Phenomenological model for predicting stationary and non-stationary spectra of wave turbulence in vibrating plates

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Phenomenological model for predicting stationary and

non-stationary spectra of wave turbulence in vibrating

plates

Thomas Humbert, Christophe Josserand, Cyril Touzé, Olivier Cadot

To cite this version:

Thomas Humbert, Christophe Josserand, Cyril Touzé, Olivier Cadot. Phenomenological model for

predicting stationary and non-stationary spectra of wave turbulence in vibrating plates. Physica D:

Nonlinear Phenomena, Elsevier, 2016, 316, pp.34-42. �10.1016/j.physd.2015.11.006�. �hal-01257109�

non-stationary spe tra of wave turbulen e in vibrating plates T.Humbert a,1 ,C.Josserand b ,C.Touzé ,O.Cadot a

SPEC,CNRS,CEA,UniversitéParis-Sa lay,91191 Gif-sur-Yvette,Fran e b

Institutd'Alembert,CNRS,UMR7190,SorbonneUniversités,UPMC,75005Paris, Fran e.

IMSIA,ENSTAParisTe h,CNRS,CEA,EDF,UniversitéParis-Sa lay,828 bddes Maré haux,91762Palaiseau edex,Fran e

Abstra t

Aphenomenologi almodeldes ribingthetime-frequen ydependen eofthe power spe trum of thin plates vibrating in a wave turbulen e regime, is in-trodu ed. The model equation ontains as basi solutionsthe Rayleigh-Jeans equipartitionofenergy,aswell astheKolmogorov-Zakharovspe trumof wave turbulen e. In the WaveTurbulen e Theory framework,the model is used to investigate the self-similar,non-stationary solutions of for ed and free turbu-lentvibrations.Frequen y-dependentdampinglaws aneasilybea ountedfor. Theiree ts on the hara teristi sof thestationary spe traof turbulen eare then investigated. Thanks to this analysis, self-similar universal solutionsare given,relatingthepowerspe trumtoboththeinje tedpowerandthedamping law.

1. Introdu tion

TheWave(orWeak)Turbulen eTheory(WTT)aimsatdes ribingthe long-termbehaviourofweaklynonlinearsystemswherethenonlinearity ontrolsthe ex hangesbetweens ales[1,2,3℄.Under lassi alassumptionssu h as disper-sivity,weaknonlinearitiesandtheexisten eofatransparen ywindowinwhi h thedynami sisassumedtobe onservative,akineti equation anbededu ed forthe slow dynami s of the spe tral amplitude.In addition to the Rayleigh-Jeansspe trumthat orrespondstotheequipartitionofthe onservedquantity, heretheenergy,abroadbandKolmogorov-Zakharov(KZ)spe trumof onstant energyux ispredi ted,byanalogywithhydrodynami turbulen e[1,2℄.Su h dynami shas beenrstly studied for o ean (gravity)waves[4,5, 6℄ and sin e theninsystemssu has apillarywaves[7,8℄,nonlinearopti s[9℄orplasmas[10℄.

1. Correspondingauthor.

theoreti allyandobservednumeri allyin[11℄.Thetheoreti alanalysis onsiders thedynami sofageometri allynonlinearthinvibratingplateintheframework of the Föppl-von Kármán (FVK) equations. The WTT analysis leads to the predi tion of a dire t as ade hara terized by a KZ spe trum with onstant energyux.Soonafter,twoindependentexperimentsperformedonthin elasti plates[12,13,14℄didnotre overthetheoreti allypredi tedandnumeri ally ob-servedspe tra,questioningthevalidityoftheunderlying assumptionsofWTT inthe aseof vibratingplates.Re ently,anexperimental andnumeri alstudy onsideringtheee t ofdampingontheturbulentpropertiesofthin vibrating plateshas learlyestablishedthat[15℄:

 Inexperiments,dampinga tsatalls alessu hthat theassumptionof a transparen ywindow,adomaininthewavenumberspa ewhere dissipa-tionandinje tion anbenegle ted,isquestionable.

 Modifyingthedampingalterstheshapeof thevelo itypowerspe traso that a dire t omparison with the predi ted spe tra is out of rea h in experimental onditions.

 However,byin ludingtheexperimentallymeasureddampinglawsin the numeri alsimulationsof thefull dynami s (the FVKequations), agood agreement withthe experimentsis retrieved.This suggeststhat the dis- repan iesbetweentheexperimentsandtheWTTpredi tionsaremainly dueto damping.

These on lusionshavebeen orroboratedbyanumeri alstudywherethe damp-ingwasgraduallymodied,fromtheexperimentallymeasuredlawtoavanishing valueinagivenfrequen yband[16℄,showingalsohowthespe traaremodied byasmallyet non-negligiblevaluesofdampingfoundin realplates.

A ountingfordissipationwithintheWTTframeworkremains hallenging sin etheanalyti al ulationsarebasedonthelongtimeasymptoti evolution oftheweaklynonlinearHamiltoniandynami s.Theinje tionanddissipationin this ontext anbe seenasboundary onditions imposed to the transparen y window in the wave number spa e and to the best of our knowledge, we do notknowanyanalyti alattempttointrodu edissipationwithintheWTT. An-other option would be to nd an alternative des ription of the dynami s of the power spe trum, where adding dissipation appears more straightforward. The alternative anbe provided by using a phenomenologi al model des rib-ingthetemporal evolutionofthepowerspe tra,asrstproposed byLeithfor hydrodynami isotropi turbulen e[17℄.Thesemodelsprovideanatural frame-workforinvestigatingunsteadyandself-similardynami sinavarietyof ontext [17, 18, 19, 20, 21, 22℄. They are generally derived from ad-ho assumptions, by onstru ting a model equation admitting asstationary solutions both the Rayleigh-Jeansequipartition ofenergyandtheKZspe trum.Thisresultsin a nonlineardiusionequationinthewavenumber(

k

-spa e)orthefrequen y(

ω

-spa e)domain, whi h mimi stheenergytransferwithinthemodes.Thanksto thisapproa h,idealsituations anbeinvestigated,asforinstan etheinje tion

observedin these ases.

Thegoalofthis paperisthus toderiveandinvestigatesu h a phenomeno-logi almodelinthe aseofelasti vibratingplates. Themodelequationshould ontainbothRayleigh-JeansandKZsolutions.Inje tionand dissipationterms arethenintrodu edinordertostudymoreparti ularlytheee tsofthe damp-ing.Twomain resultsareobtained.First,self-similardynami s forfor edand isolatedturbulen eintheabsen eofdissipationareretrieved.Inase ondpart, theee t ofthedampingonthe as adingturbulentspe trumisinvestigated, exhibiting a self-similar solution relating the power spe trum to the inje ted powerandthedampinglaw.

2. Model equation

Theappli ationofthewaveturbulen etheorytotheFöppl-vonKármánthin plateequationshasbeenperformedin[11℄(seeAppendixAforthedimensional and non-dimensional forms of these equations. Note that for this se tion, all valuesare dimensionless). Without re alling the details of the derivation and the omplexform of the kineti equation, oneonly needs to remind that the twostationarysolutionsofthekineti equation,writtenhereunder theformof adensityofenergy

E

ω

,fun tion ofthefrequen y

ω

,are:

 TheRayleigh-Jeansequilibriumsolution,where theenergy

E

ω

is equally partedalongalltheavailablemodes.Consequently,thedensityofenergy

E

ω

isa onstantthat isdenotedas

C

:

E

ω

= C.

(1)

 The Kolmogorov-Zakharovsolution,for whi h anenergy ux

ε

is trans-ferredalongthe as adeuntilitsdissipationnear

ω

⋆

,the ut-ofrequen y ofthespe trum.Referringto[11℄,theenergyspe truminthis aseissu h that

E

ω

KZ

= Aε

1

3

log

1

3

 ω

⋆

ω



,

(2)

where A is a onstant. The spe i form of this solution, onsisting in a logarithmi orre tion of the Rayleigh-Jeans spe trum, omes from a degenera yoftheequilibriumsolutioninasimilarmannerasforthe non-linearS hrödingerequation [9℄.Infa t,thislogarithmi orre tionis ob-tainedusingaperturbativeexpansionandisvalidfarfrom

ω

⋆

.Therefore, although Eq. (2) exhibits asteep ut-obe auseof the non-existen eof the mathemati alsolution above

ω

⋆

(negative energy), experimentsand numeri al simulations do notshow su h a behaviour, and the spe trum de reasesmoresmoothlyas

ω

in reasesinthevi inityof

ω

⋆

[15,24,25℄. Thephenomenologi almodelisdire tlydedu edfromthesestationary solu-tionsoftheenergyspe trum.Letus onsiderthefollowingdiusion-likeequation

inthe

ω

-spa efortheenergyspe trum

E

ω

(ω, t)

:

∂

t

E

ω

= ∂

ω

(ωE

ω

2

∂

ω

E

ω

),

(3)

where

∂

t

and

∂

ω

refer respe tively for the partial derivatives with respe t to timeand angularfrequen y. Theenergyux asso iatedto thisequation reads straightforwardly

ε = −ωE

ω

2

∂

ω

E

ω

.

(4) Thankstotheidenti ationoftheenergyux

ε

,theproportionality onstant

A

ofEq. (2)isthenuniquelydenedas

A = 3

1

3

.Hen e,forthephenomenologi al modeltheKZsolutionnallyreads:

E

KZ

ω

= (3ε)

1

3

log

1

3

 ω

⋆

ω



.

(5)

Themodelequation,Eq.(3),is onstru tedsothatEq.(1)and(2)arestationary solutions (

∂

t

E

ω

= 0

). The Rayleigh-Jeans equilibrium is a trivial solution to Eq.(3)inthestationary asesin e

∂

ω

E

ω

= 0

.FortheKZspe trum,onehasjust toverify,byderivingEq.(2)withrespe tto

ω

,that

ωE

2

ω

∂

ω

E

ω

is onstantwith respe tto

ω

.Be ausethismodelequationhasbeendedu edinthedimensionless framework, only a numeri al prefa tor, whi h ould be easily absorbed by a res alingofthetime,should bepresentintheright-handsideof Eq.(3).

The phenomenologi al equation is nothing else than a nonlinear diusion equationinthefrequen yspa e,inthespiritoftheRi hardson as adeviewof turbulentpro esses[23℄. However,adire t derivation ofthis equationstarting from thekineti equation annot bedoneformally, and onlyqualitative argu-ments anbededu edfromalo alapproa honthekineti equation[1℄(Se tion 4.3).In fa t,attempts to dedu e su h simplied Fokker-Plan kequation from theweakturbulen eequationsgo ba ktothepioneering worksdoneforo ean waves by Hasselmann [31, 32, 33℄, although additional approximations were neededto dedu esu hlo al modelsinfrequen y.

Nonlineardiusionequations anexhibitimportantdieren esas ompared to diusion one.Inparti ular, singularity anbeformed bythe nonlinear dy-nami sand ompa tsupportsolutions analsobepresent,byoppositiontothe the lineardiusion where disturban es propagate at innite speed [34℄. Here, whileasingular ut-owillbeobservedforthespe tra,theequation doesnot orrespondaprioritothesituation were ompa tsupport solutionshavebeen provedtoexist[35℄.Finally,itshouldbesaidthatotherphenomenologi al mod-elsexhibiting thesamestationary solutions ould bededu ed and thepresent model anbe onsideredasoneofthesimplestamongotherones.

Numeri alsimulationsofthismodelequationwillnowbe ondu ted in var-ious asesinorderto investigatedierentdynami alsituations.Webeginwith the lassi al ase where an energy ux is imposed at low frequen y and for whi h the lassi al KZspe trum should be observedwhen dissipation a ts at highfrequen y.

3.1. For edturbulen e

3.1.1. Non-stationaryandstationary spe tra

In order to simulate numeri ally Eq. (3) , a nite volume method is used. The ux

ε

is omputed at ea h frequen y in rement and the value of

E

ω

is dened at the entre of the mesh element. A onstant value

ε

I

overtime for theux at

ω = 0

is applied and strongdissipation is in ludedupon

ω = 10

3

. Remarkably,thanksto thismodel equationalongwiththis numeri almethod, simulationsexa tly orresponding to theideal onguration ofturbulen e an belaun hed,withaux ofenergyimposedat

ω = 0

,anddissipationofenergy realizedwithasinkat highfrequen y.A typi alrun onsists in

2048

pointsin the

ω

dire tion, a time step equal to

10

−

7

time unit and a total duration of

2

time units.Whenthedissipatives aleisrea hed,the as ade frontstopsits evolutionandastationaryregimearises.

Fig. 1(a) displays the energy spe trum every

0.2

time unit in the onsid-eredframework.At the beginning (for

t < 1

), the as ade growstowardhigh frequen iessuggestingaself-similarbehaviour.Morepre isely,a hara teristi frequen ymaybedenedas

ω

c

=

R

∞

0

E

ω

ωdω

R

∞

0

E

ω

dω

,

(6)

inordertoobtainamorequantitativeanalysis.Fig.1(b)showstheevolutionof

ω

c

versustime,exhibitinga learlinearbehaviourinthetransparen ywindow. When the as ade front rea hesthe dissipative s ale xed here arbitrarily at

ω = 10

3

,the hara teristi frequen ydoesnotevolveanymoreandis onstant. Letusrst onsider thenon-stationaryregimewherethe hara teristi fre-quen y of the as ade evolves linearly with time for a onstant xed ux. Fig. 1( ) displays the non-stationary spe tra of Fig. 1(a) taken before

t < 1

asfun tions of thenon-dimensionalfrequen y

ω/ω

c

. Allthe urvesmergeinto auniquefun tion, onrmingtheself-similargrowthofthe as ade.Theshape of this fun tion will be dis ussed later but an already be ompared to the Kolmogorov-Zakharovspe trum Eq.(5), the solutionof thephenomenologi al equationforthe onservative ase,displayedbyagreendashedlineinFig.1( ). Althoughthetwofun tions arequite losetoea hother, theself-similar fun -tionof thenon-stationaryregime is steepernear the ut-o.This dis repan y hasalreadybeennotedin [24℄, wherethe aseoffor edturbulen e within the framework of the Föppl-von Kármán equations (dire t simulation) has been studied.

In thestationary regime,shown in Fig.1(d), thephenomenologi al model re overtheKolmogorov-Zakharovsolutionforthinplates,asawaited.The s al-ingoftheamplitudeofthespe trumby

ε

1/3

I

,astheoreti allypredi ted,isalso veriedbyourdata.Thetypi albehaviourofenergyspe traofvibratingplates inthestationaryregime istherefore orre tlydes ribedbythe phenomenologi- alEq.(3). Moreover,ourmodelre oversthefa t thatthe as adegrowswith

10

0

10

2

10

−1

10

0

ω

E

ω

(a)

0

0.5

1

1.5

2

0

100

200

300

400

500

600

700

t

ω

c

(b)

10

−2

10

−1

10

0

10

0

ω

/

ω

_c

E

ω

(c)

10

−2

10

−1

10

0

10

0

ω

/

ω

_c

E

ω

/

ε

I

1/3

(d)

Figure 1: For ed turbulen e. (a) Energy spe trum

E

ω

as a fun tion of the frequen y

ω

, for times in reasing from the left to the right,and with

ε

I

= 1

. (b)Chara teristi frequen y

ω

c

dened by Eq. (6) asa fun tion of time. Red dashed line :

ω

c

∝ t

. ( ) Energy spe trum

E

ω

(

ω

ω

c

)

omputed from the non-stationaryspe tra(before

t = 1

)shownin (a),and omparedtothestationary Kolmogorov-Zakharov spe trum

E

KZ

ω

= (3ε

I

)

1

3

log

1

3

(

ω

⋆

ω

)

(green dashed line). (d)Stationaryregime.Energyspe trum

E

ω

,dividedby

ε

1/3

I

,plottedasa fun -tionoftheres aledfrequen y

ω/ω

c

forseveralenergyuxes

ε

I

= 0.5, 1, 2, 5

and omparedtotheKZtheoreti alspe trum

E

KZ

whereastationaryregime inagreementwiththetheoreti alpredi tionsarises. 3.1.2. Self-similar analysis

In order to re over the numeri al behaviour of the non-stationary regime observedinFig.1(a)(b)( ),theself-similarsolutionsofEq.(3)areinvestigated. Thesolutionsarethus writtenundertheform

E

ω

= t

α

g(

ω

t

β

),

(7)

with

α

and

β

tworealunknownsand

g

afun tionto bedetermined.Inserting Eq.(7)intoEq.(3), onendsthat

α

and

β

mustfulll therelationship

2α = β − 1.

(8)

If we assume further that when inje ting with a onstant ux over time, the totalenergyoftheplateisgrowinglinearlywithtime,theequality

Z

+∞

0

E

ω

dω = Bt,

(9)

where

B

is a onstant, leads to a se ond relationship

α + β = 1

. This yields

α = 0

and

β = 1

sothatnally theself-similarsolutionsarene essarilyunder theform

E

ω

= g(

ω

t

).

(10)

The previous observation that the hara teristi frequen y of the self-similar solutions of Eq. (3) in ase of for ed turbulen e grows linearly with time is retrieved.

Inserting Eq. (10) into Eq. (3), the equation for the self-similar fun tion

g

η

= g(ω/t)

nallyreads

−ηg

′

η

= (ηg

2

η

g

′

η

)

′

,

(11) where

′

standsforthederivativewithrespe ttotheself-similarvariable

η = ω/t

. Thisequation is solvedusing Matlab algorithmode45 whi h applies a fourth-order Runge-Kutta s heme with a variable time step [26℄. For this purpose, Eq.(11)iswritten attherstorder:

Y

′

=

1

0

0 −

g

1

2

η

−

1

η

− 2

g

′

η

g

η

!

Y,

with

Y =

g

η

g

′

η



and

Y

′

=

g

′

η

g

′′

η



.

(12)

The initial valueproblem onsists in hoosing,for

η

0

given and small (in the simulations,

η

0

= 0.01

is sele ted),thevaluesof

g

η

and

g

′

η

that determine the desired initial ux

ε

I

. Whereas the value of

g

η

(η

0

)

is sele ted for omparison withagivendataset,

g

′

(η

0

)

isretrievedfromEq.(4).Astheux

ε

I

isxed,one obtains

g

′

η

(η

0

) = −

_η

₀

_g

ε

2

I

η

(η

0

)

.

Fig. 2(a) ompares the self-similarsolution dedu ed from the phenomeno-logi almodel (andalreadydisplayedinFig.1( ))withtheself-similarsolution

a ut-o abovewhi h the solution vanishes. As shown in [24℄, the self-similar solution an be obtained dire tly from the kineti equation. However in this ase,the generalshapeof the fun tion is notprovided bythe theory. Thanks to Eq. (11), the phenomenologi al model is able to predi t the shape of the self-similarfun tion.

Letusnow omparethissolutionwithdire tnumeri alsimulations.Fig.2(b) shows the obtained results, res aled a ording to the self-similar relationship proposed in Eq. (11). Two dierent numeri al s hemes have been used for a better omparison.On theonehand,a nite-dieren eand energy- onserving s hemesimulatesaperfe tre tangularplatewithsimply-supportedout-of-plane boundary onditionsandin-plane movableedges [24℄. Theplate has asurfa e of 0.4

×

0.6 m

2

, the thi knessis 1 mm, and the material parameters are that of a metal, see [24℄ for more details. The other solution is obtained thanks to the pseudo-spe tral method used in previous works [11, 15℄, where su h a spe tral approa h leadsto periodi boundary onditions. Thesimulated plate has also the material properties of a metal and orresponds to a square of

0.4 × 0.4 m

2

and its thi kness is

1

mm [11℄. In both numeri s, the plate is ontinuouslyex itedatlarges ale, orrespondingroughlytoa onstantinje tion ofenergywithtime. Forthenite-dieren esimulation, thisisrealizedwith a pointwisefor ing,thefrequen yofwhi hissele tedinthevi inityofthefourth eigenfrequen y.Forthepseudo-spe tral ode,thisisrealizedintheFourierspa e dire tlythrougharandom noisea tingat smallwavenumbersonly.With the twonumeri als hemes,a learself-similarbehaviourhasbeenobserved.Hen e weare in position to ompare themaster urvesofthe self-similarpro essfor thephenomenologi almodelwiththosefoundinthenumeri alsimulations.For the detailed presentation of the self-similar pro ess found in dire t numeri al simulations,theinterestedreaderisreferredto[24℄.

Fig. 2(b) shows that the two dierent numeri al methods exhibit similar res aled spe tra. Comparing to Fig. 2(a), one an observe two dis repan ies betweenthetwosolutions:

 Inthe dire tnumeri alsimulations, theslopein theturbulent as ading regimeisabitsteeper.This anbeassignedtothepresen eofthefor ing term in the very-low frequen y part of the spe trum, whi h reates a small prominen e that has already been observed and ommented, see e.g.[24,28℄.

 Near the ut-o, it appears that numeri al spe tra of the full dynami s de rease ontinuously and smoothly, whereas theoreti alspe tra display asteep ut-obe auseofthenon-existen eofthemathemati alsolution. ThisobservationissimilartowhathasbeenobtainedfortheKZstationary spe trum.

Despite these two dieren es, the general shape of the self-similar solutions in the aseof non-stationaryfor edturbulen e shows averygood agreement, validatingtheresultsprovidedbythephenomenologi almodel.

10

−1

10

0

10

−4

10

−2

10

0

ω

/

ω

c

g

η

(a)

10

−1

10

0

10

−4

10

−2

10

0

(b)

ω

/

ω

c

g

η

Figure2: Self-similarfun tion

g

η

in aseofnon-stationaryfor edturbulen e. (a)Bluepoints:numeri alsimulationofEq.(3)with

ε

I

= 1

.Reddashedline: solutionofEq.(11).(b)Dire tsimulationsoftheFöppl-vonKármánequations. Bla k line : nite-dieren e and energy- onserving s heme [24℄. Green line : pseudo-spe tralmethoddetailedin [11℄.

3.2. FreeTurbulen e

The aseoffreeturbulen e,i.e.theevolutionofthe as adewithoutexternal for ing, for a given amount of energy as initial ondition, is now onsidered. Asshownin [24℄ from thekineti equationand onrmedby dire t numeri al simulation,the as adefrontmustevolvetohighfrequen iesas

t

1/3

.Theability ofthephenomenologi almodeltoretrievethis dynami sisnowinvestigated. 3.2.1. Self-similar analysis

Consideringfreeturbulen e leadsto withdraw for inganddampingterms. Thesystembeing onservative,theamountofinitialenergy

K

is onserved,so thatEq.(9)isrepla edby:

Z

+∞

0

E

ω

dω = K.

(13)

The se ond relationship that links the unknowns

α

and

β

now turns to be

α = −β

, leading to

α = −1/3

and

β = 1/3

. Theself-similar solution for the energyspe trum

E

ω

readsin this ase

E

ω

= t

−

1/3

_h(

ω

t

1/3

).

(14)

Inorder tosimulatenumeri allythe framework offreeturbulen e, the dis-sipationintrodu ed earlierat high frequen y, is nowremoved. An energy ux

ε

I

is imposed for a few time steps and then an elled, thus xing the origin of time. Then, the simulation is run by imposing a vanishing energy ux at

ω = 0

, ensuring free turbulen e. Fig. 3(a) shows the evolution for an initial amount of energy K ( orresponding to the spe trum in red) asa fun tion of time. Fig.3(b)( ) des ribethe evolution of the hara teristi frequen y

ω

c

as

therst meshelement(

ω = 0.5

).Twobehavioursrespe tivelyproportionalsto

t

1/3

and to

t

−

1/3

aredisplayed.These twoobservationsare in agreementwith theself-similarsolutiongivenbyEq.(14).

In thesame manner asfor the for ed ase, the solution given by Eq. (14) anbeinsertedintoEq.(3)inordertoobtaintheevolutionequationofthe self-similarfun tion

h

ν

= h(ω/t

1/3

₎

.TheanalogueofEq.(11)forthefreeturbulen e asethenreads

−

1

₃

νh

′

ν

= (νh

2

ν

h

′

ν

)

′

,

(15) where

′

stands herefor thederivativewith respe t to the self-similarvariable

ν = ω/t

1/3

. Thenumeri almethod used in orderto solve Eq. (11)is now ap-plied to Eq. (15). Fig. 4(a) displays the self-similar fun tion built from the spe tra al ulatedbythephenomenologi almodelatmultipletimesands aled aspres ribedbyEq.(14).For omparison,thesolutionoftheself-similar equa-tionEq.(15)isalsorepresented.Agoodagreementisobserved, onrmingthe self-similarevolutionof thespe trum.Fig. 4(b) displaysthenumeri al results fromthedire tnumeri alsimulationsoftheFöppl-vonKármánequations.On e again,thetwonumeri als hemesleadstofun tionsthatarevery losefromea h other.A mu hbetteragreementisobservedbetweenthesolutionsfrom dire t simulations and the one from the phenomenologi al model, in parti ular the slopein the as ade regime are really thesame.This onrms on e again the ee tofthefor ingwhi h reatesasmallbumpin theverylow-frequen ypart of the spe trum and alters the dire t omparison between the dierent solu-tions.Hereinthefreeturbulen e ase,aperfe tagreementisobserved,theonly dieren ebeingthebehaviournearthe ut-ofrequen ywherethede reaseof the spe trum is mu h slower for the dire t numeri al simulations, as already ommented.

Twosituations belonging tothetheoreti al onservativeframeworkofwave turbulen e in thin vibratingplates have been investigated through numeri al simulationsofthephenomenologi alequation.Self-similarbehaviourspertaining to thephenomenologyof theFöppl-vonKármán equationshavebeen su ess-fullyre overed.Notealsothatthebehaviours

E

ω

= g(

ω

t

)

forfor edturbulen e and

E

ω

= t

−

1/3

_h(

ω

t

1/3

)

forfreeturbulen e anbederivedbyananalysisof the kineti equation,asshownin[24℄.However,inthis asetheself-similarfun tions

g

and

h

areleftunknown.Thankstothephenomenologi almodel,twodierent ordinarydierentialequations havebeen dedu ed, the solutions of whi h are fun tions

g

and

h

.Hen e,themodelgivesfurtherinformationswhi hhavebeen foundtoberelevantby omparisonswiththedire tnumeri alsimulations.All theseresultsshowtheabilityofoursimpleequationtore over omplexfeatures ofthephysi softheproblem.

10

0

10

1

10

2

10

−3

10

−2

10

−1

10

0

ω

E

ω

(a)

10

−2

10

−1

10

0

10

1

10

2

_(b)

t

ω

c

10

−2

10

−1

10

0

10

−1

10

0

(c)

t

E

ω

(

ω

=0.5)

Figure3: (a)Energy spe trum

E

ω

asafun tionof thefrequen y

ω

at(from topto bottom)

t = 0, 1, 2, 3, 4

[nondim℄. (b) Chara teristi frequen y

ω

c

as a fun tionof time.Reddashed line:

ω

c

∝ t

1/3

.( )

E

ω

(ω = 0.5)

asafun tion of time.Reddashedline:

E

ω

(ω = 0.5) ∝ t

−

1/3

.

10

−1

10

0

10

−4

10

−2

10

0

ω

/

ω

c

(a)

10

−1

10

0

10

−4

10

−2

10

0

ω

/

ω

c

h

ν

(b)

Figure4: Self-similarfun tion in aseoffreeturbulen e. (a)Points:spe tra of Fig.3 res aledby theself-similar law given byEq. (14). Red dashed line : solutionofEq.(15).(b)Resultsofthedire tsimulationsoftheFöppl-von Kár-mánequations.Bla kline:nite-dieren eandenergy- onservings heme[24℄. Greenline:pseudo-spe tralmethod detailedin[11℄.

4.1. Model equation

Physi al dissipation an be introdu ed in the phenomenologi al model by addingalineardissipationtermto Eq.(3):

∂

t

E

ω

= ∂

ω

(ωE

ω

2

∂

ω

E

ω

) − ˆγE

ω

,

(16) where

ˆ

γ

an be hosen as a fun tion of

ω

for the sake of generality. In thin plates, the damping depends strongly on parameters su h as the size of the plate,its thi kness, the boundary onditions. Regarding these valuesand the frequen yrangeofinterest, either thermoelasti ,vis oelasti , a ousti al radia-tion,or lossesthroughthe boundary onditions, andominate[36,37, 38, 39℄. Intheframeworkof ourexperimental set-up, theimportan eof mostof these ontributionshasbeenestimatedandrelatedtotheoreti alpredi tionsin [40℄. Asastartingpoint,letus onsiderthedampinglawsobtainedfrom experi-ments.Asobservedin[15℄whereexperimentalmethodshavebeenusedinorder toin reasetheamountofdampingintheplate,thedampinglawsforfour dif-ferent ongurationswerefoundtofollowthepower-law

γ = ξω

ˆ

0.6

,withrelative valuesof

ξ

(withrespe ttothesmallestone)rangingfrom1to5.Thisdamping lawwith varying

ξ

isrstused forinvestigatingthe solutionsofEq. (16). Ap-pendixBgivesthefull orresponden ebetweenexperimentallymeasuredvalues of

ξ

andtheirrespe tivedimensionless ounterpartsusedin thenumeri al sim-ulationsofEq. (16). Thesamenitevolumemethodis usedasin theprevious se tions, and the ux of energy

ε

I

is xed at

ω = 0

. After a ertain number of time steps(depending on the sele ted damping oe ient

ξ

), a stationary regimeisrea hed.

Fig.5exhibitsthestationaryenergyspe trumobtainedforea hofthefour damping ases retrieved from [15℄, with an amount of damping oe ient

ξ

multipliedby5betweenthesmallestandlargestones.Theenergyuxat

ω = 0

is thesameforea hsituation.Forverylowfrequen ies(say

0 < ω < 5

),allspe tra showsroughlythesamebehaviour.Forlargerfrequen ies,themoredampedthe system,thesteeperthespe trumandthesmallerits hara teristi frequen yare. Moreover,it appearsthat thedissipation ae ts the energytransfersbetween s ales,sothatsummingupthestationaryspe tratopowerlawsisnotpossible anymore.

Fig. 6 shows the previous spe tra as fun tions of the res aled frequen y

ω/ω

c

. Theres alingofthefrequen yaxismakesallspe tra ollapseintoa sin-gle urve,whi happearstobesteeperthantheKolmogorov-Zakharovspe trum (displayedbyagreendashedlineinFig.6).Thisresult,obtainedwiththe phe-nomenologi almodel,issimilartothe on lusionsalreadyreportedin[15℄from experiments only : damping plays an important role in the dis repan ies be-tweentheoreti alandexperimentalspe tra.However,thisuniquemaster urve hasneverbeenobservedbeforeandtendstoprovideasimpleexplanationonthe behaviourofthe as adeinpresen eofdamping.Indeed,itshowsthattheee t

10

0

10

2

10

−4

10

−3

10

−2

10

−1

ω

E

ω

Figure5: Stationaryenergyspe trum

E

ω

in thedamped ase,asafun tion of the frequen y

ω

and for

ε

I

= 1 × 10

−

5

. Red :

ξ = 1.908 × 10

−

5

. Bla k :

ξ = 3.0528 × 10

−

5

.Magenta:

ξ = 5.9359 × 10

−

5

.Blue:

ξ = 9.3279 × 10

−

5

.

10

−1

10

0

10

−4

10

−3

10

−2

ω

/

ω

c

E

ω

Figure 6:Energy spe tradisplayed in Fig.5asfun tions ofthe res aled fre-quen y

ω/ω

c

. Green dashed line : Kolmogorov-Zakharov spe trum

E

KZ

ω

=

(3ε

I

)

1/3

log(

ω

⋆

ω

)

1/3

for

ε

I

= 1 × 10

−

5

betweenthe onservative term

∂

ω

(ωE

2

ω

∂

ω

E

ω

)

and the dissipativeterm

γ

ˆ

ω

E

ω

, sin eonly these termsare present in thephenomenologi al model, and allows one to retrieve the experimental observations. There is obviously no inertial rangeso that the stationary solutiondepends on the shape of thedissipation fun tionanddiersfromtheKolmogorov-Zakharovspe trum.

Finally, the ollapse suggestsaself-similarbehaviourof the spe trum asa fun tionoftheinje tedux

ε

I

andthedamping oe ient

ξ

.Inordertoderive theequation orrespondingtothisself-similarsolution,theenergyspe trum

E

ω

isthus writtenundertheform

E

ω

= ε

µ

I

ξ

x

f

η

 ω

ω

c



with

ω

c

= ε

y

I

ξ

z

,

(17)

where

f

η

is an unknown fun tion of the self-similar variable

η = ω/ω

c

and

µ, x, y, z

are onstants to be determined. Re alling that the inje ted ux

ε

I

orrespondsinthephenomenologi almodelto

ε

I

= lim

ω→0

(−ωE

2

ω

∂

ω

E

ω

),

(18)

oneobtains,afterinsertingEq. (17)into Eq.(18),thefollowingrelationship:

ε

I

= −ε

3µ

I

ξ

3x

_η→0

lim

(ηf

η

2

∂

η

f

η

),

(19)

sothat

µ = 1/3

and

x = 0

.Theenergyspe trummustthuswrite :

E

ω

= ε

1/3

I

f

η



_ω

ε

y

I

ξ

z



.

(20)

In addition, inserting Eq. (17) in the phenomenologi al equation (16) with a dampingoftheformofanunknownpowerlaw

γ = ξω

ˆ

λ

_{= ξη}

λ

_ω

λ

c

yields:

∂

t

E

ω

= 0 = ε

1−y

I

ξ

−

z

_∂

η

(ηf

η

2

∂

η

f

η

) − ε

λy+1/3

I

ξ

λz+1

_ηf

η

.

(21)

Thus,theunknowns

y

and

z

mustfulllthefollowingrelationshipsthatdepends onthefrequen ydependen e ofthedamping:

z = −

_{1 + λ}

1

,

y =

2

3(1 + λ)

.

(22)

AlltheunknownsofEq.(17)havebeendetermined,leadingtoanequationfor thefun tion

f

η

,

∂

η

(ηf

η

2

∂

η

f

η

) − f

η

η

λ

= 0,

(23) andtoanexpressionforthe hara teristi frequen yasafun tionofthedamping andtheinje tedux :

ω

c

= ε

2

3(1+λ)

I

ξ

−

1

1+λ

.

(24) Eq.(23)hasnoanalyti alsolutionbut anbesolvednumeri allyfollowingthe

10

−4

10

−3

10

1

10

2

α

ω

c

Figure7:Chara teristi frequen y

ω

c

asafun tionofthedamping oe ient

ξ

,

ε

I

= 1 × 10

−

5

. Bla k:

λ = 2

. Red :

λ = 1

. Blue :

λ = 0.6

. Dashed lines : evolutionlawspredi tedbyEq.(24).

displayingaperfe tagreementwiththeuniversalsolutionobtainedbyres aling allthespe tra.

To on ludethispart,thevalidityofEq.(24),whi hexpressesthebehaviour of the hara teristi frequen y, is questioned. As already observed in Fig. 5 for

λ = 0.6

, in reasing the damping oe ient

ξ

de reases the hara teristi frequen y

ω

c

.Inthis ase,thetheoreti alpredi tionprovidedbyEq.(24)reads

ω

c

= ε

5/12

I

ξ

−

5/8

_.

(25) Fig.7 omparesthispredi tionwiththe hara teristi frequen iesobtainedby solvingEq.(16)for

ˆ

γ = ξω

λ

and

λ = 0.6

.Thesamestudyfor

λ = 1

and

λ = 2

is alsodisplayed. A perfe t agreementis found, showing that the evolutionof the hara teristi frequen y an be fully explained thanks to the self-similar behaviouroftheenergyspe trumwithdampingandinje tedux.

4.2. Dis ussion

The results of the previous se tion, obtained with the phenomenologi al model,haveshowntheexisten eofauniquemaster urveonwhi h allspe tra ollapse when res aling the frequen y with respe t to the hara teristi fre-quen y. This feature has not been noti ed before in the experimental results reported in [15℄, where four dierent ongurations of damping for the same platehavebeenmeasured.Itispotentiallyaveryimportantresultsin eit sug-geststhat the hange in the as ade slope observedwhen the dampingvaries (following the strongerthe dissipation, the steeperthe energy spe tra are) is simplya onsequen eofthemaster urvewhi hdoesnotexhibitasingleslope.

10

−1

10

0

10

−6

10

−5

10

−4

10

−3

10

−2

f/f

c

P

v

[m

2

.s

−1

]

(a)

10

−1

10

0

10

−3

10

−2

10

−1

10

0

f/f

c

P

v

[m

2

.s

−1

]

(b)

Figure 8: Powerspe traldensity of thetransverse velo ity

P

v

asa fun tion ofthe res aledfrequen y

f /f

c

. Red :

ξ = 0.045

. Bla k:

ξ = 0.072

. Magenta :

ξ = 0.14

. Blue :

ξ = 0.22

. (a) Experiments. Red :

ε

I

= 0.56 × 10

−

3

m

3

_.

s

−

3

. Bla k :

ε

I

= 0.54 × 10

−

3

m

3

_.

s

−

3

. Magenta :

ε

I

= 0.52 × 10

−

3

m

3

_.

s

−

3

. Blue :

ε

I

= 0.48 × 10

−

3

m

3

_.

s

−

3

. (b) Numeri al simulations. Green :

ξ = 0

,

ε

I

=

0.057 × 10

−

3

m

3

_.

s

−

3

.Other ases:

ε

I

= 0.024 × 10

−

3

m

3

_.

s

−

3

.

Depending on the dissipation a dierent region of the master urve is domi-nating, exhibiting dierent "apparent" slope. It is thus ru ial to investigate whetherthisfeatureisalsopresentinexperimentsandinnumeri alsimulations oftheplate equations.

Fig. 8 displays pre isely the experimental and numeri al (pseudo-spe tral method)powerspe traldensities

P

v

from [15℄ asfun tionsof theres aled fre-quen y

f /f

c

for dierent damping oe ients

ξ

. Asfor the phenomenologi al model, theproposed res aling ausesall urvesto ollapseinto aunique mas-ter urve.InFig.8(b),thespe trafrom thedamped aseare omparedto the KZspe trumobtainednumeri allywhenthedissipationisonlylo atedathigh frequen y,showingthatthespe traare learlysteeperthantheusualKZ spe -trum.Moreover,boththeexperimentalandthenumeri al asesexhibitsimilar proles,but are verydierentfrom themaster urveofthe phenomenologi al model,in thesamevein thantheothersituations studiedabove.Nevertheless, thesegures onrmherethattheobservationsbroughtbythe phenomenolog-i almodel des ribeatruefeature ofthephysi al system.

Finally,therelationEq.(24)betweenthe hara teristi frequen y,the damp-ingandtheinje tedpower analsobequestionedusingtheexperimentalresults. Fig.9displays,forthreeinje tedpowers,theevolutionoftheratio

ω

c

/ε

5/12

I

as

afun tion of the dampingparameter

ξ

. The predi teddependen e of

ω

c

with

ξ

is also drawn for omparison :

ω

c

/ε

5/12

I

∝ ξ

−

5/8

. The a ordan e is good, onrmingthattheresultsofthemodelareinagreementwiththebehaviourof theexperiments.

10

−1

10

3

10

4

α

ω

c

Figure 9:Ratio

ω

c

/ε

5/12

I

asafun tion of thedamping oe ient

ξ

.

λ = 0.6

. Red:

ε

I

= 0.52 × 10

−

3

m

3

_.

s

−

3

. Green :

ε

I

= 0.16 × 10

−

3

m

3

_.

s

−

3

. Blue:

ε

I

=

0.58×10

−

4

m

3

_.

s

−

3

.Line:evolutionlawpredi tedbyEq.(24):

ω

c

/ε

5/12

I

∝ ξ

−

5/8

.

5. Con lusion

Aphenomenologi almodeldes ribingthetime-frequen ydependen eofthe powerspe trumforwaveturbulen ein thinvibratingplates,hasbeenderived. Intheframework ofnon-stationaryturbulen e,the modelequation hasshown itsability inpredi tingtheself-similarbehavioursfortwodierent ases:free andfor edturbulen e.Thesetwoexamplesshowtheabilityofourmodelto ap-turethemostsalientfeaturesofthedynami softhinelasti plates.Themodel equationpossessesanumberofattra tingfeaturesforfurtherstudies,the promi-nentonebeingitssimpli ityinhandling ompli atingee tssu hasfor ingand dissipation.Besides itsability in re overingtheself-similarbehavioursalready derived from thekineti equation [24℄, astep further hasbeen obtainedwith thederivation of two equations, (11) and (15), the solutions of whi h are the self-similaruniversalfun tionsforthefor edandthefree ases,whi hwerenot providedbythetheorydevelopedfromthekineti equationin [24℄.

Thephenomenologi almodel hasthen beenused in order tofurther inves-tigate the ee t of damping on the spe tra of turbulen e for thin vibrating plates remindingthat,in that ase,dampinga ts at alls ales and breaks the transparen ywindowrequired by thewaveturbulen e theory. Then, no more power-law behaviour an be observed, and the slope of energy spe tra does notrepresentthemostimportantparametertoinvestigate [15℄.Thanksto the phenomenologi almodel,aself-similaranalysisprovidesnewresultsandmakes appeararelationshipbetweenthepowerspe tra,thedampinglawand the in-je tedpower.Withthemodelequationandforagivendampinglaw,all urves ollapseintoasingleonewhenin reasingthedampingfa tor,andthe hara ter-isti frequen y anbedire tlystudiedandpredi tedfromtheenergybudgetof

lentspe trawithdamping.Thisalso onrmsthatthephenomenologi almodel isausefultoolforstudying ompli atingee ts inwaveturbulen eofplates.

Appendix A. Non-dimensionalFöppl-vonKármán equations

Thedynami softhinvibratingplatesisdes ribedbytheFöppl-vonKármán equationswithtwounknownsthatarethetransversedispla ementeld

ζ(x, y, t)

andtheAirystressfun tion

χ(x, y, t)

.Forathinplateofthi kness

h

,madefrom amaterialwithPoissonratio

ν

,density

ρ

andYoung'smodulus

E

,theequations ofmotionread[27,29,30℄

ρh

∂

2

_ζ

∂t

2

= −

Eh

3

12(1 − ν

2

₎

∆

2

_{ζ + L(χ, ζ),}

(A.1)

∆

2

χ

= −

Eh

₂

L(ζ, ζ).

(A.2)

Theoperator

L

isbilinearsymmetri ,andreadsinCartesian oordinates

L(f, g) =

f

xx

g

yy

+ f

yy

g

xx

− 2f

xy

g

xy

.

Thefollowing hangeofvariablesisappliedtoobtaindimensionlessvariables

x

′

=

x

l

,

ζ

′

=

ζ

l

,

t

′

=

t

τ

,

χ

′

=

χ

C

,

(A.3) wherethe hara teristi length

l =

h

√

3(1−ν

2

₎

,time

τ = l

p

ρ

E

and

C = Ehl

2

,have beenintrodu ed. Thisleadsto thefollowingset of non-dimensionaldynami al equations:

∂

2

_ζ

∂t

2

= −

1

4

∆

2

ζ + L(χ, ζ),

(A.4)

∆

2

χ

= −

1

₂

L(ζ, ζ).

(A.5)

Appendix B. Corresponden ebetweenexperimentaland phenomeno-logi al values ofthe damping oe ient

ξ

In theFöppl-vonKármán equations, vis ous dissipation anbetaken into a ountwiththeterm

ρhγ

∂ζ

∂t

,sothat theequationsofmotionwrites:

ρh

∂

2

_ζ

∂t

2

=

−

Eh

3

12(1 − ν

2

₎

∆

2

ζ + L(χ, ζ) − ρhγ

∂ζ

_∂t

,

(B.1)

∆

2

χ

=

−

Eh

₂

L(ζ, ζ),

(B.2)

∂

2

_ζ

∂t

2

= −

1

4

∆

2

ζ + L(χ, ζ) − ˆγ

∂ζ

_∂t

,

(B.3)

∆

2

χ

= −

1

₂

L(ζ, ζ).

(B.4)

ˆ

γ

isthenon-dimensionaldampingfa tor:

ˆ

γ = γτ = γh

r

_ρ

3E(1 − ν

2

₎

.

(B.5) In[7℄,thedampinglawhasbeenmeasuredandbehavesas

γ = ξf

0.6

,where

ξ

isaparametertakingdierentvalues,obtainedby hangingthe onguration of the plate in a given manner. In order to use the same range of damping valuesin thephenomenologi almodelasintheexperiment,onehastoexpress the relationship between the dimensional values of

ξ

and their dimensionless ounterparts

ξ

ˆ

.Thanksto Eq.(B.5),wehave

ˆ

γ = τ (ξf

0.6

) = ˆ

ξ ˆ

ω

0.6

with ˆ

ξ =

τ

0.4

(2π)

0.6

ξ.

(B.6)

Table1sumsupthenumeri al

ξ

valuesobtainedfromtheexperiments(rstline, from[15℄)andtheirequivalentnon-dimensionalvalues

ξ

ˆ

usedpreviouslyforthe simulationsofthephenomenologi almodel.Notethatinthepresentpaperand forthe sakeof simpli ity,the oe ientsused in the phenomenologi almodel werenamed

ξ

.

ξ

0.045 0.072 0.14 0.22

ˆ

ξ × 10

5

1.908 3.0528 5.9359 9.3279

Table B.1: Corresponden e between the experimentally measured values of damping oe ients

ξ

and theirdimensionless ounterparts

ξ

ˆ

used in the sim-ulations.

[1℄ V.Zakharov,V.L'vovandG.Falkovi h,KolmogorovSpe traofTurbulen e I:WaveTurbulen e,Springer,1992.

[2℄ S.Nazarenko,WaveTurbulen e,Springer,2011.

[3℄ A. Newell and B. Rumpf, Wave Turbulen e, Annual Review of Fluid Me- hani s,43(2011)59-78.

[4℄ V.ZakharovandN. Filonenko,Theenergyspe trumfor sto hasti surfa e levelos illations,Dokl.ANSSSR,170(1966)1292.

[5℄ A. Dya henko, A. Korotkevi h and V. Zakharov, Weak turbulent Kol-mogorov spe trum for surfa e gravity waves, Phys. Rev. Lett., 92 (2004) 134501.

turbulen ein wavetanks :spa e andtimestatisti s,Phys.Rev.Lett.,103 (2009)044501.

[7℄ E. Fal on, C. Laro heand S. Fauve,Observation ofgravity- apillary wave turbulen e,Phys.Rev.Lett.,98(2007)094503.

[8℄ G.DüringandC.Fal ón,Symmetryindu edfour-wave apillarywave tur-bulen e,Phys.Rev.Lett.,103(2009)174503.

[9℄ S. Dya henko, A. Newell, A. Pushkarev and V. Zakharov, Opti al turbu-len e:weakturbulen e, ondensatesand ollapsinglamentsinthe nonlin-earS hrödingerequation,Physi aD, 57(1992)96-160.

[10℄ S.Galtier,S.Nazarenko,A.NewellandA.Pouquet,Aweakturbulen e the-oryforin ompressiblemagnetohydrodynami s,J.PlasmaPhys.,63 (2000) 447.

[11℄ G.Düring,C.JosserandandS.Ri a,Weakturbulen eforavibratingplate: anonehearaKolmogorovspe trum?,Phys.Rev.Lett.,97(2006)025503. [12℄ A. Boudaoud, O. Cadot, B. Odille and C. Touzé, Observation of wave

turbulen einvibratingplates,Phys.Rev.Lett.,100(2008)234504. [13℄ N.Mordant,Aretherewavesinelasti waveturbulen e?,Phys.Rev.Lett.,

100(2008)234505.

[14℄ B.MiquelandN.Mordant,Nonlineardynami sofexuralwaveturbulen e, Phys.Rev.E,84(2011).

[15℄ T. Humbert, O. Cadot, G. Düring, C. Josserand, S. Ri a and C. Touzé, Waveturbulen einvibratingplates:theee tofdamping,Europhys. Let-ters,102(3)(2013)30002.

[16℄ B. Miquel, A. Alexakis and N. Mordant, Role of dissipation in exural wave turbulen e : from experimental spe trum to Kolmogorov-Zakharov spe trum,Phys.Rev.E,89(2014)062925.

[17℄ C. Leith, Diusion approximation to inertial energy transfer in isotropi turbulen e,Phys.ofFluids,10(1967)1409-1416.

[18℄ S.HasselmannandK.Hasselmann,Computationsandparametrizationsof the nonlinear energy transfer in a gravity-wave spe trum. Part I : a new method for e ient omputations of the exa tnonlinear transfer integral, JournalofPhysi alO eanography,15(1985)1369-1377.

[19℄ G.Falkovi handA.Shafarenko,Non-stationarywaveturbulen e,Journal ofNonlinearS ien e, 1(1991)457.

inadiusionmodelofturbulen e, Phys.Rev.Lett.,92(2004)044501. [22℄ C. Josserand, Y. Pomeau andS. Ri a, Self-similarsingularities in the

ki-neti sof ondensation,J.LowTemp.Phys.145,231(2006) [23℄ U. Fris h, Turbulen e.CambridgeUniversityPress,1995.

[24℄ M. Du es hi, O. Cadot, C. Touzé and S. Bilbao, Dynami s of the wave turbulen espe trumin vibratingplates :anumeri alinvestigationusing a onservativenitedieren es heme,Physi aD,280(2014)73-85.

[25℄ B. Miquel, A. Alexakis, C. Josserand and N. Mordant, Transition from wave turbulen e to dynami al rumpling in vibrated elasti plates, Phys. Rev.Lett.,111(2013)054302.

[26℄ L.ShampineandM.Rei helt,TheMatlabODEsuite,SIAMJ.ofS ienti Computing, 18(1)(1997)1-22.

[27℄ L.LandauandE.Lifshitz,TheoryofElasti ity,PergamonPress,NewYork 1959.

[28℄ B. Miquel and N. Mordant, Nonstationary waveturbulen e in an elasti plate,Phys.Rev.Lett.,107(3)(2011)034501.

[29℄ T. von Kármán, Festigkeitsprobleme im Mas hinenbau, En yklopadie Math.Wiss.,4(1910)311-385.

[30℄ O. Thomas and S. Bilbao, Geometri ally nonlinear exural vibrationsof plates : in-plane boundary onditions and some symmetry properties, J. SoundVib., 315(2008)569-590.

[31℄ K.Hasselmann, Onthenon-linearenergytransferin agravity-wave spe -trum,J.FluidMe h, 12(15)(1962)481-500.

[32℄ K.Hasselmann, Onthenon-linearenergytransferin agravity-wave spe -trum Part 2.Conservationtheorems;wave-parti leanalogy;irreversibility, J.FluidMe h,15(02)(1963)273-281.

[33℄ K.Hasselmann,D.B.Ross,P.MullerandW.Sell,Parametri wave predi -tionmodel,JournalofPhysi al O eanography,6(2)(1976)200-228. [34℄ J.I. Diaz Diaz, Solutions with ompa t support for some degenerate

paraboli problems, Nonlinear Analysis, Theory, Methods & Appli ations 3(1979)831-847.

[35℄ Y.B. Zeldovi h and Y.P. Raizer, Physi s of Sho k Waves and High-TemperatureHydrodynami Phenomena,Dover.

[36℄ A.N. Norris,Dynami softhermoelasti thinplates:a omparisonoffour theories,JournalofThermalStresses,29(2)(2006)169-195.

pa ted plates. I. Theory and experiments, The Journal of the A ousti al So ietyofAmeri a,109(4)(2001)1422-1432.

[38℄ K.Ar as,Simulation numériqued'unréverbérateuràplaque.Thesede l'é- olepolyte hnique,2009.

[39℄ A.Cara ioloandC.Valette,Dampingme hanismsgoverningplate vibra-tion,A taA usti a,3(5)(1995)393-404.

[40℄ T. Humbert, Turbulen e d'ondes dans les plaques min es en vibration : étude expérimentale et numérique de l'eet de l'amortissement. These de l'UniversitéPierre etMarieCurie,2014.