5) coincides with the maximum of the carrier wave, we generate the high‐amplitude rogue wave. In the opposite case when this position coincides with the minimum of the carrier, we generate the hole. The carrier‐envelope phase in our experiment is controlled by the computer program.

[8] The experiments are performed in a 15 × 1.6 × 1.5 m water wave tank with 1 m water depth which is the same used by Chabchoub et al. [2011]. The tank is depicted schematically in Figure 2 of Chabchoub et al. [2011]. A single‐flap paddle activated by a hydraulic cylinder is located at one end of the tank. It can be seen on a photo in Figure 1. A capacitance wave gauge with a sensitivity of 1.06 V cm−1 and a sampling frequency of 500 Hz is used to measure the surface elevation of the water. The wave gauge can be seen at the middle part of the photo in Figure 1. A wave absorbing beach at the opposite end of the tank is installed to inhibit wave reflections.

Figure 1 Open in figure viewer PowerPoint The photo of the wave generating end of the water wave tank used in the experiment. The flap is located at the far end of the tank while the wave gauge can be seen in the middle.

[9] The choice for the parameters of the waves is determined by the size of the tank. In all tests, the frequency of the carrier has been set to f 0 = 1.7 s−1, corresponding to an angular frequency of about ω 0 = 2πf 0 = 10.7 s−1, a wave number of about k 0 = 11.63 m−1, and a wavelength of λ 0 = = 0.54 m. The dimensional far‐field amplitude of the background varies from a 0 = 0.005 m to a 0 = 0.01 m. These values have been chosen in order to guarantee that the wavelength is large enough to inhibit capillary effects, but still small enough to have sufficient tank length to develop the wave evolution described by equation (5) and to ensure deep water conditions. To satisfy these conditions, the depth d and the wave number k 0 should satisfy the condition k 0 d ≫ 1 [Mei, 1983]. Our parameters d = 1 m and k 0 = 11.63 m−1 are well within this requirement. These values of a 0 also serve as the scaling parameter in the transformations (4).

[10] The parameters of the wave, i.e., frequency and the amplitude are controlled by the flap motion that in turn is controlled by the computer program. The wavelength is derived from the dispersion relation. We have checked that the growing rogue hole moves with the group velocity c g . We arranged a carrier wave yielding the maximum breather amplitude about 9 m along the tank. This is an ideal position for observation as there is another 1 m of propagation left before the waves meet the beach.

[11] As rogue waves are growing due to an instability, the accurate initial conditions are the key for their precise generation. Indeed, periodic initial conditions with small amplitude on top of the carrier wave lead to modulation instability and the periodic waves with growing amplitude provided that the frequency is within the instability band [Yuen and Lake, 1975; Lake et al., 1977]. Eventually, after the maximum amplitude has been reached, periodic waves decay and subsequently disappear [Akhmediev and Korneev, 1986]. The reverse process is known as Fermi‐Pasta‐Ulam recurrence [Akhmediev, 2001]. Periodic initial conditions with frequencies above the instability band do not grow in amplitude and propagate without changes. For generating a rogue wave, we need a single localized perturbation of the carrier, i.e., the modulation frequency has to be at the zero limit within the instability band. This limit is described by the Peregrine solution [Peregrine, 1983]. Any other localized initial condition would generate a wider spectrum of waves that will generally disperse.

[12] In order to program the paddle and compare the experimental results to the theoretical solutions, the Peregrine solution In order to program the paddle and compare the experimental results to the theoretical solutions, the Peregrine solution (5) has to be written in dimensional units. Applying the transformations (4) to (5) , the Peregrine solution becomes

[13] For deep water waves the relation between the phase of the carrier and the maximum of the envelope plays crucial role in the way the rogue wave appear on the surface. Indeed, the expression for surface elevation modeled with the Peregrine breather has an important parameter, the carrier‐envelope phase ϕ For deep water waves the relation between the phase of the carrier and the maximum of the envelope plays crucial role in the way the rogue wave appear on the surface. Indeed, the expression for surface elevation modeled with the Peregrine breather has an important parameter, the carrier‐envelope phaseThis formula is used both to determine the initial conditions for the wave maker's paddle motion, and for comparing measured surface‐height time series with theoretical predictions. The position where the hole in the water surface develops its maximum amplitude depends on the initial conditions at the wave maker.

[14] The experiments described below were performed with a flap motion which was directly proportional to the surface elevation given by (7) at the side where we define the initial conditions. Namely, the water motion created by the flap should correspond to the initial conditions given by the Peregrine breather solution at the starting side of the tank.

[15] When the maximum of the envelope is at the point of the maximum of the carrier (ϕ = 0), the rogue wave appears as an elevated high‐amplitude rogue wave. This case has been studied in our previous work [Chabchoub et al., 2011]. It is presented in Figure 2. Conversely, when the maximum of the envelope coincides with the position of a minimum of the carrier (ϕ = π), the rogue wave appears as a deep hole on the surface of water. This case is shown in Figure 3. This rogue wave hole was measured at the distance 9.20 m from the paddle. In order to do that, the paddle motion has been programmed to provide the surface elevation η(x, t) given by the exact solution back at the point x = −9.2. Comparing the theoretical solution with the measurement, in the present case the measurement depression is somewhat smaller than would be expected from theory. A similar behavior was already observed previously for the case of the high‐amplitude rogue wave (see Figure 2). The discrepancy seems to decrease for decreasing the steepness (see below) and thus seems to originate in the small steepness assumption implicit to the NLS theory, and can be explained by the high‐order terms which are not taken into account in this analysis. Figure 4 shows four stages of breather evolution along the channel. Remarkably, due to the differences in group velocity of the breather and the phase velocity of the carrier, the hole appears only at the exact position x = 9.2. In each of the above cases the amplitude of the background wave is 1 cm. All experimental conditions are similar except for the initial conditions applied at the position of the paddle.

Figure 2 Open in figure viewer PowerPoint Comparison of measured surface height near the position of maximum rogue wave amplitude (solid blue line) with theoretical Peregrine solution (dashed red line) evaluated at X = 0. Initial conditions are chosen to have an elevated rogue wave (ϕ = 0).

Figure 3 Open in figure viewer PowerPoint Comparison of measured surface height near the position of maximum hole depth (solid blue line) with theoretical Peregrine solution (dashed red line) evaluated at X = 0. The background wave amplitude is 1 cm. Initial conditions are chosen to create a hole (ϕ = π).

Figure 4 Open in figure viewer PowerPoint Temporal profiles of the carrier and the growing rogue wave hole at various distances along the tank measured from the wave maker. The first distance is 3.2 m, i.e., when the breather is already well developed.

[16] According to the theory (red dashed curves), there should be a mirror symmetry between the two plots relative to the zero level. This symmetry transforms the elevated peak into a deep hole. Indeed, when comparing the two experimental (solid blue) curves, we can also see this symmetry. The only difference is that the depth of the hole relative to the two neighboring crests is smaller than the height of the central peak in Figure 2 relative to the deepest troughs. We attribute this reduction in the height to the wave steepening effect which has opposite effect on two types of waves thus suppressing the depth of the hole more than the height of the elevated rogue wave.

[17] Ideally, the amplification factor, i.e., the ratio of the depth of the hole to the amplitude of the carrier should be equal to 3. In order to see the influence of the steepness of the waves on the relative depth of the hole we reduced the amplitude of the carrier wave and observed the increase of the amplification factor. Figure 5 shows experimental results for the surface elevation when the carrier amplitude is only 75% of the previous case without varying the frequency. The deepest amplitude is now closer to 3 times the amplitude of the carrier as the wave steepness is lower so that the carrier wave propagates without much distortions.

Figure 5 Open in figure viewer PowerPoint Comparison of measured surface height near the the position of maximum hole depth (solid blue line) with theoretical Peregrine solution (dashed red line) evaluated at X = 0. The background wave amplitude is 0.75 cm.

[18] Further reduction in the carrier wave amplitude makes the amplification factor even closer to the theoretical value 3. Figure 6 shows experimental results when the carrier amplitude is half of the value presented in Figure 3, i.e., it is equal to 0.5 cm, where the frequency remains unchanged, i.e., f 0 = 1.7 s−1. The wave steepness is the lowest in this case and the highest wave amplitude is close to 3 times the amplitude of the carrier wave.

Figure 6 Open in figure viewer PowerPoint Comparison of measured surface height near the position of maximum hole depth (solid blue line) with theoretical Peregrine solution (dashed red line) evaluated at X = 0. The background wave amplitude is 0.5 cm.

[19] Our experiment is a clear demonstration that deep water waves are governed by the nonlinear equation and have to be modeled accordingly. The fact that such a unique observed effect as the instability at the zero threshold and the subsequent wave amplification given by the rational expression is described by the NLS is an indication that nonlinearity sits deeply in the nature of these waves. While interference of linear waves could be another reason for occasional appearance of high‐amplitude waves or holes, their description would be different and would contradict to our present observations.

[20] Can rogue holes be observed in other media like in optics, for example? In this respect, we remind that Peregrine soliton has been previously observed in an optical fibre [Kibler et al., 2010]. For optical pulses much longer than the wavelength, i.e., in picosecond regime and longer, this is unlikely as the measurable quantity, the intensity in optics is different from the measurable elevation for water waves. However, with the move toward attosecond optical pulses carrier‐envelope phase becomes highly important and measurable parameter [Jones et al., 2000]. From this point of view, there is no principle restrictions for rogue hole observation in optics. We can also mention the so‐called “rogue black holes” that may exist in our Galaxy [Holley‐Bockelmann et al. [2008]. Possibilities for observations seem to be infinite.