Spatial evolution of an initially narrow-banded wave train

Transcription

1 DOI /s RESEARCH ARTICLE Spatial evolution of an initially narrow-banded wave train Lev Shemer 1 Anna Chernyshova 1 Received: 28 February 2017 / Accepted: 26 July 2017 Springer International Publishing AG 2017 Abstract Nonlinear evolution of narrow-banded unidirectional gravity wave trains along wave flume is studied both experimentally and numerically. The spatial version of the Zakharov equation serves as the theoretical model. The frequency domains of spatially linearly unstable disturbances to a monochromatic wave, as well as the frequencies of the most linearly unstable modes are determined theoretically as functions of the carrier wave steepness; these results serve as a reference for the following study. The effect of the spectral width on the evolution along the tank is considered for bi-modal initial spectra, as well as for spectra consisting of a carrier wave and two sidebands with small but finite amplitude. Good agreement between the experimental results and numerical simulations is obtained. The variation of the frequency spectra along the tank resulting from nonlinearity, as well as of the maximum envelope and crest height is investigated as a function of the initial conditions. Fermi Pasta Ulam recurrence is obtained for frequency spacing between the initial spectral harmonics approximately corresponding to most unstable disturbance. For narrower spectra, the evolution pattern becomes irregular, numerous additional harmonics are generated by nonlinearity; in this process very steep waves can be generated. The relevance of those findings to appearance of rogue waves is discussed. Keywords Nonlinear waves Benjamin Feir instability Spatial Zakharov equation Bi-modal wave systems Modulated wave groups Rogue waves B Lev Shemer shemer@eng.tau.ac.il 1 School of Mechanical Engineering, Tel-Aviv University, Tel-Aviv, Israel 1 Introduction Quantitative characteristics of extreme (rogue, or freak) waves that appear locally and spontaneously in the sea, and in particular, the probability of their appearance, constitute an important design parameter in marine and offshore engineering. Fortunately, in random wave field in deep sea these waves appear relatively infrequently, thus only few examples of such freak waves were recorded in field measurements (Kharif et al. 2009; Dysthe et al. 2008; Onorato et al. 2013). Therefore, in order to accumulate reliable information on rogue waves statistics, numerous experimental studies of probability of appearance of extremely high waves were carried out under controlled conditions in large experimental facilities (Onorato et al. 2006, 2009; Shemer and Sergeeva 2009; Waseda et al. 2009; Shemer et al. 2010a, b and additional references therein). Both unidirectional random wave fields and those with directional spreading were considered. A different research direction dealt with generation of extremely steep deterministic waves, thus eliminating the effect of randomness. Two main mechanisms have been considered in deep water conditions. Steep waves may be excited by focusing of multiple harmonics at a prescribed time and location. This approach has been realized experimentally for both unidirectional and 2D waves (Brown and Jensen 2001; Johannessen and Swan 2001, 2003; Shemer et al. 2007). Although high waves generated by focusing appear due to an essentially linear constructive interference of various spectral components, nonlinear effects are important, in particular near the focusing location. The second mechanism is essentially nonlinear and based on the analytical solutions of the nonlinear Schrödinger (NLS) equation, the so-called solitons, or breathers (Peregrine 1983; Akhmediev et al. 1987). These solutions represent particular cases of NLS solitons found earlier by Kuznetsov (1977) and Ma

2 (1979). The breathers are characterized by an initially nearly monochromatic wave train η (t) of specific shape that develops a growing hump. This hump growth can occur in the theoretical solutions periodically or only once. The Peregrine breather was suggested as a major route to deterministic freak waves generation (Shrira and Geogjaev 2010). Early experimental observations of envelope solitons were reported by Yuen and Lake (1975); results of measurements of the Peregrine breather for water waves were presented in Chabchoub et al. (2011) and in some subsequent publications, see, e.g. Onorato et al. (2013). These experiments indeed show some qualitative agreement with the Peregrine (1983) analytical solution of the NLS equation. It is important to stress, however, that significant quantitative and qualitative differences exist between the experimental results obtained in those studies and the theoretical solutions. To elucidate this point, careful experiments on the evolution of the wave train initially corresponding to Peregrine breather were carried out. In Shemer (2013) these experiments were accompanied by numerical solutions based on the modified NLS (Dysthe) equation, while in Shemer and Ee (2015) the fully nonlinear numerical solution of the potential unidirectional flow equations was presented. It was clearly demonstrated in those studies that the amplitude growth in the experiments, as well as in the numerical solutions, is actually notably slower and weaker than that predicted by the Peregrine breather behavior. Moreover, the wave train evolution is characterized by an apparently irreversible and essentially asymmetric spectral widening that cannot be obtained within the framework of the NLS equation. The intensity of the nonlinear interactions for a given characteristic wave steepness ε = k 0 ( η 2 ) 1/2, where k0 is the dominant wave number, depends strongly on the spectral widths (Shemer et al. 2010a), that thus plays an important role in the wave field evolution. Janssen (2003) introduced the Benjamin Feir index as a ratio of nonlinearity to the normalized spectral width, BFI = ε/( ω/ω 0 ) where ω 0 is the dominant radian wave frequency. Janssen stressed that BFI can be expected to characterize unidirectional wave field only in the framework of the NLS equation. However, as discussed in Shemer (2015), the NLS equation is derived under an assumption of vanishing spectral width (Zakharov 1968) and thus cannot be expected to describe quantitatively rogue waves that necessarily contain numerous spectral harmonics. Moreover, BFI is closely related to the Benjamin and Feir (1967) wave instability; nonlinear evolution of the wave field is commonly seen as originating due to this instability. The frequency spectrum width ωin the definition of BFI reflects the deviation from the carrier frequency of the most unstable sideband BF disturbance, or the limit of the sideband linear stability in a slightly different definition of BFI. However, BF instability parameters are obtained as a result of linear analysis of a system consisting of a carrier wave and two infinitesimal sidebands, and predict exponential growth of the amplitude of those sidebands in time. The amplitude of initial sidebands thus fast becomes finite, so that the conclusions based on the linear approach cease to be valid. Shemer (2010) considered nonlinear temporal evolution of a 3-wave system consisting of a carrier wave with a given steepness, and two side disturbances with small but finite amplitudes. Such a system allows an exact analytic solution of the wave evolution problem based on the Zakharov (1968) equation in terms of Jacobi elliptic functions (Stiassnie and Shemer 2005). Appearance of additional harmonics due to nonlinear interactions is not accounted for in this analysis. It was shown that the wave system under considerations undergoes periodic Fermi Pasta Ulam recurrence. Conclusions based on the linear stability analysis, however, proved to be are largely irrelevant for a 3-wave system in which the initial sideband amplitudes are finite. It was shown in Shemer (2010) that finite-amplitude disturbances may interact with carrier wave even beyond the linear stability limit. For short gravity-capillary waves, this conclusion was supported by experimental results on Benjamin Feir instability by Shemer and Chamesse (1999). For unidirectional waves with given nonlinearity, the sidebands wavenumber deviation from that of the carrier does not uniquely define the nonlinear evolution pattern of the whole wave system; this depends also on the initial complex sideband amplitudes. The goal of the present study is to examine the relation between the linear stability analysis and the actual evolution of an initially simple unidirectional wave system experimentally and theoretically. It should be stressed that there are two important factors that distinguish experiments in a laboratory wave tank from the theoretical studies cited above, see also Dold and Peregrine (2006) and Banner and Tian (1996). First, the evolution of the wave field in experiments occurs in space (along the tank) rather than in time. Measurements of the surface elevation η are performed at fixed locations xalong the tank as a function of time; the wave field thus can be characterized by the evolution of the frequency spectra of ˆη (ω, x) rather than by the variation in time of the wave vector spectra. Second, the initial spectrum limited to three frequencies or less may undergo widening due to nonlinear interactions and thus the number of significant harmonics in the spectrum may change in the course of evolution. The theoretical analysis that constitutes the basis for the present experiments accounts for these factors. The initial narrow-banded spectra considered here are limited to two or three harmonics. We extend the previous investigations of initially bi-modal spectra by Shemer et al. (1998, 2001) and examine experimentally and numerically

3 the effect of the initial spectral width and of the spectral shape defined by initial harmonics amplitudes, on the spatial evolution pattern of the wave field. These finding may be related to the results of the linear stability analysis of spatially evolving unidirectional gravity waves. 2 Theoretical approach and the numerical procedure For sufficiently deep water, at the lowest order, the basic unit that enables nontrivial nonlinear interactions among gravity waves is a near resonant wave quartet. The evolution of complex amplitude of each wave vector harmonic in time is described by the Zakharov (1968) equation. In order to carry out quantitative comparison of the experimental results with the theoretical predictions, the set of unidirectional discretized spatial Zakharov equations that describes evolution of amplitude of each frequency harmonic in the spectrum with the distance x rather than in time (Shemer et al. 2001, 2007) is applied: c g, j i db j(x) dx = V j,l,m,n Bl B m B n e i(k j +k l k m k n)x. (1) ω j +ω l =ω m +ω n The complex amplitudes B j (x) = B ( ω j, x ) are related to the Fourier amplitudes of the surface elevation ˆη ( ω j, x ) and of the velocity potential at the free surface ˆφ s (ω j, x): ( ) g 1/2 B(ω j, x) = ˆη(ω j, x) + i 2ω j ( ) ω 1/2 j ˆφ s (ω j, x). 2g (2) The frequency ω is related to wave number k by intermediatedepth linear dispersion relation ω 2 = kg tanh (kh), (3) where h is the constant water depth. In Eq. (1), the values of V j,l,m,n = V (k(ω j ), k(ω l ), k(ω m ), k(ω n )) represent the quartet interaction coefficients in the temporal Zakharov equation (Krasitskii 1994); c g, j is the group velocity of the j th spectral component. Nontrivial (i.e. containing at least 3 different harmonics) unidirectional resonant quartets do not exist. Nonlinear interactions are possible when the wave numbers k j = k(ω j ) in the wave quartet satisfy the near-resonance condition: k j + k l k m k n = O ( ε 2 ) k mn, where the mean quartet wavenumber k mn = ( ) k j + k l + k m + k n /4. Set of equations (1) is solved for each free spectral harmonic represented by B j, j = 1,...N, N being the total number of free harmonics considered. The summation in (1) is carried out over all near resonant quartets that contain the jth harmonic. For each initial spectrum, the number of harmonics N considered in the computations was selected so that the amplitudes at the lowest frequency, ω 1, and the highest frequency, ω N, remain vanishingly small everywhere in the course of the spatial wave train evolution. Benjamin Feir (BF) instability is a result of linear analysis of a system consisting of a carrier wave with amplitude a 0 and wave vector k 0, and two infinitesimal disturbances with wave vectors k and k +,2k 0 = k + k +. The temporal growth of these disturbances was determined using either the linearized Zakharov equation (Crawford et al. 1981; Stiassnie and Shemer 1984), or full nonlinear equations (McLean et al. 1981; McLean 1982). In the unidirectional spatial evolution approach, the corresponding 3-wave system consists of a carrier (taken twice in the near-resonating quartet) with frequency ω 0 and amplitude a 0 and two sidebands of vanishing amplitude and frequencies ω ± = ω 0 ± ω, so that 2ω 0 = ω + ω +. Following Stiassnie and Shemer (1984), the linearized (with respect to sideband amplitudes B ± ) system of the spatial Zakharov equations (1) can be written as c g,0 i db ( 0 dx = V 0,0,0,0 B 0 2) B 0 (4a) c g,+ i db ( + dx = 2V +,0,+,0 B 0 2) B + + V +,,0,0 B B2 0 e ikx (4b) c g, i db ( dx = 2V,0,,0 B 0 2) B + V,+,0,0 B+ B2 0 e ikx, (4c) where K = k + + k 2k 0 is the deviation of the wavenumbers in the quartet from the exact resonance, B 0 represents the carrier wave and B, B + the sidebands. Note that the indices +/ in Eqs. (4b, 4c) and in sequel denote the frequency components ω + and ω, respectively. For the initial amplitudes B j (0) = b j, the solution of Eq. (4) is: B 0 = b 0 e i V 0,0,0,0 C b 0 2 x g,0 B + = b + e i(0.5k 1+δ)x B = b e i(0.5k1 δ)x. In the solution (5), (5a) (5b) (5c) K 1 = 2 V 0,0,0,0 c g,0 b K (6a) and δ = b 2 0 ( V+,0,+,0 c g,+ V ) c,,0,,0 ± D. (6b) c g,

4 becomes negative. The curves corresponding to D = 0 define the instability boundaries, the disturbances with D < 0are unstable with the dimensionless spatial growth rate σ = D/k0. The most unstable disturbance corresponds to the minimum of D. Thespatialgrowthrateσ calculated using (7) for deepwater gravity waves is plotted in Fig. 1 for a number of values of the carrier wave steepness ε = a 0 k 0, where the carrier wave amplitude a 0 is related to b 0 in (5) (7) as ( ) 1 b 0 = a 0 π 2g 2 ω j,0. The curves in Fig. 1 are in general compliance with temporal growth rates presented in the works cited above. The instability domain widens with increased steepness. The growth rate of the most unstable mode and the relative frequency deviation of the most unstable sidebands from that of the carrier wave also increase with ε. However, the stability analysis based on the spatial Zakharov equation shows that, similarly to the temporal instability case (Shemer 2010), the width of the frequency domain of the unstable disturbances and the maximum spatial growth rate do not vary linearly with the carrier wave steepness ε. The nonlinear spatial evolution of the prescribed frequency amplitude spectrum was determined by solving numerically the discretized spatial Zakharov equation (1) using Runge Kutta numerical scheme. The temporal variation of the surface elevation η(t, x) was obtained at each location x from the computed complex amplitude frequency spectrum a(ω j, x) by applying inverse Fourier transform. Second-order bound waves were added to the computed time series of η(t, x) in order to carry out comparison with the measurements (Shemer et al. 2007). 3 Experimental facility and procedure Fig. 1 The dimensionless spatial growth rate σ = D/k 0 of linearly unstable sidebands as a function of the relative deviation of the sideband frequency from that of the carrier for different values of wave steepness ε The sidebands become unstable and grow exponentially with x when the discriminant D = ( ( V+,0,+,0 0.5K 1 V +,,0,0 c g,+ c g,+ V ) ) 2 c,,0,,0 b0 2 c g, V c,+,0,0 c g, b 4 0 (7) Experiments were performed in the Tel Aviv University wave tank that has a length of 18 m, width of 1.2 m and depth of 0.9 m, the water depth during the experiments was maintained at h = 0.6 m. The tank is equipped with a computercontrolled flap type wavemaker. A 3-m-long wave energy absorbing sloping beach is positioned at the far end of the tank. Measurements of the instantaneous surface elevation along the centerline of the tank were performed by four resistance-type wave gauges mounted on a bar parallel to the side walls of the tank and connected to a computer-controlled carriage that can be placed at any desired location along the wave tank. The wave gauges were statically calibrated before and after each experimental session. For details of the computer-controlled calibration procedure, see Shemer et al. (2007). Distance between the consecutive gauges is 40 cm, the closest position of first wave gauge to the wave maker is 40 cm. Each experimental run was initiated when the water surface in the tank was calm and the carriage was located at the desired position along the tank. Data sampling was initiated at the instant of the wavemaker activation by the desired driving signal. The outputs of the 4 wave gauges, as well as of the wavemaker position sensor were sampled for the whole duration of the driving signal and beyond, until the wave train passed through the measuring locations. The probe carriage was then moved by 1.6 m to the next location. Sufficient interval (usually 10 min) was maintained before the start of the next experimental run to ensure that all disturbances on the water surface disappear. Measurements were performed at 7 carriage positions, resulting in 28 locations along the wave tank. The distance of the last measuring location from the wavemaker was x = 1120 cm. The following considerations were taken into account in selecting the operational parameters. The dimensionless scaled length for third-order nonlinear spatial evolution is given by ε 2 k 0 x (Shemer et al. 2002, 2007), thus in order to increase the effective length of the experimental facility the steepness ε should be high, and the carrier wave as short as possible. On basis of previous experiments in our facility, the highest carrier wave frequency f 0 was selected that enables accurate measurements with largely negligible dissipation effects along the tank; it corresponds to the period T 0 = 1/f 0 = 0.7 s and the carrier wave length λ 0 =

5 Fig. 2 Comparison of the bi-modal wave field at the prescribed distance of 160 cm from the wavemaker (in blue), with that at the wavemaker location (in red): a amplitude spectra, b surface elevation variation with time (free waves only). Envelope steepness ε = a env k 0 = 0.21; carrier wave period T 0 = 0.7 s; sideband frequencies ω ± = (ω 0 ± ω); ω/ω 0 = 1/10. Initial amplitudes a = a + = 1/2a env, 10 harmonics used in simulations 2π/k 0 = 0.77 m. The dimensionless depth k 0 h = 4.9 >> 1, so deep water conditions for the carrier wave are satisfied. However, since longer wave harmonics may appear in the process of nonlinear evolution, dispersion relation (3) valid for arbitrary depth was used in computations. Initial amplitudes generated by the wavemaker are limited by emerging breaking that is not accounted for in the theoretical model. The wavemaker driving signal was therefore chosen to form an envelope with steepness ε = 0.21; no wave breaking was observed in the tank at the measuring locations. It should be noted that occasional breaking occurred at larger distances from the wavemaker (x > 12 m). The initial phases of all harmonics were set to zero. For this steepness, the maximum growth rate is attained at the relative sideband frequency deviation of ω/ω 0 = 0.195; the disturbances with ω/ω 0 > are linearly stable (see Fig. 1). Note also the shorter is the carrier wave length, the slower is the propagation velocity of the narrow-banded wave train determined by its group velocity. For the selected carrier wave period T 0, the group velocity c g = 0.55 m/s. In view of possible contamination of the forward-propagating waves by reflections from the far end of the tank, an effort was made to limit the duration of the wave train. Attention was given to the appearance of the evanescent modes that affect surface elevation measurements in the vicinity of the wavemaker up to distance corresponding to about 3 h (Dean and Dalrymple 1991). The experiments were therefore designed so that the initial surface elevation frequency spectrum was prescribed at a distance of 160 cm from the wavemaker. To determine the wavemaker driving signal, the required temporal variation of the surface elevation η(t) at the wavemaker was computed by backward integration of (1) from x = 160 cm to the wavemaker located at x = 0cm. The wavemaker driving signal is then calculated from η(t) employing the wavemaker transfer function. This procedure is illustrated in Fig. 2 for an initial wave given as η(t) = a 0 [cos(ω t) + cos(ω + t)], with ω + ω = ω 0 /5. As is clearly visible in this figure, in order to obtain bi-modal spectrum at the prescribed location, the wavemaker has to generate a more complicated non-symmetric spectrum that contains at least three significant harmonics of unequal amplitudes. The envelope of the resulting temporal variation at the wavemaker is therefore visibly skewed. Hilbert transform was used to calculate the envelopes of the measured wave trains; bound waves were filtered out of the measured data using a band pass filter (Shemer et al. 2007). 4 Results 4.1 Initially bi-modal wave trains Wave trains consisting of two components with initially equal amplitudes a ± at angular frequencies ω ± = ω 0 ± ω around the carrier frequency ω 0 and different frequency spacing ω around the carrier are considered first. The inverse spectral width for this case may be defined by the number of waves in the group, n = ω 0 / ω, larger values of n correspond to narrower initial spectrum and for a given n, longer duration of the single wave group. In the present study the following integer values n were considered: n = 8, 10, 16, 20, 26, and

6 Fig. 3 Comparison of measured (solid lines) and simulated (broken lines) surface elevation variation with time for an initially bi-modal system with envelope steepness ε = a env k 0 = 0.21, a = a + = 1/2a env ; carrier wave period T 0 = 0.7 s; 10 harmonics used in simulations for n = 10; 16 harmonics for n = In each run, 3.5 wave groups were generated, with the total duration ranging from 19.6 to 78.4 s, depending on n. The analysis of the recorded surface elevation was carried out for the central part of the signal that represents a single group with duration nt 0 and is least affected by the end effects. To enable comparison at various distances from the wavemaker x, the analyzed segments of the wave elevation records were shifted in time by x/c g. The results presented in Fig. 3 are in general agreement with the previous studies of bi-modal wave groups (Shemer et al. 1998, 2001). Note that the computed wave shapes in this figure contain contributions of the second-order bound waves that result in a notable crest trough asymmetry of steep waves. The measured and simulated records of the surface elevation agree well in both left and right parts of Fig. 3. While the initial wave train shapes are symmetric about the vertical axis, left right asymmetry develops as the wave train propagates along the tank; this asymmetry is more pronounced in the longer wave group (Fig. 3b). The developing asymmetry of the wave train is accompanied by steepening of the largest wave in the group. These phenomena indicate that nonlinearity plays a prominent role in the evolution of wave groups. In order to assess quantitatively the effects of nonlinearity, evolution of frequency spectra of the surface elevation along the tank is now considered. Note that while variations in the wave train shape can occur also due to linear effects such as constructive or destructive interference, spectral changes can only be caused by nonlinear interactions. The computed amplitude spectra of the surface elevation are compared in Fig. 4 with the corresponding measured spectra at a number of locations along the tank for different numbers of waves in the group, n. Note that in all cases the computations are performed for the same range of frequencies; a number of harmonics increases with n due to finer spectral resolution for longer wave groups; it is determined by the duration of the group, ω = 2π/nT 0. Good agreement is obtained between the numerical and experimental results. In all spectra, the adopted experimental procedure indeed allowed to generate at the prescribed location x = 160 cm wave trains with an essentially bi-modal spectrum for all values of n. Additional harmonics are generated by nonlinear interactions as the wave train propagates along the tank as is clearly visible in simulations as well as in the experimentally obtained spectra. The number of harmonics in the spectra varies with n and the spectra become more complicated for larger values of n. Note that a plateau appears at the simulated spectra for n 26. Nevertheless, the spatial evolution patterns of spectra for different values of n have significant similarities. In experiments as well as in computations, the initially symmetric spectra become notably asymmetric as the wave train evolves along the tank. The amplitudes of the harmonic with frequency ω remain nearly constant along the tank, while those of ω + decrease notably; this harmonic apparently loses energy to the additional spectral components generated by nonlinear interactions, most of them at frequencies ω>ω +. This effect is somewhat more pronounced in the experiments than in the simulations. Nonlinearly generated harmonics with frequencies ω<ω have significant amplitudes only when the frequencies are very close to ω and therefore are only visible at higher resolution spectra for larger values of n. To get a closer insight into the spectral changes in the process of nonlinear wave train evolution, the variation along

7 Fig. 4 Spectral evolution of an initially bi-modal wave trains for ε = 0.21, a = a + = 1/2a env and various numbers of waves in the group n. Measured amplitudes are plotted by open symbols, simulations by filled symbols the tank of the amplitudes of four harmonics that contribute essentially to wave energy is studied in Fig. 5 for two extreme lengths of the wave group, n = 8 and n = 32. In addition to the harmonics with frequencies ω and ω + present in the initial spectrum, the variation with x of the amplitudes of the two closest to them harmonics with angular frequencies ω ω

8 Fig. 5 Evolution of amplitudes of an initially bi-modal wave trains ε = 0.21, a = a + = 1/2a env and various numbers of waves in the group n along the wave tank. Simulated amplitudes are plotted by lines, measured by markers and ω + + ω is plotted as well. These additional harmonics indeed have vanishingly small amplitudes at the prescribed location x = 160 cm, thus demonstrating the effectiveness of the adopted procedure. However, in order to attain the required spectral shape at x = 160 cm, at the wavemaker those harmonics need to have nonzero amplitudes. There is a reasonably good agreement between the simulations and experimental results that both show an essential qualitative difference in the evolution patterns of the amplitudes of the selected harmonics in the two panels of Fig. 5. In both cases the amplitudes of ω nearly do not vary with x in the simulations; in experiments some oscillations in the amplitude about the nearly constant value are observed. The amplitudes of ω + initially decrease with the distance and then seem to grow again in the simulations for n = 8; no such growth is obtained in the experiments. For n = 32 (Fig. 5b), the amplitudes of ω + decrease with x in simulations as well as in experiments. The variation of the amplitudes of two additional side harmonics in both panels of Fig. 5 shows good agreement between computations and experimental results. The agreement between experiments and simulations in Fig. 5 and indications that the variations of the harmonic amplitudes along the tank may exhibit periodic, or quasiperiodic behavior that are clearly identifiable in Fig. 5a, prompted us to extend the investigation of the evolution of these harmonics to simulations performed along much larger distances, up to x = 100 m; see Fig. 6. In this figure, the energy conservation in the course of wave train evolution along the tank and the contribution of the selected harmonics to the total wave energy is assessed as well. To the leading order, the total wave energy is represented by the sum of the squared amplitudes of all spectral components, 1 a 2 j. The contribution of the four selected harmonics to the total energy is given in the linear approximation by the sum of squared amplitudes of those harmonics, 2. Both those quantities, normalized by the total initial energy of the bi-modal spectrum represented by a 2 + a2 +, are plotted in Fig. 6 as well. In all panels of Fig.6, the total wave energy 1 is nearly constant during the whole spatial evolution process, thus confirming the accuracy of the numerical simulations. The deviations of 1 from exact constant may be attributed to contribution of the higher order terms not accounted for in the energy calculations based on the linear theory (Stiassnie and Shemer 1987). The relative contribution of the four selected harmonics to the total energy, 2, is dominant for the relatively short duration of the wave group, n = 8, and decreases with increase in n. Figure 6 confirms the conclusion obtained from Fig. 4 that for n 10, the wave energy beyond those four harmonics is quite small (the only essential spectral harmonic not accounted for is that at the frequency ω ω). In these cases (Fig. 6a, b), the amplitudes of those harmonics vary nearly periodically in space, exhibiting evolution pattern reminding Fermi Pasta Ulam recurrence. Note that in Fig. 6a, b the harmonics ω ω, ω and ω + + ω vary in phase (at the slow length scale), while the phase of the variation of ω + is opposite, so that the total energy of the evolving waves 1 is nearly conserved. The period of modulation for n = 8 is shorter than that obtained for n = 10, while the depth of modulation decreases for larger n and thus smaller frequency spacing between the harmonics. The findings of the analytical solution of the temporal evolution problems for a three-wave system by Shemer (2010) are in qualitative agreement with the results plotted in Fig. 6a, b. Stiassnie and Shemer (2005) demonstrated analytically using the Zakharov equation that a four-wave system undergoes periodic slow time modulation of amplitudes, although

9 Fig. 6 As in Fig. 5 for numerical simulations only, carried out for long distances exceeding the length of wave tank. 1 is the normalized total wave energy in the linear approximation, 2 represents the normalized energy of the four harmonics contrary to a three-wave system, the periodicity in this case in not exact since the phase behavior is not periodic. The lack of exact periodicity of amplitudes in Fig. 6a, b can be attributed in part to differences between the spatial and the temporal formulations of the Zakharov equation; the spatial formulation (1) does not allow obtaining an analytical solution for a three- or four-wave system. Even more important, the accumulated effect of additional harmonics that are present in the spectrum further contributes to the lack of exact periodicity. Indeed, the deviations from periodicity are more visible for n = 10 where the frequency component corresponding to ω ω that is not plotted in Fig. 6 is more significant; see Fig. 4. The effect of additional harmonics becomes much more pronounced in Fig. 6c, d that correspond to longer initial wave groups with a smaller distance ω between the harmonics and thus narrower initial bi-modal spectrum. While some quasi-periodicity in long-distance modulation of the selected four harmonics can still be identified in Fig. 6c, d, the overall pattern is much more chaotic. The decrease in ω is accompanied by longer characteristic evolution distances in this case. The variation with x of the maximum of the group envelope is plotted in Fig. 7 for two values of n. Since the envelope only represents the contribution of free waves, the evolution along the tank of the maximum crest heights and of minimum trough depths is also plotted in this figure to demonstrate the contribution of bound waves. The variation of the all these parameters in Fig. 7a is consistent with the spatial quasiperiodicity in the evolution of the four leading harmonics in Fig. 6b; the spatial extent of the measurements domain roughly corresponds to the modulation period. In the case of the narrow initial spectrum (n = 26, Fig. 7b), the evolution is aperiodic and strong amplification of the envelope and of the crest maxima is obtained in experiments as well as in simulations. This is consistent with wave form evolution plotted for a somewhat longer wave train with n = 32 in Fig. 3b.

10 Fig. 7 Evolution of maximum of crest (dashed line simulated, triangles measured), maximum amplitude of envelope (solid linesimulated, squares measured), minimum of trough (dotted line simulated, circles measured) We now consider initially bi- modal spectra with unequal amplitudes; the sum of both amplitudes is retained as in the previous figures so that the wave steepness and thus the characteristic wave group nonlinearity remain unchanged. In the spatial evolution of all spectra presented in Fig. 8 for an intermediate group length with n = 20, notable similarity is observed between simulations and experiments. In Fig. 8a, the case with the initial amplitude of the lowfrequency ω component constitutes half of the amplitude of the ω + component is considered. In experiments as well as in simulations, the initial spectrum rapidly evolves to the spectral shape in which the low frequency harmonics grows while the amplitude of the ω + component decreases. This process is accompanied by spectral widening, mainly on the high-frequency side. The evolving spectral distributions in Fig. 8a are very similar to those plotted in Fig 4d. Figure 8b, c examines the case when the ratio of initial amplitudes in the spectrum is 1:3. In Fig. 8b higher frequency has larger amplitude, while in Fig. 8c the leading initial component has angular frequency ω. While certain differences between the spectral shapes persist up to the most remote location x = 11.2 m, both those spectra are characterized by the low-frequency component becoming significantly larger than that corresponding to ω +, and qualitatively similar spectral distributions. It thus can be concluded that although the initial spectral shape affects somewhat the spectral evolution along the tank, the prevalent factor that determines the spectrum at larger distances from the wavemaker is the initial spectral width and nonlinearity. In order to follow more closely the spectral variation along the tank, the amplitudes of four harmonics with frequencies ω ω, ω,ω + and ω + + ω are plotted in Fig. 9 for the conditions of Fig. 8. An additional panel corresponding to the case with close initial amplitudes with a /a + = 6/5 is presented as well. Independently of its initial relative amplitude, in all panels of this figure, the harmonic with frequency ω becomes the largest one after evolution along about 10 dominant wave lengths, λ 0. The numerical and experimental results agree well in all cases. The variation of the harmonic amplitudes is mostly non-monotonic and differs notably in various panels of this figure. In all cases presented, however, the amplitudes of initially nonexistent harmonics ω ω and ω + + ω become significant in the course of spatial evolution and may exceed the amplitudes of the harmonics in the initial bi-modal spectrum. Typical examples of the computed evolution of the same harmonics along much longer distances are presented in Fig. 10. The evolution patterns in Fig. 10 for n = 20 are irregular, although some quasi-periodicity can be noticed, similarly to the behavior observed in Fig. 6c, d for longer bi-modal wave trains with identical initial amplitudes of the spectral components. Qualitatively similar evolution of the important harmonics was obtained for other ratios of the initial amplitudes in the bi-modal spectrum and therefore is not shown here. The variation with distance of the maximum envelope, crest and trough values for some of the conditions studied in Figs. 8, 9 and 10 is examined in Fig. 11. The simulations and the experimental results clearly show that the maximum crest height recorded for n = 20 may exceed by a factor of 2 the initial maximum envelope; this amplification is even stronger than that observed for the symmetric initial conditions; see Fig. 7. The evolution of an initially bi-modal nonlinear wave group with sufficiently narrow spectrum thus may result in appearance of quite steep waves.

11 Fig. 8 Spectral evolution of an initially bi-modal wave trains for ε = 0.21, n = 20 and different ratios of initial amplitudes a /a +. For legend see Fig Initial wave system consisting of a carrier wave and sidebands The initially bi-modal spectra considered so far represent the simplest possible wave system beyond the Stokes wave with a single free wave component. The evolution of such a system, however, cannot be analyzed directly in the framework of Benjamin Feir linear stability approach since both initial harmonics have comparable amplitudes, with zero amplitude at the carrier wave frequency ω 0 = (ω +ω + )/2. We now investigate the spatial evolution of a system that initially consists of a dominant component at the carrier frequency ω 0, and one or two sidebands at frequencies ω ± = ω 0 ± ω of relatively small compared to the carrier but finite amplitude. Following the approach adopted in the investigation of initially bi-modal spectra, the carrier wave frequency ω 0 corresponds to the wave period T 0 = 0.7s; the sum of amplitudes of all initial harmonics a env = a 0 + a + + that determines the maximum envelope is also kept constant, so that the characteristic nonlinearity ε = a env k 0 = Based on the results presented in Fig. 1, for ω/ω 0 > 0.275, the sidebands are linearly stable. While nonlinear interactions for finite-amplitude sidebands are still possible beyond the linear stability limit, they are quite weak (Shemer 2010) thus preventing their quantitative experimental investigation. The present study is therefore limited to relatively narrow band with sideband frequencies within the linear instability domain. Experiments and simulations were therefore carried out for n = 4, 5, 8, 10 and 16, where the inverse relative width of the initial spectrum is defined as n = ω 0 / ω, so that n represents the number of waves in the group with duration nt 0. The evolution of the frequency spectra along the tank is presented in Fig. 12 for four values of n and the initial symmetric spectrum with the relative amplitude of both sidebands a ± /a env = 0.1 and of the carrier a 0 /a env = 0.8. The results in this figure are consistent with the spatial evolution of ini-

12 Fig. 9 Evolution of amplitudes of an initially bi-modal wave trains for ε = 0.21, n = 20 and different ratios of initial amplitudes a /a +. Simulated amplitudes are plotted by lines, measured by markers Fig. 10 As in Fig. 9 for n = 20, simulations carried out for distances exceeding the length of wave tank. For legend see Fig. 6

13 Fig. 11 Evolution of the maximum of the crest (dashed line simulated, triangles measured), maximum amplitude of the envelope (solid line simulated, squares measured), minimum of the trough (dotted line simulated, circles measured) for n = 20 tially bi-modal spectra in Fig. 4 and exhibit good agreement between the numerical simulations and experiments. As the initial spectrum becomes narrower with increase in n, the spectral variations along the tank become more pronounced, with significant spectral widening and increase in the amplitude of the high-frequency harmonics at the expense of that of the carrier wave. Note that the spectral evolution for n = 5 that approximately corresponds to the linearly most unstable disturbance is qualitatively different. The amplitudes of additional harmonics generated by nonlinear interactions remain small at all locations, and the variations of the amplitudes of the carrier wave and sidebands, while existing, are notably smaller than that obtained in simulations and measured in experiments for higher values of n and thus initially narrower spectra. In Fig. 13 the effect of asymmetry of the initial spectrum is examined for an even wider initial spectrum with n = 4, both sidebands are generated by the wavemaker in Fig. 13a, while Fig. 13b, c only one sideband is initially present in the spectrum. Note that the sideband disturbances for ω/ ω 0 = 0.25 and ε = 0.21 are still linearly unstable but have frequencies close to the instability limit; see Fig. 1. For initially symmetric spectrum in Fig. 13a, the spectral evolution is qualitatively similar to that obtained for n = 5 in Fig. 12, with no essentially new harmonics being generated. The variation of amplitudes of the three harmonics in Fig. 13a is somewhat weaker than that observed in Fig. 12 for n = 5. When only one sideband is excited, the initial spectrum is bi-modal and the spectral evolution pattern has common features with that observed for a much narrower spectrum in Fig. 8 that was characterized by a stronger activity on the high-frequency side of the spectrum. In Fig. 13b only low-frequency sideband is excited initially by the wavemaker, the high-frequency sidebands grows and its amplitude becomes comparable and may even exceed that at the frequency ω 0 ω. Contrary to that, when the high-frequency sideband is excited, the amplitude of lower sideband remains relatively small along the whole evolution distance. The initial spectral asymmetry can also affect the evolution of the wave group envelope as examined in Fig. 14 for a narrower initial spectrum with n = 10. This figure demonstrates that for initially asymmetric spectra with only one sideband excited, the maximum of the group envelope, as well as the maximum of crest heights and trough depths do not change notably along the tank length in experiments and in simulations. For the initially symmetric spectrum however, sharp increase in the maximum crest height was obtained in experiments; in simulations this effect was less pronounced. Evolution of wave groups along distances exceeding significantly that of the tank is examined numerically in Fig. 15 for sidebands of initially equal amplitude, a ± = 0.1 and ε = For n = 4, no additional harmonics with significant amplitude are generated and the energy of the three harmonics is very close to whole wave field energy along the entire spatial simulations domain. The wave field everywhere thus contains only those harmonics that were initially excited by the wavemaker; nonlinear interactions among the three waves lead to a nearly perfectly periodic Fermi Pasta Ulam recurrence, in agreement with the analytical 3-wave solution by Shemer (2010). The strongest modulation is experienced by the high-frequency sideband, whereas the low-frequency sideband remains nearly constant. Note also that for the case with n = 4 only, the actual amplitudes of both sidebands remain mostly below their initial level; the carrier wave amplitude is increased accordingly, so that total energy is conserved. This spatial behavior can be identified in the experimentally measured amplitude spectra plotted in Fig. 13b. When the sideband frequencies correspond to the

14 Fig. 12 Spectral evolution of an initially three-modal wave system with amplitude ratios a ± /a env = 0.1 anda 0 /a env = 0.8 for different spectral widths characterized by n. For legend see Fig. 4 linearly most unstable case, n = 5, the initial three harmonics remain dominant, but new harmonics with non-negligible amplitudes are excited at certain locations and then decay in a nearly periodic fashion, as can be seen from the deviations of the total wave energy from that of the three initially excited waves. The spatial evolution pattern is still nearly periodic; it closely reminds the solution of the temporal evolution of a three-wave system in terms of Jacobi elliptic functions (Shemer 2010). Deviation in the shape of the spatial dependence from the smooth behavior characteristic for the analytically obtained temporal evolution is mainly visible in Fig. 15b for the high-frequency sideband. The modulations for n = 5are notably stronger than those for n = 4; the period of spatial modulations is quite long and corresponds to about 70 λ 0 as compared to the characteristic modulation periodicity length for n = 4 of about 20λ 0, compatible with the effective length of the experimental facility. For n = 8, the spatial modulation pattern is qualitatively similar to that obtained for n = 5, but is notably less regular. The amplitude modulation of the three main harmonics in this case is stronger than that for n = 5; whereas their characteristic period about of 35λ 0 is twice shorter. For an even narrower initial spectrum with n = 16, the modulation ceases to be regular. The initial three harmonics vary apparently randomly; their amplitudes become of the same order and their relative contribution to the total energy decreases sharply. At distances from the wavemaker exceeding about 70λ 0, the energy in the initial spectral harmonics constitutes less than 50% of the total energy at most locations. The total energy of all components is still well conserved. 5 Discussion and conclusions Results of numerical simulations, coupled with measurements in a wave tank, of the spatial evolution of unidirectional deep gravity wave trains with spectra containing initially only two or three harmonics are presented. The numerical simu-

15 Fig. 13 Spectral evolution for different initial spectral shapes for n = 4, for legend see Fig. 4 lations based on the spatial Zakharov equation yield results that are consistent with the experiments. The evolution length scale at which near-resonant third-order nonlinear interaction occur depends on the nonlinearity ε and on the characteristic wave number k 0 as (ε 2 k 0 ) 1. To extend the effective length of the 18-m-long facility (14 m if wavemaker and beach are excluded), experiments were therefore carried out for the highest wave steepness that does not result in significant breaking, and shortest possible characteristic waves with length λ 0 = 2π/k 0 that are not affected significantly by dissipation and can be measured accurately. Nevertheless, in most cases considered, nonlinear evolution of the wave train continued at distances exceeding the length of the tank. To study the spatial evolution patterns at larger distances, the domain of simulations was therefore extended considerably beyond the physical dimensions of the tank. Nonlinear evolution of water gravity waves is often related to the temporal Benjamin Feir instability of Stokes waves. Since in any experimental facility the evolution occurs in space rather than in time, the linear spatial rather than temporal stability of monochromatic wave train was studied first in the framework of the spatial version of the Zakharov equation. This analysis allowed establishing the frequencies of the linearly most unstable disturbances and the domain of frequencies where the infinitesimal disturbances grow exponentially with distance. This preliminary analysis provided the background for study of nonlinear spatial evolution of simplest possible wave systems with finite initial amplitudes. Analytical study of nonlinear evolution of simple wave systems carried out by Shemer (2010) suggested that extension of conclusions based on Benjamin Feir instability to long-time evolution of wave trains initially containing few finite amplitude components may be inappropriate. The present study expands and modifies this earlier investigation in several important aspects. First, a combined experimental and theoretical study is performed, with consistent results obtained in measurements and in the numerical simulations. Second, in order to enable quantitative comparison between

16 Fig. 14 The variation of the maximum of envelope, crest height and trough depth within the group along the tank for n = 10 and different amplitudes in the initial spectrum. For legend see Fig. 7 experiments and calculations, spatial rather than temporal evolution of unidirectional nonlinear wave field is considered. Third, the present theoretical analysis is not limited to the spectral harmonics initially excited by the wavemaker, but accounts for appearance of additional components that may be excited in the process of the nonlinear wave train evolution along the tank. In this study, the wave groups are defined by their discrete frequency spectrum, which is consistent with the approach adopted in Benjamin Feir stability analysis. For an initially bi-modal spectrum, the spacing between the frequencies prescribes the temporal periodicity of the group that is retained along the entire spatial evolution domain. In experiments, however, the generated wave train necessarily has finite duration, and the exact periodicity defined by ω = ω + ω _ is violated; it is rather defined by the actual duration of the wavemaker driving signal. For a three-wave system, the exact periodicity may exceed significantly the effective group duration nt 0. A constant number of wave groups generated by the wavemaker in each experimental run was maintained in present experiments. The effective frequency resolution therefore increases with n. The similarity between the spectrum of the wavemaker driving signal with finite duration and the frequency spectrum assumed in the theoretical analysis thus improves as the values of n become larger, resulting in a better agreement between the numerical simulations and the measurements for longer group durations nt 0. The present experiments and numerical simulations exhibit common features characterizing the spatial wave group evolution with the initial spectra containing no more than three free harmonics within the linear stability domain. As long as the frequency spacing between the initial harmonics remains sufficiently large, the effect of the spectral widening is limited, and the evolution at the slow spatial scale is dominated by the three major harmonics present in the initial spectrum. For the carrier wave and two sidebands system, these harmonics govern the evolution along the whole extent of simulations; while for an initially bi-modal spectrum the gov-