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Extraction of solitons and harmonic wave from the time series: ͑ a ͒ initial time series and series with solitons cut off on the interval 1 + 2 ; ͑ b ͒ harmonic wave and “harmonic wave+ soliton;” ͑ c ͒ a sequence of solitons
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We have investigated the spatiotemporal properties of solitons generated on the shallow water surface over a background of a large-scale mode in a hydrodynamic resonator when it is forced near the second frequency mode. We have used the space-time diagrams to highlight the spatiotemporal dynamics of nonlinear fields for two solitons colliding in a...
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... pulses in the resonator are excited against the back- ground of a harmonic wave. The problem is how to separate pulses and harmonic oscillations. Obviously, this cannot be done by linear filters, as the repetition rate of pulses is equal exactly to the frequency of harmonic oscillations. Conse- quently, we determined the position of pulse maxima Fig. 5a, and then the resulting data were replaced in the inter- val 1 + 2 by the linear dependence of the free surface dis- placement on time. In this fashion we were able to separate the pulses. Further, all the harmonics with the frequency larger than the excitation frequency were separated from the obtained signal by filtering. The ...
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... inter- val 1 + 2 by the linear dependence of the free surface dis- placement on time. In this fashion we were able to separate the pulses. Further, all the harmonics with the frequency larger than the excitation frequency were separated from the obtained signal by filtering. The resulting signal as well as the initial time series are depicted in Fig. 5b. Figure 5c illustrates separately the harmonic mode of the resonator and the sequence of pulses. Apparently, the results of filtering depend significantly on the way we choose 1 and 2 . Differ- ent trials for quantities 1 and 2 showed that, with a reason- able choice of these parameters 1,2 T / 6, where T is the period of external ...
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... resulting signal as well as the initial time series are depicted in Fig. 5b. Figure 5c illustrates separately the harmonic mode of the resonator and the sequence of pulses. Apparently, the results of filtering depend significantly on the way we choose 1 and 2 . ...
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... , k are the frequency and wavenumber of harmonic waves, 0 is the wave amplitude, and s is the phase shift between the soliton and the harmonic wave Fig. 5c. Near the reflecting endwall x =0, where the stationary sensor is located, the displacement of the free surface can be repre- sented in the ...
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... series of experiments, we have varied the ratio of the coefficients of frequency-dependent and frequency-independent dissipation. Under the assumption that the dissipation is largely due to frequency-independent losses, e.g., 1 =10 −4 , =15 10 −4 , an oscillating tail compo- nent appears on the intervals between solitons Fig. 14; com- pare with Fig. 5 for experimental data. This phenomenon can be explained as follows: when the damping is propor- tional to k 2 , energy losses at higher harmonics increase with the number of harmonics, and high-frequency oscilla- tions are suppressed. Consequently, the soliton changes smoothly at the periphery. In the case of frequency- independent ...
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... Experiments are performed in the 33.73 m-long and 55 cm-wide LEGI wave flume with side glass panels. A schematic of the flume is given in Fig. 1 and a picture in Fig. 2. At one end the waves are generated by a pistontype wavemaker and opposite a vertical wall ends the flume, in a similar configuration to that of [26,30] but in a longer flume. Waves propagate back and forth in the flume reflecting on the wall and the wavemaker. ...
... al [33] and ii) of the amplitude amplification of the non-linear modes by their interactions with the moving wavemaker. Mention should be made of the experiments by [30] who, by slightly detuning the wavemaker motion with respect to the periodic longitudinal seiching mode of the channel, were able to find a route towards the generation of integrable turbulence. ...
We analyze a set of bidirectional wave experiments in a linear wave flume of which some are conducive to integrable turbulence. In all experiments the wavemaker forcing is sinusoidal and the wave motion is recorded by seven high-resolution side-looking cameras. The periodic scattering transform is implemented and power spectral densities computed to discriminate linear wave motion states from integrable turbulence and soliton gas. Values of the wavemaker forcing Ursell number and relative amplitude are required to be above some threshold values for the integral turbulence to occur. Despite the unavoidable slow damping, soliton gases achieve stationary states because of the continuous energy input by the wavemaker. The statistical properties are given in terms of probability density distribution, skewness and kurtosis. The route to integrable turbulence, by the disorganization of the wavemaker induced sinusoidal wave motion, depends on the non-linearity of the waves but equally on the amplitude amplification and reduction due to the wavemaker feedback on the wave field.
... Empirical confirmation that soliton gases can be generated were given in optics [21], for deep water waves [22] and for shallow water wave motion [23][24][25][26]. In the experiments of [25], energy dissipation cannot be avoided and this seems at first glance strongly incompatible with the concept of integrability. ...
... al [32] and ii) of the amplitude amplification of the non-linear modes by their interactions with the moving wavemaker. Mention should made of the experiments by [23,24] who by slightly detuning the wave maker motion with respect to the periodic longitudinal seiching mode of the channel where able to find a route towards the generation of weakly integrable turbulence. ...
We analyze a set of bidirectional wave experiments in a linear wave flume of which some are conducive to integrable turbulence. In all experiments the wavemaker forcing is sinusoidal and the wave motion is recorded by seven high-resolution side-looking cameras. The periodic scattering transform is implemented and power spectral densities computed to discriminate linear wave motion states from integrable turbulence and soliton gas. Values of the wavemaker forcing Ursell number and relative amplitude are required to be above some threshold values for the integral turbulence to occur. Despite the unavoidable slow damping, soliton gases achieve stationary states because of the continuous energy input by the wavemaker. The statistical properties are given in terms of probability density distribution, skewness, and kurtosis. The route to integrable turbulence, by the disorganization of the wavemaker induced sinusoidal wave motion, depends on the nonlinearity of the waves but equally on the amplitude amplification and reduction due to the wavemaker feedback on the wave field.
... A similar approach can be applied in regards to more complex sets of several solitons that interact with each other and with the long carrier wave. A similar process was observed in experiments with solitons excited by a standing wave in wave tanks and in electromagnetic transmission lines [67][68][69] ...
... La durée caractéristique de l'onde solitaire est définie pour = cosh −2 (1) = 0.41997, où A désigne l'amplitude (Ezersky et al. (2006)). La figure ...
Nous présentons les résultats expérimentaux sur la propagation des vagues solitaires et des trains d’ondes fortement non-linéaires se propageant en eau intermédiaire vers la zone côtière. Trois différents spectres ont été utilisés lors de la génération des groupes de vagues : spectre gaussien, spectre de Pierson-Moskowitz et spectre de JONSWAP. Les essais ont été réalisés dans le canal à houle du laboratoire M2C (UMR 6143 Caen). Les signaux obtenus par les sondes résistives sont utilisés pour la détermination de l’évolution spatiale de : la fréquence de pic, la fréquence caractéristique, les transferts énergétiques, les paramètres de forme et les interactions non-linéaires entre les composantes fréquentielles. La fréquence de pic demeure constante au cours de la propagation du train d'ondes de Pierson-Moskowitz ; cependant, une diminution est observée dans le cas des trains d’ondes de JONSWAP. Un partitionnement spectral a été effectué sur chaque spectre de référence pour quantifier les variations spatiales de l’énergie dans chaque gamme de fréquences. Des transferts énergétiques en amont et en aval de déferlement ont été mis en évidence. Une approche bispectrale par ondelettes a été utilisée dans le but de quantifier les interactions non-linéaires entre les composantes fréquentielles grâce à un paramètre appelé bicohérence. Au cours de la propagation, ce paramètre augmente progressivement et atteint des valeurs proches de 1 au moment du déferlement et décroit en aval de ce déferlement. Des simulations numériques ont été effectuées grâce au code mPeregrine, basé sur les équations de Boussinesq modifiées (Dutykh et al. (2011)). Les résultats expérimentaux concordent de façon satisfaisante avec les prédictions numériques, en particulier dans le cas des trains d'ondes gaussien (étroit) et de JONSWAP.
... cm f =0.177 Hz, et séries temporelles correspondantes prises au milieu du canal (aa,bb,cc,dd). Figure tirée de[20]. ...
... On verra que cela n'est pas antagoniste avec un comportement intégrable dans la mesure où cette répartition uniforme n'est pas liée à une équi-partition de l'énergie entre modes (thermalisation), mais au mélange des phases de ces modes qui conduit à une répartition aléatoire de l'énergie dans le système. Ce phénomène a été théorisé par Gurevich, Zybkin et El [27] qui, s'intéressant à l'évolution d'une condition initiale apériodique et oscillante à grande échelle, avaient constaté une désorganisation des grandes oscillations en de petites oscillations chaotiques due à un phénomène de mélange des phases et qualié ce comportement de turbulence solitonique.On s'intéressera donc à l'évolution de l'organisation de nos trains de solitons an d'établir que le chaos observé résulte d'une dynamique intégrable (mélange de phases) catalysée par un défaut de périodicité des premiers trains de solitons.On verra également le rôle que peuvent jouer, dans cette désorganisation, les phénomènes de transferts d'énergie (dynamique non-intégrable) décrits par Ezersky et al.[20] [21] entre modes propres longs du canal et modes localisés. ...
... L'excitation de la seiche du canal (mode propre 1) semble catalyser le phénomène.Transfert des modes du canal vers des modes localisésAprès 2000 secondes d'évolution, on commence à exciter le mode de seiche du canal f 1 = 0.0208 Hz. Celui-ci est susamment long pour être considéré comme un courant par un soliton.On observe alors un transfert d'énergie du mode long vers le mode localisé par un mécanisme mis en évidence dans[20].La gure 8.16 illustre l'évolution à l'aide d'une série temporelle de ce régime, agrandie dans 3 zones d'intérêt : -à t = 500 s (8.16 b), le régime semble avoir atteint un équilibre, on observe une houle périodique avec quelques harmoniques ; -à t = 2500 s (8.16 c), on excite un mode localisé. On peut analyser le phénomène comme un transfert du mode de seiche vers un soliton tant que celui-ci est en phase. ...
La turbulence d'onde est un état statistique impliquant un grand nombre d'ondes couplées par effet non linéaire. Cet état générique est potentiellement décrit, dans la limite de faible non linéarité, par la théorie de la turbulence faible. Cette théorie prédit un phénoménologie proche de celle de la turbulence hydrodynamique avec en particulier une cascade d'énergie au travers des échelles. La thèse de doctorat se concentrera sur le cas d'ondes de gravité-capillarité à la surface d'un fluide dans la limite des grandes longueurs d'onde. Ce système est potentiellement représentatif d'états de mer lorsque la houle est développée. Des expériences seront menées dans le canal à Houle 1D de 36m du LEGI. Le cas des ondes unidirectionnelles est assez particulier dans le cadre général de la théorie de la turbulence faible. En effet il est prédit que la turbulence faible est instable dans ces conditions et que l'on devrait observer plutôt l'apparence de structures cohérentes de type soliton. Au cours de cette thèse, on testera différentes conditions de forçage pour tester les prédictions théoriques concernant les cascades d'énergie et de quantité de mouvement. On sortira également du cadre strict de la théorie pour étudier des régimes de forçage fort ou de faible profondeu
... As observed experimentally in [37,38] and numerically in [39,40], a sine wave in shallow water spontaneously steepens and decomposes, after a certain propagation distance, into a train of solitons of various amplitudes. Our generation setup is similar to [41,42] but with a longer flume to ensure soliton fission. These solitons then interact with solitons emitted earlier that survive until dissipated by viscosity. ...
We report on an experimental realization of a bidirectional soliton gas in a 34-m-long wave flume in a shallow water regime. We take advantage of the fission of a sinusoidal wave to continuously inject solitons that propagate along the tank, back and forth. Despite the unavoidable damping, solitons retain their profile adiabatically, while decaying. The outcome is the formation of a stationary state characterized by a dense soliton gas whose statistical properties are well described by a pure integrable dynamics. The basic ingredient in the gas, i.e., the two-soliton interaction, is studied in detail and compared favorably with the analytical solutions of the Kaup-Boussinesq integrable equation. High resolution space-time measurements of the surface elevation in the wave flume provide a unique tool for studying experimentally the whole spectrum of excitations.
... The spots in Figs. 17 and 18(b) are discrete in both mode and frequency. The frequency discretization is due to the regular character of the record, and its interval is larger than the resolution. ...
... The nonlinear wave dynamics in a shallow water resonator was investigated by means of the numerical solution of the Boussinesq system in [17]. Importance of three-wave interactions revealed in the present study may prove that simplified models (like Boussinesq of Korteweg-de Vries equations) could be able to describe the observed phenomena, and be helpful in obtaining quantitative estimates. ...
Arising of modulations of surface gravity waves in a shallow water resonator under harmonic forcing is investigated in laboratory experiments. Different types of modulations are found, when the wave amplitude exceeds a certain threshold. Bifurcation diagram on the plane “amplitude of excitation – frequency of excitation” is determined. Numerical simulations of the Euler equations within the frameworks of the High-Order Spectral Method are performed with the purpose of reproducing the modulational regimes observed in the laboratory experiments. The simulations allowed us to determine physical mechanisms responsible for the occurrence of modulated waves.
... This method is used in the present study and contra-propagative solitary waves are excited on the background of a standing harmonic wave. The spatiotemporal properties of these solitary waves were investigated in [16]. From one to four pulses propagating in each direction of the flume may be generated on the time period of the flow, depending on the frequency and the amplitude of horizontal displacement of the oscillating paddle. ...
... From one to four pulses propagating in each direction of the flume may be generated on the time period of the flow, depending on the frequency and the amplitude of horizontal displacement of the oscillating paddle. These pulses were identified as solitons (one pulse) and bound states of solitons (several pulses) [16]. To the authors' knowledge, the only study on ripple formation induced by solitons was carried out by Marin et al. [17]. ...
... For values of the amplitude of horizontal displacement of the oscillating paddle averaged over depth a h lower than 2 cm, only standing harmonic waves are generated in the flume. For values of a h greater than 2 cm, pulses propagating from one end of the channel to the other end are excited on the background of the standing harmonic wave [16]. The generation of solitons in the flume results from the excitation of higher harmonics. ...
This paper reports the results of an experimental study of the formation of sand bedforms under solitons and bound states of solitons in a wave flume used in resonant mode. A numerical modeling of Boussinesq equations is carried out to simulate the generated flow and a good agreement is obtained between the measured and calculated fluid velocities. An original method based on image processing is developed to obtain the morphology of the sandy bed over large areas. The size and the shape of ripples which form on the bed significantly change from one end of the flume to the other end. Accretion zones are observed beneath the nodes of the standing harmonic wave. These bed features result from the variation of the amplitude of the first harmonic of fluid velocity, and of the time interval between the passage of contra-propagative solitons and bound states of solitons, with the distance from the wave paddle along the flume. An equation obtained from a phenomenological analysis is proposed for the description of the temporal growth of the ripples.
... For values of the amplitude of horizontal displacement of the oscillating paddle averaged over depth a h lower than 2 cm, only standing harmonic waves are generated in the flume. For values of a h greater than 2 cm, pulses propagating from one end of the channel to the other end are excited on the background of the standing harmonic wave [Ezersky et al, 2006]. The shape of these pulses was closed to the shape of soliton described by the classical Korteweg de Vries (KdV) ...
... The values of the frequency f and of the amplitude a h are chosen such as one or two pulses (one or two solitons regime) propagate in each direction of the flume on the time period of the flow. The procedure of signal separation into two parts corresponding to the harmonics and nonlinear wave solitons has been discussed in [Ezersky et al, 2006;Marin and Ezersky, 2008]. ...
In this report we present experimental results on sand bottom profile formation and particles segregation under strongly nonlinear surface (solitons) waves. We reveal the influence of bottom profile on dissipation rate of surface waves. Theoretical models for the interaction of solitons with sandy bottoms and the segregation of particles are proposed.
