Obrvation of star-shaped surface gravity waves
我要购物Jean Rajchenbach 1,Didier Clamond2and Alphon Leroux1
1Laboratoire de Physique de la Mati`e re Condens´e e(CNRS UMR7336)
Universit´e de Nice–Sophia Antipolis,
Parc Valro,06108Nice Cedex2,France
2Laboratoire Jean-Alexandre Dieudonn´e(CNRS UMR7351)
Universit´e de Nice–Sophia Antipolis,
Parc Valro,06108Nice Cedex2,France
Abstract
We report a new type of standing gravity waves of large amplitude,having alternatively the shape of a star and of a polygon.This wave is obrved by means of a laboratory experiment by vibrating vertically a tank.The symmetry of the he number of branches)is independent of the container form and si
ze,and can be changed according to the amplitude and frequency of the vibration.We show that this wave geometry results from nonlinear resonant couplings between three waves,although this possibility has been denied for pure gravity waves up to now.
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Nonlinear and dispersive effects in water waves give ri to remarkable phenomena,such as solitary and freak waves.The wave phenomena,originally obrved at a liquid surface, turned out to have analogues in number of other domains involving nonlinear waves.For example,solitary waves have also been recognized in opticalfibers[1],and’freak’waves, which are giant waves of very short life time[2–4]have been identified infibre optics[5] and in plasmas[6].Another remarkable effect of nonlinearities is to give ri to patterning [7].For example,‘hor-shoe’waves[8]have been shown to result from the nonlinear interactions betweenfive waves[9,10].Nevertheless,although the existence of a large variety of different waves is expected as a result of nonlinearities,experimental evidences of new types of waves are noticeably scarce.In this paper,we report the obrvation of a new type of standing waves,displaying alternatively a star-like and a polygonal shape.The waves are obrved at the free surface of a liquid submitted to vertical sinusoidal vibrations.
Experimental tup and obrvations.The system studied is afluid layer of about 1cm deep;the liquid chon for the investigations is a silicon oil,which,like water,displays a Newtonian rheological behavior.The kinematic viscosity is10−5m2/ times that of water),and the surface tension is0.02m/s.Experiments are conducted with containers of various shapes(rectangular,circular)and of various sizes(from7to20cm in size or in diameter).Thefluid vesl is mounted on a shaker and experiences a vertical sinusoidal motion,with a frequencyΩ/2πranging typically from7to11Hz.The amplitude of the cell oscillations can be driven up to20mm and the surface deformations are recorded by means of a fast camera(250fps).
For the sake of clarity,we describefirst the results obtained in a cylindrical container(with diameter9cm)vibrated with a frequencyΩ/2πequal to8Hz,and with afilling level of7mm. For small oscillation amplitudes,we obrve at the free surface of the liquid layer“meniscus ripples”originating from the contact line between the free surface and the inner wall of the container and propagating toward the center of the cell.The ripples oscillate with the same frequency as the driving,and the damping lengths are small compared to the radius of the container.Increasing the vibration amplitude up to1.55mm,we obrve(e Fig. 1and movie1in[11])two contra-propagative,axisymmetric gravity waves,with a period T which is twice that of the T=4π/Ω)as it is expected for parametricall
y-forced waves[12,13].When the circular crest of the centripetal wave focus to the center
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of the container,an upward jet is formed which breaks into a droplet.It is interesting to point that,when the crests of the two centrifugal and centripetal axisymmetric waves are crossing,they do not simply superimpo,but they also experience a pha shift(e Fig.
2).More precily,the crests remain in spatial coincidence during a typical time of0.05s for the above experimental parameters.The pha delay phenomenon during crossing has been recognized in the ca of two crossing plane solitary waves,and testifies to a strong nonlinear coupling between the waves[14].Still increasing the vibration amplitude to1.85mm,we notice the appearance offive corners in the crest line when the centrifugal and centripetal waves are crossing(e Fig.3and movie3in[11]).The tips sign the breaking of the rotational symmetry.At last,for a typical vibration amplitude of1.95mm,we obrve a drastic change in the wave geometry.The surface pattern displays alternatively a star and a pentagonal shape,parated by a time interval of of2π/Ω(e Figs. 4.a,4.b and movie4in[11]).A remarkable feature is that the alternate star-polygon-shaped waves are independent of the container size and shape.Identical patterns are obrved in larger circ
ular or rectangular containers(Figs. 5.a,5.b).Note that we have also obrved stars and polygons with other symmetries(3,4and6),merely by varying the frequency and the amplitude of vibration(e Fig.6and movies6a and6b in[11]).Note also that the system exhibits hysteresis,meaning that for the same forcing parameters different patterns can be obrved according to the forcing history.It is therefore not possible to establish a pha diagram related to the symmetry as a function of the forcing parameters.
It must be emphasized that the waves are extreme:(i)the wave amplitude can be of the order of two times the liquid mean depth;(ii)in the trough,the depth is reduced to afilm of less than1mm thick.Thus,the are highly nonlinear waves appearing in the context of shallow he wavelength/depth ratio is large).In other words,we deal with large standing cnoidal waves.
Theoretical explanation.Our interpretation is inspired of tho of Mermin et Troian [15]and Pomeau and Newell[16]for quasi-crystals,and that of Edwards and Fauve[17] for the formation of quasi-patterns in capillary waves.It is noteworthy that in the prent experiments we have|k| 1/ c( c is the capillary length),so that here surface tension effects are negligible compared to gravity effects,and therefore we are dealing with pure gravity waves.Our explanatory scheme involves a nonlinear resonant interaction between
sound是什么意思
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three surface waves.The three wave resonance conditions read asω1±ω2±ω3=0and k1±k2±k3=0(ωi and k i are the angular frequencies and the wave vectors)[18,19]. The conditions can be simultaneously satisfied in capillary-gravity waves[20–23],but the three-wave resonance mechanism was considered up to now as irrelevant for the pure gravity waves that we are facing[18].The reason is that the relation of dispersion of undamped,unforced gravity waves reads asω∝|k|αwithα 1(α=1/2in deep water,α=1in shallow water),so that the above resonance conditions cannot hold.However,we show that this three-wave resonance mechanism is actually relevant to trigger the reported phenomenon,becau the relation of dispersion is significantly modified by the dissipation and forcing.We will explicit below the dispersion relation taking into account dissipation and forcing,and then we will briefly explain how the amended relation of dispersion allows a three gravity wave resonant interaction and how the latter can lect a m-fold symmetry.
It is well-known[12,13,24–26]that the amplitudeζ(k,t)of parametrically-driven in-finitesimal surface waves infinite depth,or Faraday waves[27],can be modeled by a damped Mathieu equation
∂2ζ∂t2+2σ
美剧
∂ζ
∂t
+ω2
[1−F cos(Ωt)]ζ=0,(1)
whereσis the associated viscous attenuation,Ωis the forcing angular frequency,F cor-responds to a dimensionless forcing(amplitude of the vertical acceleration divided by the gravity acceleration g),andω0=ω0(|k|)is the angular frequency of linear waves without damping and forcing(for linear water waves infinite depth h we haveω2
=gk tanh(kh) with k=|k|[19]).The viscous attenuation termσaccounts both for the bulk dissipation (proportional toνk2[28])and friction with bottom(proportional to(νk2)12[29]).It must be emphasized that Eq.(1)is linear,and is derived for infinitesimal waves infinite depth ( in shallow water).Here we deal with large amplitude,cnoidal waves,so that the validity of Eq.(1)is very limited.Nonetheless,Eq.(1)is worth providing insights on the mechanism triggering the formation of the patterns that we report here.
Systems obeying a damped Mathieu equation like Eq.(1)exhibit a ries of resonances angular frequencies nΩ/2(the integer n is the order of the resonance)[30,31].According to Floquet theory,bounded periodic solutions of Eq.(1)exist under some special relations between the parameters[32],the relations providing a dispersion relation(that cannot
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理臣教育be expresd in term of elementary functions).Numerical investigations,using various expressions for σ,show that there are at most two wave numbers solutions of the dispersion relation for each n .This can be easily en from the analytical expressions that we can derive in the limit of small F and small σ,that read as
ω0≈Ω2 1± F 216−4σ2Ω
2 ,(2)for the subharmonic respon,and
ω0≈Ω
1+F 212± F 464−σ2
Ω2 ,(3)
for the fundamental one.Note that the damping introduces a threshold in the forcing amplitude giving ri to the formation of surface waves.In the limit of small F and σ,the threshold is F 1≈8σ/Ωfor the subharmonic respon (n =1)and F 2≈ 8σ/Ωfor the fundamental respon (n =2).
Unlike the ca of undamped,unforced waves,relations (2)and (3)show that two modes with different wave numbers can oscillate at the same frequency.Therefore,according to the forcing amplitude,different cas must be distinguished:
rapidfire(i)For F <F 1,there are no solutions of the dispersion relation (2).Physically,it means that there are no formations of parametric waves becau the input of energy is not sufficient to overcome the viscous dissipation.
(ii)For F 1<F <F 2the excited modes are only tho corresponding to subharmonic ,they oscillate with angular frequency Ω/2.If an infinite number of subhar-
monic waves with the same wave number (say k −1)are prent,we obrve an axisymmetric
wave becau,in a circular basin,the vertical wall boundary condition do not privilege any particular direction.
(iii)For F 2<F <F 3,both subharmonic modes (oscillating at Ω/2)and fundamental
modes (oscillating at Ω)are excited.There are two wavenumbers k −1and k +1(k −1
k +1)corresponding to the subharmonic mode,and two wave numbers k −2and k +2(k −2
k +2)for the harmonic one.All the modes interact nonlinearly.The simplest mechanism to be considered to explain the formation of waves with a m -fold rotational symmetry is the three-wave resonant coupling mechanism.Two subharmonic waves,of different wave vectors
k −1and k +1and of identical angular frequencies ω1=Ω/2,interact between them and also
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interact with one fundamental mode,of wave vector k−
2
and of angular frequencyω2=
Ω.Thus,the conditionω1(k−
1)+ω1(k+
1
)=ω2(k−
2
)is automatically met.The additional
condition to be fulfilled by wave vectors is k−巴巴里人
1+k+
1
=k−
2
.This three-wave resonance
condition gives naturally ri to the lection of a peculiar angle(k−
1,k+
1
),which breaks
the rotational invariance.Physically,the lf-tuning of the angle between the wave vectors allows a continuous energy supply from two wave numbers to the third one.We have
mentioned a three-wave resonant mechanism with wavenumbers k−
1,k+
1
and k−
2
,but another
possible three-wave resonance involves k+
dazy
apply是什么意思2instead of k−
2
2011年12月英语四级
.This multiplicity of possible3-
wave resonances may be one cau of the obrved hysteresis.Another cau is that,in viscousfluid,the parametric instability is subcritical,due to nonlinear effects,inducing thus a memory effect[33,34].
The m-branched stars and m-sided polygonal patterns correspond to the lection of an angleθ=2π/m,with m integer.Clearly,the above resonance criterion leads in general to m non-integer.In the latter ca,the surface pattern appears unstationnary,until a surface mode(not perfectly resonant)corresponding to m integer is locked.Once this mode(with m integer)is locked,it is en to survive to moderate changes in the forcing parameters. This is a another possible origin for the obrved hysteresis.
Although the above model is worth showing the three wave resonance as a mechanism capable to g
enerate m-fold symmetric gravity waves,it is insufficient to predict with accuracy the obrved symmetries as a function of the forcing parameters.The reason is that Eqs.2and3are derived within the hypothes of infinitesimal amplitude waves,while we are facing large amplitudes cnoidal waves.Actually,the wave amplitudes intervene certainly in the dispersion relations.Moreover,considering sinusoidal waves as eigenmodes is a too crude approximation,unable to capture numerous physical properties[35].The design of a highly nonlinear theory suited too large and steep cnoidal standing waves in shallow water remains a theoretical challenge for future studies.ious
ACKNOWLEDGMENTS.
We acknowledge Gerard Iooss for fruitful discussions.
This project has been partially supported by CNRS and R´e gion PACA.
( )Corresponding author Jean.Rajchenbach@unice.fr
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