The application of WT to **vibrating** **plates** started with the theoretical derivation of
the kinetic equation from the dynamical von K´ arm´ an equations [vK10, CH56, LL59] that
describe large-amplitude motions of thin **plates** [DJR06]. Since this date, numerous papers have been published covering experimental, theoretical and numerical materials. **In** fact, it appears that the **vibrating** plate is a perfect candidate for a thorough comparison of experiments with theoretical predictions. As compared to other physical systems such as capillary or gravity waves for example, an experimental set-up with a fine control of energy injection and a confortable range of wavelength is not too difficult to put **in** place. Secondly, the available measurement techniques allow one to get a complete and precise picture of the dynamics through the scales, both **in** the space and frequency domains. Finally, numerical codes with good accuracy have been developed so that all the underlying assumptions of the theory as well as its predictions have been tested, both on the experimental and the numerical levels.

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t ) (8)
which shows that the self-similar solutions of Eq.(2) have a characteristic frequency growing linearly with time.
Non-stationary **wave** **turbulence** **in** **plates** has been studied numerically **in** [10] with a finite-difference, energy-conserving scheme, showing a cascade front propagating to higher frequencies and leaving **in** its wake a self-similar spectrum. When the plate is continuously excited with an harmonic forcing, the front cascade evolves linearly with time. These behaviours are perfectly recovered **in** the present study, showing the ability of the phenomenological model to determine thanks to a simple equation complex features of the turbulent regime. The case of free **turbulence** has also been considered **in** [10], where the cascade front then evolves to higher frequencies as t 1/3 . This framework is here investigated through the use of

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ENSTA-UME, Unit´e de Recherche en M´ecanique, Chemin de la Huni`ere, 91761 Palaiseau, Cedex, France
(Dated: October 4, 2008)
The nonlinear interaction of waves **in** a driven medium may lead to **wave** **turbulence**, a state such that energy is transferred from large to small lengthscales. Here, **wave** **turbulence** is observed **in** experiments on a **vibrating** plate. The frequency power spectra of the normal velocity of the plate may be rescaled on a single curve, with power-law behaviors that are incompatible with the weak **turbulence** theory of D¨ uring et al. [Phys. Rev. Lett. 97, 025503 (2006)]. Alternative scenarios are suggested to account for this discrepancy — **in** particular the occurrence of **wave** breaking at high frequencies. Finally, the statistics of velocity increments do not display an intermittent behavior.

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.
5. Con
lusion
A phenomenologi
al model des
ribing the time-frequen
y dependen
e of the power spe
trum for **wave** turbulen
e **in** thin **vibrating** **plates**, has been derived. **In** the framework of non-stationary turbulen
e, the model equation has shown its ability **in** predi
ting the self-similar behaviours for two dierent
ases : free and for
ed turbulen
e. These two examples show the ability of our model to
ap- ture the most salient features of the dynami
s of thin elasti
**plates**. The model equation possesses a number of attra
ting features for further studies, the promi- nent one being its simpli
ity **in** handling
ompli
ating ee
ts su
h as for
ing and dissipation. Besides its ability **in** re
overing the self-similar behaviours already derived from the kineti
equation [24℄, a step further has been obtained with the derivation of two equations, (11) and (15), the solutions of whi
h are the self-similar universal fun
tions for the for
ed and the free
ases, whi
h were not provided by the theory developed from the kineti
equation **in** [24℄.

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UPMC, 75005 Paris, France, christophe.josserand@upmc.fr 2 IMSIA, ENSTA-ParisTech, CNRS, CEA, EDF, Universit´ e Paris-Saclay
91762 Palaiseau, France, cyril.touze@ensta.fr, olivier.cadot@ensta.fr
Abstract Thin **plates** **vibrating** at large amplitudes may exhibit a strongly nonlinear regime that has to be studied within the framework of **wave** **turbulence**. Experimental studies have revealed the importance of the damping on the spectra of **wave** **turbulence**, which precludes for a direct comparison with the theoretical results, that assumes a Hamiltonian dynamics. A phenomenological model is here introduced so as to predict the effect of the damping on the **turbulence** spectra. Self-similar solutions are found and the cut-off frequency is expressed as function of the damping rate and the injected power.

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{michele.ducceschi,cyril.touze,olivier.cadot}@ensta-paristech.fr
(2) Acoustics and Audio Group, University of Edinburgh, James Clerk Maxwell Building, Edinburgh,
UK, stefan.bilbao@ed.ac.uk
Summary. Nonlinear (large amplitude) vibrations of thin elastic **plates** can exhibit strongly nonlinear regimes characterized by a broadband Fourier spectrum and a cascade of energy from the large to the small wavelengths. This particular regime can be properly described within the framework of **wave** **turbulence** theory. The dynamics of the local kinetic energy spectrum is here investigated numerically with a finite difference, energy-conserving scheme, for a simply-supported rectangular plate excited pointwise and har- monically. Damping is not considered so that energy is left free to cascade until the highest simulated frequency is reached. The framework of non-stationary **wave** **turbulence** is thus appropriate to study quantitatively the numerical results. **In** particular, numerical simulations show the presence of a front propagating to high frequencies, leaving a steady spectrum **in** its wake, which has the property of being self-similar. When a finite amount of energy is given at initial state to the plate which is then left free to vibrate, the spectra are found to be **in** perfect accordance with the log-correction theoretically predicted. When forced vibrations are considered so that energy is continuously fed into the plate, a slightly steeper slope is observed **in** the low-frequency range of the spectrum. It is concluded that the pointwise forcing introduces an anisotropy that have an influence on the slope of the power spectrum, hence explaining one of the discrepancies reported **in** experimental studies.

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Pioneering analytical work **in** the analysis of rectangular thin plate vibrations with geometrical nonlineari- ties was carried out **in** the 1950s by Chu and Herrmann [ 17 ], demonstrating for the first time the hardening-type nonlinearity that has been confirmed by numerous experiments; see, e.g. [ 1 , 28 ]. Restricting the attention to the case of rectangular **plates**, the work by Yamaki [ 55 ] confirms analytically the hardening-type nonlinearity for forced **plates**. The case of 1:1 internal resonance for rectangular **plates** (where two eigenmodes have nearly equal eigenfrequencies) has been studied by Chang et al. [ 14 ] and by Anlas and Elbeyli [ 3 ]. Parametrically excited nearly square **plates**, also displaying 1:1 internal resonance, have also been considered by Yang and Sethna [ 56 ]. All these works focus on the moderately nonlinear dynamics of rectangular **plates** where only a few modes (typically one or two) interact together. **In** these cases, the von Kármán plate equations are projected onto the linear modes, and the coupling coefficients are computed with ad-hoc assumptions that appear difficult to generalise. Finite element methods have also been employed—see, e.g. the work by Ribeiro et al. [ 42 – 44 ], and Boumediene et al. [ 10 ] to investigate the nonlinear forced response **in** the vicinity of a eigenfrequency. Re- cently, numerical simulations of more complex dynamical solutions, involving a very large number of modes **in** the permanent regime, have been conducted, **in** order to simulate the **wave** **turbulence** regime and to reproduce the typical sounds of cymbals and gongs. For that, Bilbao developed an energy-conserving scheme for finite difference approximation of the von Kármán system [ 5 ], which allows the study of the transition to **turbulence** [ 49 ] and the simulation of realistic sounds of percussive **plates** and shells [ 6 , 7 ]. Spectral methods with a very large number of degrees of freedom have also been employed **in** [ 20 ] to compare theoretical and numerical **wave** **turbulence** spectra.

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166 En savoir plus

DOI: 10.1103/PhysRevLett.125.254502
Introduction. —Weak **turbulence** theory (WTT) addresses the statistical properties of weakly nonlinear ensembles of waves **in** large domains [1 –3] . The theory provides a rigorous analytical framework for deriving quantitative predictions for the kinetic energy spectrum, a task that remains extremely challenging for standard hydrodynamic **turbulence**. WTT appears of utmost interest for 3D fluid systems **in** which bulk waves can propagate, such as rotating or stratified fluids [4,5] where such quantitative predictions could pave the way for better **turbulence** parametrizations **in** coarse atmospheric and oceanic models [6] . WTT has already proven a valuable conceptual tool for understanding energy transfers **in** 2D **wave** systems, such as surface waves [7 –10] and bending waves **in** elastic **plates** [11 –13] . Comparatively, 3D fluid systems present additional complications that have hin- dered progress at the experimental and numerical level. For instance, rapidly rotating fluids support inertial waves associated with the restoring action of the Coriolis force

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When a set of stochastic waves, propagating on a free surface, have large enough amplitudes, interactions be- tween nonlinear waves can generate a **wave** **turbulence** regime. These interactions transfer the **wave** energy from the large scales, where it is injected, to the small scales where it is dissipated. This generic phenomenon concerns various domains at different scales: Surface and inter- nal waves **in** oceans, elastic waves on **plates**, spin waves **in** solids, magnetohydrodynamic waves **in** astrophysical plasma (for reviews, see [15–18]). Weak **turbulence** theory developed **in** the 60’s [19–21] leads to predictions on the **wave** **turbulence** regime **in** almost all domains of physics involving waves [16, 17]. The past decade has seen an im- portant experimental effort to test the validity domain of weak **turbulence** theory on different **wave** systems (e.g. hy- drodynamics, optics, hydro-elastic or elastic waves) [22].

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PACS numbers: 47.35.Tv,47.65.Cb,47.27.-i
**Wave** **turbulence** is an out-of-equilibrium state where waves interact with each other nonlinearly through N - **wave** resonance process. The archetype of **wave** turbu- lence is the random state of ocean surface waves, but it appears **in** various systems: capillary waves [1, 2], plasma waves **in** solar winds, atmospheric waves, optical waves, and elastic waves on thin **plates** [3]. Recent laboratory experiments of **wave** **turbulence** have shown new obser- vations such as intermittency [4], fluctuations of the en- ergy flux [5], and finite size effect of the system [2, 6]. Some of these phenomenon have recently been considered theoretically [7]. **Wave** **turbulence** theory allows to ana- lytically derive stationary solutions for the **wave** energy spectrum as a power law of frequency or **wave** number [3]. The spectrum exponent and the number N of resonant waves depend on both the **wave** dispersion relation and the dominant nonlinear interaction. Several theoretical questions are open, notably about the validity domain of the theory [8], and the possible existence of solutions for non dispersive systems [9]. **In** this context, finding an experimental system where the dispersion relation of the waves could be tuned by the operator should be of primary interest to test the **wave** **turbulence** theory.

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3 Experimental Physics, Saarland University, D-66123 Saarbrücken, Germany 4 Gravity Simulation Laboratory, ESTEC, ESA, 2201 AZ Noordwijk, Netherlands
5 ACTA, VU University, 1081 LA Amsterdam, Netherlands
(Received 24 April 2019; revised manuscript received 12 July 2019; published 10 December 2019) We report on the observation of gravity-capillary **wave** **turbulence** on the surface of a fluid **in** a high- gravity environment. By using a large-diameter centrifuge, the effective gravity acceleration is tuned up to 20 times Earth’s gravity. The transition frequency between the gravity and capillary regimes is thus increased up to one decade as predicted theoretically. A frequency power-law **wave** spectrum is observed **in** each regime and is found to be independent of the gravity level and of the **wave** steepness. While the timescale separation required by weak **turbulence** is well verified experimentally regardless of the gravity level, the nonlinear and dissipation timescales are found to be independent of the scale, as a result of the finite size effects of the system (large-scale container modes) that are not taken currently into account theoretically.

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of superfluid vortices are therefore lost. However, this model remains useful for describing the large scale dynamics of fi- nite temperature superfluid helium. An alternative model was introduced by Schwarz [ 9 ], where vortices are described by vortex filaments interacting through regularized Biot-Savart integrals. However, the reconnection process between lines needs to be modeled **in** an ad-hoc manner and by construction the model excludes the dynamics of a superfluid at scales smaller than the vortex core size. Finally, for weakly interact- ing BECs **in** the limit of low temperature, a model of different nature can be formally derived which is the Gross-Pitaevskii (GP) equation, obtained from a mean field theory [ 1 ]. This model naturally contains vortex reconnections [ 10 , 20 ], sound emission [ 21 , 22 ], and is known to also exhibit a Kolmogorov turbulent regime at scales much larger than the intervortex dis- tance [ 11 ]. Although this model is expected to provide some qualitative description of superfluid helium at low tempera- tures, it lacks of several physical ingredients. For instance, **in** GP, density excitations do not present any roton minimum as it does superfluid helium, where interactions between boson are known to be much stronger than **in** GP [ 23 ]. However, there have been some successful attempts to include such effects **in** the GP model. For instance, a roton minimum can be easily introduced **in** GP by using a nonlocal potential that models a long-range interaction between bosons [ 24 – 26 ]. The stronger interaction of helium can also be included phenomenologi- cally by introducing high-order terms **in** the GP Hamiltonian. Note that these terms can be derived as beyond mean field corrections [ 27 ]. Some generalized version of the GP model has been used to study the vortex solutions [ 28 , 29 ] and some dynamical aspects such as vortex reconnections [ 26 ]. Intu- itively, for a turbulent superfluid, we can expect that such generalization of the GP model might be important at scales smaller than the intervortex distance and with less influence at scales at which Kolmogorov **turbulence** is observed.

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For instance, an operator can easily change the dispersion relation of linear waves propagating on the surface of a magnetic fluid by applying a uniform magnetic field [ 5 ]. Magnetic **wave** **turbulence** can then be achieved on a ferrofluid surface submitted to a vertical field [ 6 ]. When the vertical field is high enough, the so-called Rosensweig instability occurs, leading to a hexagonal pattern of peaks on the surface of the ferrofluid [ 5 ]. Waves on the surface of a magnetic fluid submitted to a horizontal magnetic field are much less studied. **In** this geometry, the dispersion relation of linear waves is always monotonous whatever the field intensity, and the Rosensweig instability is absent [ 5 ]. To our knowledge, only one experimental test of the dispersion relation has been performed **in** this configuration [ 7 ] although striking phenomena occur: strong anisotropy of propagating waves, shifting onsets of the Kelvin-Helmholtz and Rayleigh-Taylor instabilities [ 5 , 7 ], and a new pattern of the magnetic Faraday instability [ 8 ].

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Beyond these order of magnitude estimates, fully quan- titative predictions would require an exact knowledge of the base flow, together with an optimization of the bifurcation threshold over all possible combinations of resonant triads and associated resonant quartets, a challenging task **in** general. Nevertheless, we can test the qualitative predictions of the theory further: we performed three complementary experiments with a honeycomb structure at the bottom of the tank (mesh ¼ 1.7 cm, height ¼ 2 cm). We first sent an inertial-**wave** beam towards this rough bottom boundary and observed that it reflects with negligible losses: the honey- comb structure has little effect on wavelike motion. By contrast, we expect it to induce a strong damping of the geostrophic modes, through a drastic increase **in** bottom drag. **In** some sense, this honeycomb structure is an experimental means of achieving the specific damping of the geostrophic modes included numerically by Le Reun et al. [32] . **In** line with these expectations, we observe **in** Fig. 5 that the onset of the triadic instability remains unaffected, while the threshold of the secondary quartetic instability is shifted to prohibitively large values of Re [42] . These observations are qualitatively captured by expression

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Finally, we expect that all the processes outlined above **in** the 3D astrophysical case—condensation via the dual cascade, fragmentation by modulational instability, soliton formation, and soliton interaction and merger via the exchange of weakly nonlinear waves—will be qualitatively the same **in** 2D. This makes them all amenable to direct observation by nonlocal nonlinear optics experiments. As mentioned **in** Sec. I B 2 the- oretical [ 44 , 45 ] and experimental [ 42 , 43 ] comparisons have been made between astrophysical phenomena and experi- ments **in** thermo-optic media. To observe the **wave** **turbulence** cycle of condensation, collapse, and soliton interaction that we describe here one could also look to using nematic liq- uid crystals and modifying the one-dimensional experiments of [ 55 , 60 ] to 2D. Any such experiment would need to have fine control over losses and nonlinearity strength **in** order to keep within the **wave** **turbulence** regime while the con- densate is being built up. Liquid crystals are an attractive optical medium **in** this respect due to several inherently tun- able parameters [ 88 ] that would assist **in** achieving conditions relevant to **wave** **turbulence** studies.

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PACS numbers: 47.35.Tv,47.65.Cb,47.27.-i
INTRODUCTION
**Wave** **turbulence** concerns the dynamical and statisti- cal properties of numerous nonlinearly interacting waves within a dispersive medium. Contrary to hydrodynami- cal **turbulence**, out-of-equilibrium solutions for the power spectrum of waves can be analytically computed by the **wave** **turbulence** theory **in** nearly all fields of physics (e.g., oceanic surface or internal waves, elastic waves on a plate, plasma waves, and spin waves) [1–3]. This theory as- sumes strong hypotheses such as weak nonlinearities and infinite size systems. Although the number of **wave** tur- bulence experiments has strongly increased **in** the last decade (see [4] for a review), they are still lagging be- hind the theory. New experimental systems are thus ex- pected to check the validity domain of the theory **in** ex- periments and to benefit from the **wave** **turbulence** theo- retical framework.

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Keywords — **Wave** propagation; P-FGM plate; Thermal effects; Higher order theory; Neutral surface position
I. I NTRODUCTION
Functionally graded materials (FGMs) are new materials which are designed to achieve a functional performance with gradually variable properties **in** one or more directions (Koizumi, 1992) [1]. This continuity prevents the material from having disadvantages of composites such as delamination due to large interlaminar stresses, initiation and propagation of cracks because of large plastic deformation at the interfaces and so on. Typically, FGMs are made of a mixture of ceramics and a combination of different metals (Bennoun et al ., 2016 [2]; Ebrahimi and Dashti, 2015 [3]; Sallai et al., 2015 [4]; Meradjah et al., 2015 [5]; Kar and Panda, 2015 [6]; Pradhan and Chakraverty, 2015 [7]; Bakora and Tounsi, 2015 [8]; Bouchafa et al., 2015 [9]; Arefi, 2015 [10]; Akbaş, 2015 [11]; Mansouri and Shariyat, 2015 [12]; Belabed et al., 2014 [13]; Khalfi et al., 2014 [14]; Mansouri and Shariyat, 2014 [15]; Hadji et al., 2014 [16]; Fekrar et al., 2014 [17]; Tounsi et al., 2013a [18]. So the key point is an accurate description of the variables and the material properties **in** the thickness direction, to perform a satisfactory analysis of the mechanical behavior

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with double polarisation are known to have close values, thus showing a 1:1 internal resonance. It is also known from other studies on nonlinear vibrations [10, 11] that thanks to a 1:1 **in**- ternal resonance, energy can be exchanged between vibration modes so that even if the motion is initiated along one polari- sation only, the nonlinearity can make this motion unstable, so that eventually a coupled vibration arises with the two polari- sations involved. The objective of the present contribution is thus to clearly establish if the nonlinearity can be the cause of this coupling, as well as to highlight the main parameters gov- erning the transfer of energy. The 1:1 internal resonance has already been studied **in** the case of forced vibrations, see e.g. [12, 10, 11]. Here our interest is **in** the case of free vibration for which only Manevitch and Manevitch present a detailed investi- gation [13]. The complete problem will hence be fully revisited and applied to the specific case of strings.

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better understanding of the various phenomena involved [2].
The light fluid loading perturbation method (involving high density structures **in** contact with a small density fluid such as aluminum plate **in** contact with air) was recently extended [8] to cases where the perturbation parameter becomes large (which corresponds to a very light and/or very thin structure **in** contact with a high density fluid). During this study multiple resonance processes were observed where a single **in** vacuo resonance mode can have several resonance frequencies under heavy fluid loading. The spectrum of the operator can no longer be described **in** terms of the discrete spectrum based on a countable series of resonance mode/resonance frequency pairs. This typically nonlinear phenomenon is closely linked to the concept of non-linear frequency modes [1], where the non linearity depends on the time parameters (such as those involved **in** porous materials) rather than on the more usual geometrical parameters. This frequency non- linearity is related to problems which are written **in** the form O(ω)U = S, where u is the unknown factor and S is the source term. O(ω) is an operator which depends non linearly on the angular frequency ω, as **in** the case of porous environments or **in** the context of vibroacoustics where the coupling depends non-linearly on the frequency via the Helmholtz equation Green’s kernel. When the coupling is weak (as **in** air), this process is not easily observed.

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