V Vidal

J.-C Géminard

T Divoux

F Melo

Acoustic signal associated with the bursting of a soap film which initially closes an overpressurized cavity Experiment and theory

Keywords: . 68.15.+e Liquid thin films -43.20.Mv Waveguides, wave propagation in tubes and ducts -43.20.Ks Standing waves, resonance, normal modes -91.40.Yt Remote sensing of volcanoes

We report an experimental study of the sound produced by the bursting of a thin liquid film, which initially closes an overpressurized cylindrical cavity. There is a need for a deep understanding of the phenomenon, which can be very useful in numerous practical cases. For instance, in the nature, the volcanologists observe the bursting of large, elongated, gas-bubbles at the surface of lava lakes and record the associated sound emission. One can wonder which pieces of information they can get from such acoustic measurements. For a didactic purpose, we provide also the reader with all the theoretical background necessary for the understanding of the physical processes that govern the various characteristics of the acoustic signals: the cavity geometry governs the frequency; the viscous dissipation and the radiation are responsible for the damping; the acoustic energy informs about the characteristic time associated with the film-rupture more than about the energy initially loaded in the cavity.

Introduction

The sound produced by the quick opening of an under-or overpressurized cavity is a common phenomenon. It can be encountered in many situations, from everyday life to large-scale natural systems.

One of the most well-known examples is certainly the characteristic 'pop' sound when opening a champagne bottle: the pressure inside the bottle, initially larger than the atmospheric pressure, drops down when the cork is released. Excited by the sudden change in pressure, the free volume close to the bottleneck resonates. In addition, the initial overpressure is due to a large amount of gas dissolved in the liquid, which, once the pressure has decreased, escapes the system as the famous champagne a Permanent address: Laboratoire de Physique, École Normale Supérieure de Lyon, 46 Allée d'Italie, 69364 Lyon Cedex 07, France.

b e-mail: jean-christophe.geminard@ens-lyon.fr bubbles: by listening carefully to a glass full of champagne (or whatever 'bubbling' liquid), one can notice the soft popping sound generated by each bubble-bursting at the liquid surface. In this latter case, the resonators are the bubbles themselves.

To date, the studies of the sound emission have limited to the bursting of bubbles at the free surface of a newtonian fluid [1] in which, because of the rounded shape of the bubble reaching the free surface, the resonator consists of a liquid lens, the depth comparing with the bubble diameter [START_REF] Princen | Forces and structure in surfactant-laden thinliquid films[END_REF]. By contrast, the body of a bubble rising in a nonnewtonian fluid is likely to be strongly elongated along the vertical because of specific fluid rheological properties [START_REF] Kulkarni | [END_REF]. One can wonder about the characteristics of the sound emission in this latter case, which is relevant in various practical situations. For instance, the bursting of elongated bubbles can be observed in the kitchen, when cooking sauces [START_REF] Ichihara | Japan Earth and Planetary Science Joint Meeting[END_REF]. Volcanoes are large-scale natural systems which exhibit very similar phenomena; a large amount of dissolved gas is released while lava rises up the magmatic conduit [START_REF] Dingwell | Magma degassing and fragmentation: recent experimental advances, in From Magma to Tephra -Modelling physical processes of explosive volcanic eruptions[END_REF]; bursting of large gas bubbles is sometimes observed at the surface of lava lakes or at vents. The characteristics of the acoustic signal beeing likely to provide useful pieces of information on the system geometry or dynamics, geophysicists have long been recording sound produced by volcanoes [6,7]. Because of the gas release, lava at the surface forms a foam having non-newtonian rheological properties. We can guess that one observes, at the surface of lava lakes, the bursting of bubbles that are elongated along the vertical.

One can then wonder what pieces of information can be extracted from acoustic measurements. We thus performed preliminary experiments in polymeric solutions and observed that, in some cases, the bursting of the thin film that initially closes the bubble at the free surface excites the elongated bubble body, which does not deform significantly during the sound emission and resonates like a tube [START_REF] Divoux | Listening to singing bubbles at the free surface of a non-newtonian fluid, Stage de recherche de deuxième année du MIP[END_REF]. We also observed a puzzling dependence of the acoustic energy on the bubble volume as well as, in the same experimental conditions, a large scatter of the acoustic energy. In order to account for the observations, it is fundamental to deeply understand the sound emitted as the consequence of the bursting of a well-controlled thin-film which initially closes an overpressurized cavity of well-defined geometry. Thus, we consider a resonating cavity consisting of a simple cylindrical-tube, closed at the bottom, which is excited by the bursting of a soap film at the top (Fig. 1). Previous studies of soap film bursting show that the typical film rupture is very fast [9], usually faster than all the physical processes inside and outside the cavity (acoustic resonance, radiation and sound wave propagation). Therefore, immediately after bursting, the characteristics of the sound waves, recorded inside and outside the resonating tube, depend generally on the tube geometry only. With this very simple experiment, we want to analyze the different physical mechanisms responsible for the temporal and spectral characteristics of the signal. We pay special attention in describing carefully the experimental procedures and propose in appendices all the theoretical tools and references necessary for the understanding of the phenomenon. We intend thus to provide the reader with useful and didactic material which, we hope, depending on his own interest, will help him in the interpretation of more complex systems, from the crackling of the champagne bubbles to the bursting of giant elongated bubbles on volcanoes.

Experimental setup and procedure

The experimental setup consists of a vertical tube (diameter φ = 6, 8 or 10 mm) drilled in a Plexiglass block (Fig. 1). The head of a first microphone (microphone ATM33a, Audio-Technica + preamplifier Eurorack UB802) is located d = 5 cm away from the upper-output plane and oriented at α ∼ 45 • from the system axis. At bottom, the system is either closed by means of a solid rod or by a second microphone (microphone 377A10 + preamplifier 426B03 + amplifier 482A16, PCB Piezotronics Inc.). Both Fig. 1. Sketch of the experimental setup. The experimental setup consists of a tube (inner diameter φ) drilled in a plexiglass block. The length L of the cavity is tuned by changing the position of the microphone 2 (or rod) at bottom. The pressure difference δP across the film is controlled by means of a syringe pump and a pressure sensor. After the film bursting, the sound emitted outside the cavity is recorded with the help of the microphone 1 whereas the pressure at the bottom, which results from the standing waves inside the cavity, is monitored by using the microphone 2.

the rod and the microphone at bottom are movable, in order to tune the cavity length L, from the output plane to the lower wall (from 2 to 23 cm).

At the top, an initially-flat soap-film is produced by stretching a droplet of a water-soap mixture (Quix Limón) with the help of a razor blade. Then, a capillary connected to the main vertical tube makes it possible to introduce a chosen volume of air from a syringe pump into the cavity limited by the lower wall and the film. A differential pressure transducer (PX277-01D5V, OMEGA Engineering Inc.) is used to measure the resulting pressuredifference, δP , across the soap film which then takes the shape of a part of sphere. In addition, the system can be imaged either from top or side with the help of a fastcamera (HiSIS 2002, KSV Instruments Ltd).

Subjected to gravity, the film thins (drainage, [START_REF] Mysels | Soap films: studies of their thinning and a bibliography[END_REF]) and, after a few tens of second, spontaneously bursts [START_REF] Frankel | [END_REF], which produces a characteristic sound. A digital oscilloscope (54602B, Hewlett Packard) is used to monitor and digitize the signals from the microphones (Fig. 2) and to transfer the raw data to a computer through the HPIB interface. Subsequent analysis is performed numerically (Matlab, The MathWorks, Inc. and Igor Pro, Wavemetrics, Inc.).

Thus, the experimental setup makes possible the analysis of the sound emission associated with the bursting of a curved soap film, which initially ends an almost cylindrical cavity of length L and diameter Φ filled with air at the overpressure δP above the pressure of the outside atmosphere. We shall report our experimental results and interpretations in the following Section 3 and draw conclusions in Section 4. The theoretical background necessary for the interpretation of our observations is presented in Appendices A to D.

Experimental results and interpretation

In the present section, we report our experimental observations, starting with the spectral analysis of the sound emitted after the bursting of the film (Sect. 3.1). We then discuss the temporal evolution of the amplitude (Sect. 3.2), focusing on the physical processes causing the temporal damping of the signal. In Section 3.3, we show that the analysis of the initial amplitude of the harmonics provides us with interesting pieces of information about the film rupture. Finally, the last Section 3.4 is dedicated to the energy balance.

Resonant frequencies ν n

The fundamental frequency ν 0

From the experimental signals P (t) provided by the microphone 1, we measure easily the fundamental frequency ν 0 of the acoustic wave in air, which corresponds to the lowest-frequency peak in the power spectrum (for instance, ν 0 1 kHz in Fig. 2). In order to analyze the effect of the cavity geometry, we shall report the results in terms of the associated wavelength λ 0 , defined to be λ 0 ≡ c/ν 0 . Note that the velocity of sound in air, c, depends on the temperature T according to [START_REF] Backus | The acoustical foundations of music[END_REF]:

c(T ) = γRT M ( 1
)
where γ = 1.4, R = 8.14 J K -1 mol -1 , and M = 29 g mol -1 . In our experimental conditions, the room temperature is about T 298 K and c 346 m s -1 .

Let us now tune the length L by changing the position of the microphone 2 (or rod) and determine the corresponding wavelength λ 0 of the fundamental. We observe that λ 0 depends linearly on L according to:

λ 0 = ξ(L + δL) ( 2
)
where ξ = 4.00 ± 0.05 (Fig. 3) whatever the tube diameter Φ in our experimental range (6 to 10 mm). On the other hand, the length correction, δL, is observed to depend on the tube diameter Φ: for instance, we measure δL = (4.2 ± 0.1) mm in the case Φ = 10 mm.

The tube, open at one end, constitutes a resonator which is excited by the pressure drop associated with the sudden breaking of the soap film (Sect. A.5). What one can hear in air is the acoustic wave radiated outside the cavity (Appendix B) that selects the frequency (Appendix A). In a first approximation (Sect. A.1), we expect the pressure in the output plane to continuously equal the outside pressure (thus, a pressure node). At the bottom, we expect a velocity node and, accordingly, a pressure antinode. In this case, the cavity length equals one fourth of the fundamental wavelength and λ 0 = 4L. The experimental slope ξ is in fairly good agreement with this theoretical expectation (Fig. 3). However, the correction length δL is finite, as the pressure node does not exactly locate in the output plane. Indeed, because of the radiation of the acoustic wave in the output plane, one must consider the effective tube length L = L + δL where one expects δL = 4Φ/3π in the case of a flanged aperture [START_REF] Kinsler | Fundamentals of acoustics[END_REF][START_REF] Kemp | Theoretical and experimental study of wave propagation in brass musical instruments[END_REF] (Eq. (A.11)). One obtains a similar correction δL = 0.3 Φ, which is only slightly smaller than the latter, in the case of an unflanged aperture [START_REF] Kinsler | Fundamentals of acoustics[END_REF][START_REF] Levine | [END_REF]. Experimentally, the correction length is found to be in fairly good agreement with the correction expected for a flanged aperture (we measure δL 0.42 cm for Φ = 10 mm).

Higher-order harmonics ν

n (n = 0)
We point out that, in the theory (Sect. A.2.2), the correction δL is expected not to depend on the wavelength λ, so that the larger resonant frequencies ν n are harmonics of the fundamental ν 0 , ν n = (2n + 1)ν 0 associated with λ n = 4L /(2n + 1), where L = L + δL. The prediction is well satisfied experimentally. Indeed, we observe clearly that the higher harmonics have frequencies that are odd multiples of the fundamental frequency: 3ν 0 , 5ν 0 , 7ν 0 , . . . (Fig. 4). Fig. 4. Energy spectrum (microphone 1). We report, in log scale, the data from microphone 1 (Fig. 2). The vertical dashed lines clearly point out that the harmonics frequencies satisfy νn = (2n + 1)ν0 (δP = 13.1 Pa, Φ = 10 mm, L = 8 cm, microphone 2 at the bottom, sampling rate 20 kHz).

Conclusion on the resonant frequencies

The characteristic frequencies of the sound wave emitted after the film bursting depend on the geometry of the resonant cavity only. Indeed, the signal recorded by the microphone 1 mainly contains the several harmonics of the fundamental ν 0 , which is associated with the effective length L = L + δL: L is the length of the cavity, from the bottom to the aperture plane, and δL a length correction, due to the radiation at the open end, which mainly depends on the tube diameter φ. In the next Section 3.2, we discuss the temporal envelope of the signal and, more specifically, how the amplitude of each of the harmonics vanishes.

Amplitude-damping characteristic-times τ n

Qualitative observation

As can be seen on the signals P (t), especially from microphone 1 (Fig. 2), the amplitude of harmonics associated with larger frequencies decreases faster. Indeed, the signal clearly contains several harmonics at short time (typically t < 8 ms) and only the fundamental afterwards (typically t > 8 ms). Measurements performed with a long enough cavity make it possible to determine with accuracy the characteristic time τ n , over which the amplitude of the harmonic n vanishes, as a function of the corresponding frequency ν n .

Analysis procedure

The analysis procedure is as follows: from the position of the peaks in the power spectrum (Fig. 4), we determine the fundamental resonant frequency ν 0 . We then write the signal from microphone 1 for t > 0:

P (t) = ∞ n=0 A n sin (ω n t + φ n ) exp (-t/τ n ) (3)
where ω n = 2π(2n + 1)ν 0 . The interpolation of the experimental data with equation (3) provides us with the characteristic damping-time, τ n , of each of the harmonics that exhibits a significant amplitude (thus generally up to n = 2). The agreement between equation (3) and the experimental data is excellent as shown, as an example, in Figure 5.

Experimental results

Each of the τ n gives the experimental value of the characteristic damping time, τ d , at the corresponding frequency ν = (2n + 1)ν 0 . Collecting the whole set of experimental measurements obtained for the various hamonics and different tube lengths L, we obtain the experimental τ d as a function of the frequency ν. We checked that the experimental data reported in Figure 6 do not depend on the initial overpressure δP .

Discussion

The radiation at the open end alone would lead to τ d ∝ ν -3 (Sect. A.2.3), which is not observed experimentally, especially for small frequencies. We have evaluated other possible sources of dissipation [among them, the partial reflection at the closed end (Sect. A.3) and the thermal and viscous dissipation in the boundary layer at the vertical wall of the cavity (Sect. A.4)]. We estimated that, in the case of a rigid wall at bottom (rod), the damping of the acoustic wave due to the partial reflection at the closed end is negligible (this contribution must be taken into account when microphone 2 is placed at the bottom). Thus, taking into account only the radiation at the open end and the viscous and thermal dissipation in the boundary layer at the vertical wall of the cavity, we estimate the characteristic damping-time to be:

1 τ d = 2 πη[1 + (γ -1)P -1 2 r ] Φ ν 1/2 + 2π 2 Φ 2 c 2 ν 3 2n + 1 (4)
where η = 1.5 × 10 -5 m 2 s -1 denotes the kinematic viscosity of air, P r 0.7, the Prandtl number, and γ = 1.4, the specific-heat ratio. Equation (4) gives, without any adjustable parameter, the characteristic dampingtime which we find to be in rather good agreement with our experimental measurements (Fig. 6).

The damping of the standing wave inside the cavity is mainly governed by both the viscous and thermal dissipation at the wall, which dominates at small frequencies, and the radiation at the open end, which dominates at large frequencies. Whatever the dominant dissipation process, the harmonics having larger frequencies are damped faster. This is the reason why the sound, recorded outside, contains several harmonics right after the bursting of the film whereas only the fundamental is sensed later. At this point, we know that the frequencies recorded by the microphone correspond to the resonant frequencies of the cavity and how the amplitude of the various harmonics vanishes when time increases. In the following Section 3.3, we discuss how the resonator is excited by the pressure drop associated with the film bursting and we show that a detailed analysis of the acoustic signal makes it possible to determine accurately the characteristic time of the resulting pressure drop in the aperture plane. Fig. 7. Signals P (t) from the microphones (top: microphone 1, bottom: microphone 2). Left: in the case of a short rupture-time, one observes that the overpressure at the closed end of the cavity goes from δP to -δP immediately after bursting of the film. Right: if the rupture time is larger, the energy repartition among the acoustic modes is different, and the overpressure does not drop to -δP immediately after bursting. Accordingly, the amplitude of the acoustic signal outside is smaller (δP ∼ 13 Pa, Φ = 10 mm, L = 8 cm, microphone 2 at the bottom, sampling rate 20 kHz).

Spectral analysis of the pressure drop associated with the film bursting

In this section, we obtain, from the detailed analysis of the coefficients A n and φ n which appear in equation ( 3), values of the Laplace transform of the pressure drop in the output plane and, consequently, the associated characteristic time, τ .

Qualitative observations

For identical initial conditions (given L, φ and δP ), values of the initial amplitude of the acoustic signal can be very scattered. Even if the resonant frequencies are not affected, the acoustic signal varies significantly both in amplitude and shape, inside and outside the cavity (Fig. 7). This feature points out the fundamental role of the film breaking process in the excitation of the acoustic waves.

Using the fast camera, we observe the dynamics of the soap-film breaking, when not too fast, and we monitor the acoustic signal. We report in Figure 7, the acoustic signals obtained for two very different characteristic times of the film rupture. If the film rupture time is short, short enough for the soap film to disappear in the time between two images (1/4000 s), the amplitude of the signal inside the cavity initially drops down from +δP to -δP (Fig. 7, left). We note that this case corresponds to the rupture of a very thin film, which has reached the minimum accessible thickness [common black film (CBF, [START_REF] Mysels | Soap films: studies of their thinning and a bibliography[END_REF]) in our experimental conditions] before bursting. If the film rupture time is longer (in the same experimental conditions but, here, the drainage is not complete when the film breaks), the amplitude of the acoustic signal inside the cavity, right after bursting, does not drop down to -δP anymore. Accordingly, the amplitude of the acoustic signal outside is smaller (Fig. 7, right).

In order to point out the variability of the amplitude of the acoustic signal from one experiment to another if one performs the experiment without controlling the thickness of the soap film when it breaks, we report measurements of the initial amplitude of the acoustic signal inside the cavity for two sets of experiments performed in the same experimental conditions (Fig. 8). One observes that the amplitude of the first oscillation is often much smaller than δP , especially for the shorter cavity.

In the next Section 3.3.2, we report experimental results obtained for the bursting of CBFs. In this latter case, the measurements are reproducible and we report a thorough analysis of the signal and draw conclusions.

Spectral analysis

The theoretical analysis (Sect. A.5.4) demonstrates that the amplitude and phase of the acoustic signal inside the cavity are related to the Laplace transform of the pressure drop P (t) in the output plane which results from the film bursting. In what follows, we define f (s) ≡ L[P (t)/δP -1] where L denotes the Laplace transform. As the acoustic wave outside the cavity results from the radiation at the open end of the acoustic wave inside (Sect. B), we can estimate that the amplitude A n and the phase φ n of the signal from the microphone 1, located outside, satisfy:

A n ∝ ω n f (jω n ) ( 5
)
φ n = arg[ f (jω n )] -ω n t 0 + d c ( 6
)
where d denotes the distance between the microphone and the output plane [we introduce the time delay t 0 as, experimentally, the origin of time is taken when the signal exceeds a given threshold value and not when the film breaks. We thus expect t 0 to be of the order of magnitude of the signal period 2π/ω 0 ]. From the interpolation of the acoustic signal outside, we measure A n and φ n and, then, report the amplitude (Fig. 9) and the phase (Fig. 10) of f (jw) as a function of the frequency ω. At small frequency, both the amplitude and the phase are constant, whereas the amplitude decreases significantly (like ω -k with k 5) above a typical value 1/τ of the frequency ω. We observe a decrease in the phase θ ≡ arg[ f (jω n )] around the value 1/τ but the scatter of the experimental data at large frequency avoids an accurate determination of the limit reached when ω → ∞. Nevertheless, the experimental results clearly exhibit a characteristic time τ associated with the pressure drop at the open end. We estimate τ ∼ 40 µs.

At this point, one could be tempted to recover the characteristic time τ in the signal recorded at the closed end of a long cavity. However, because of the dissipation at the side walls (Sect. A.4) which leads to a dispersive propagation (the harmonics have different velocity), the width of the initial pressure front, generated by the bursting of the soap film, increases during the propagation toward the closed end. As a consequence, the temporal behavior of the pressure from microphone 2 does not image what happened at the open end. As an experimental proof, we report results obtained for two different cavitylengths (8 and 23 cm) and different viscosities of the soap film (Fig. 11).

The viscous soap films are obtained by mixing water, soap, sugar and glycerine. We observe with the fast camera that the characteristic rupture time is, at least, 2 to 3 times larger in this case. However, the first pressure drop measured at the closed end of the cavity looks very similar in the cases of the regular and viscous soap films, in the case of the longest cavity (L = 23 cm): the initial pressure drop associated with the film bursting is "hidden" by the widening of the pressure front during the propagation towards the closed end. Note also that the initial amplitude of the signal does not depend on the characteristic time τ of the pressure drop as, because of the choice of a long cavity, τ remains smaller than 1/ω 0 . Making use of a different cavity length (L = 8 cm), we estimate from the data reported in Figure 11 the widening rate to be ζ ∼ 15 µs/cm.

Even if the pressure drop at the bottom does not exactly image the pressure drop at the open end, we can get an estimate of the rupture time τ in the case of a short cavity. For each signal, we determine the time τ drop necessary for the pressure at the bottom to reach the first minimum. We estimate τ to be about (τ drop -ζL)/2. We report in Figure 12 the initial amplitude of the pressure signal, normalized to δP , as a function of the estimated τ . We observe, whatever the initial δP , a drastic decrease of the amplitude when the rupture time increases, which proves that the scattering of the data observed in Figure 8 is due to the variability of the rupture time.

Finally, we can compare the characteristic time τ measured in the experiment to an estimate of the typical time, τ typ , expected for the opening of a film stretched over an aperture of radius Φ/2. The opening of the film is driven by the surface tension and is limited either by the viscous dissipation [16] or by the inertia, this latter case being relevant in our experimental conditions. As a simple determination of the opening velocity appeared to be rather subtle [START_REF] Frankel | [END_REF], we detail here the model [17]. Let us now con- sider a pore of radius r which grows with the velocity v from the center of a frame (diameter φ/2, Fig. 13). The liquid, initially located in the central disk of radius r, forms a rim of mass m = πr 2 eρ, which grows with time (e stands for the film thickness and ρ for the density of the liquid). One could be tempted, at this point, to write the energy conservation by simply balancing the gain of surface free energy 2Γ πr 2 (Γ 30 ± 2 mN/m stands for the surface free energy of the liquid-air interface, the surface energy of the film is twice larger) with the kinetic energy 1/2mv 2 . However, if one considers the problem in the frame of reference of the rim (Fig. 13, inset), one can see that kinetic energy is continuously injected in inner flows (these flows are likely to be damped by the viscous dissipation but it is not nessecary to solve the hydrodynamical problem inside to go farther). We can easily see that the amount of kinetic energy which is introduced per unit time in the rim writes 2πerv×1/2ρv 2 (2πerv corresponds to the volume of liquid, having the kinetic energy 1/2ρv 2 per unit volume, which enters the rim per unit time). As a consequence, the energy balance must be written:

2Γ × 2πrv = d dt 1 2 mv 2 + 2πerv × 1 2 ρv 2 . ( 7
)
Taking into account the dependence of m on r, we obtain the constant velocity of the rim v = 2Γ/ρe. Note that one half of the energy is dissipated and that an estimate of the velocity obtained without introducing the dissipation would be wrong by a factor √ 2 [START_REF] Frankel | [END_REF]. Assuming that τ typ corresponds to the time the torus needs to reach the edge of the aperture, we get τ typ = Φ 2 ρe/2Γ . We can make use of τ typ = τ = 40 µs to determine an estimate of the unknown film thickness. We get e = 5.7 nm, which is small but of the order of magnitude of the CBF thickness reported in the literature. We do not expect a quantitative agreement between τ typ and τ as, first, there is no experimental proof that the bursting of the soap film corresponds to the opening of a pore at the center; second, as the relation between the dynamics of the pore opening and the associated pressure drop can be rather complex; and, finally, because the initial curvature of the soap film is not taken into account. Nevertheless, we note that τ typ and τ are, at least, of the same order of magnitude, which is an indication that the acoustic measurements do provide a realistic estimate of the characteristic time associated with the pressure drop at the open end.

Energy transferred to the acoustic wave

The sudden breaking of the soap film produces a pressure drop inside the cavity. The total potential energy E p released can be evaluated by considering the adiabatic expansion of the volume V = π(Φ/2) 2 L of gas when the inner pressure drops down by δP . One obtains

E p = 1 2 V δP 2 ρc 2 .
In the present section, we report how the energy E p is transferred to the acoustic signal after the bursting of the soap film. Before reporting our experimental results, we discuss in Section 3.4.1 what the microphone 1 senses. The behavior of the total energy is analyzed in Section 3.4.2.

Preliminary remarks

The microphone 1 senses the acoustic signal P (t) at a given point outside the cavity. As the acoustic wave outside the cavity is produced by a source of finite spatial extension (the surface area of the open end), the acoustic intensity I is likely to depend on the position of the microphone (distance and orientation) in a rather complex way.

Experimentally, the microphone is located at a finite distance d = 5 cm from the open end. One could worry the distance between the open end and the microphone to alter the amplitude of the various harmonics relative to each other. However, in our experimental conditions, the wavelength associated with the resonant frequencies is large compared to the cavity diameter and, as shown in Section B.1, the far field approximation applies. Thus, the amplitude associated with each of the modes decreases like 1/d where d is the distance to the open end, independent of the frequency. Moving the microphone toward or outward the tube end only rescales the power spectrum (as 1/d 2 ) and does not modify the shape of the envelope.

On the other hand, the microphone is oriented at about 45 degrees from the cavity axis and we could also worry the spatial structure of the acoustic wave to alter the measurements. We show in Section B.2, that the angular dependence of the acoustic wave outside the cavity leads to a slight dependence of the acoustic intensity on the frequency for a given orientation of the microphone [Eq. (B.5)]. However, the correction is less than 10% in all our experimental conditions (Φ < 10 mm, ν < 10 kHz). In addition, we estimate from the first zero of the form factor that the modes having frequencies higher than about 40 kHz will not be sensed by the microphone.

The microphone is located at a finite distance from the tube aperture. One must, in principle, consider the spherical nature of the acoustic wave outside when estimating the acoustic energy from the pressure signal P (t) provided by the microphone 1. However, we show in the appendix (Sect. A.5.7) that the total acoustic energy measured outside the cavity is given, to within better than 2.5% in all our experimental conditions, when neglecting the curvature of the wave front.

Finally, the typical lateral size of the experimental setup is less than d. It seems reasonable to assume that the spatial structure of the outside wave is a sphere and, thus, that the acoustic intensity is constant on the sphere of radius d. As a consequence, we estimate the total acoustic energy outside E out T to be simply given by

E out T ≡ 4πd 2 ρc ∞ t=0 P 2 (t)dt, (8)
where P (t) is the pressure signal from the microphone 1.

We expect the main source of error to be the integration over the sphere which leads to the prefactor 4π. Even if the potential error is there difficult to estimate, we can guess that E out T cannot overestimate the acoustic energy by more than 15% as we checked experimentally that the amplitude of the acoustic signal does not depend significantly on the angle α the microphone makes with the cavity axis, up to α 135 • .

Measurements and discussion of the energy balance

We already noticed that the soap film is likely to burst before having reached its minimum accessible thickness which leads to a scatter in the initial amplitude of the acoustic signals. We measured the acoustic energy E out T for different initial δP and report the data obtained when the film has reached its minimum accessible thickness (CBF) before bursting. In this case, the energy is maximum and the measurements are reproducible (Fig. 14).

We observe that E out T ∝ δP 2 for small δP in agreement with E p ∝ δP 2 . However, we note a departure from the quadratic law for larger δP . This feature is qualitatively explained by a change in the geometry of the film. Indeed, the film is almost flat at small δP and its bursting results in an almost-planar pressure-front propagating toward the cavity. In this case, the geometry of the front matches that of the planar modes inside the cavity and the energy transfer is efficient. By contrast, at large δP , the film bends significantly and its bursting results in a curved pressure-front propagating toward the cavity. The geometry of the front does not match that of the planar modes inside the cavity anymore and the efficiency of the energy transfer drops down.

More precisely, in the appendix A.5.7, we determine the input transmission coefficient for the energy:

T i = 1 - x 6j + x 1 -cos θ 0 1 -e jx tan θ 0 2 2 (9)
where θ 0 is defined by δP = (8Γ/Φ) sin θ 0 and x = Φ/λ. The dependence of T i on δP accounts for the non-trivial behavior of E out T . Indeed, from the dynamics of the pressure drop at the open end and from E p , the total energy transferred to the resonant modes inside the cavity, E in T , can be obtained by taking into account the transmission of each of the harmonics. In addition, we expect

E out T ∝ E in
T , provided that only δP is varied. We observe that the dependence of E out T on δP , predicted from T i , compares nicely with the experimental observations (lines vs. initial overpressure δP . Whatever the cavity diameter Φ, the acoustic energy scales like δP 2 for small δP whereas a departure from the quadratic law is observed for large δP . The bending of the film accounts nicely (lines) for the experimental observations (L = 5.5 cm). Black arrows indicate the maximum δP = 8Γ/Φ for each of the diameters. In inset is plotted the percentage of energy we would measure outside (from the fit of the experimental points in this figure), as a function of the percentage of the energy expected to be measured, when excluding the geometrical effects. Note the good agreement between theory and the experimental observations. in Fig. 14). The agreement proves that the geometry of the film is responsible for non-trivial dependence of the acoustic energy on the initial overpressure.

To go farther, we deduce from the interpolation of the experimental data the fraction β = (1/T i )E out T /E p of the total energy E p , that would be transferred to the acoustic modes outside the cavity in absence of the geometrical effects due to the bending of the film. Measured values β m are reported in Table 1.

We can compare β m to another estimate of β obtained, in addition, by considering the viscous dissipation and radiation only. On the one hand, assuming that the bursting of the film results in an exponential decrease of the pressure in the output plane with the characteristic time τ , one can estimate the amount of energy initially transferred to the acoustic modes inside the cavity (Sect. A.5.3) in absence of geometrical effects:

E in T = 1 - 2ω 0 τ π tanh π 2ω 0 τ E p ≡ β τ E p . ( 10
)
Note that the fraction β τ depends on the rupture time τ , which explains the scatter of the experimental data obtained when the film thickness is not controlled. In the case of a common black film, we estimated τ ∼ 40 µs for Φ = 8 mm, which leads to β τ 0.75. As we expect τ to depend linearly on Φ, we can also report estimates of β τ for Φ = 6 and 10 mm in Table 1.

On the other hand, the total acoustic energy recovered outside, E out T , corresponds to the part of the energy E in T which is radiated and not dissipated at the walls. One can show that the fraction τ Z n /(τ Z n + τ v n ) of the energy initially transferred to the mode n inside the cavity is dissipated at the cavity wall. We remind that τ v n and τ Z n are the characteristic times associated with the dissipation at the wall and with the radiation at the open end, respectively. Defining

β v = E out
T /E in T and summing the contribution of all the harmonics, we obtain the expected value of β v for Φ = 8 mm. Taking into account the scaling behaviors of τ v n and τ Z n on Φ, we get the estimates reported in Table 1 for Φ = 6 and 10 mm. Thus, from the dissipation at the wall and from the finite rupture time τ , we expect, in absence of the geometrical effects due to the bending of the film, E out T β e E p , with β e = β τ β v , which compares nicely to β m (see inset Fig. 14). However, note that the agreement must be considered with caution, considering the uncertainty in measuring the characteristic time τ and in the surface of integration in equation [START_REF] Divoux | Listening to singing bubbles at the free surface of a non-newtonian fluid, Stage de recherche de deuxième année du MIP[END_REF].

As a conclusion of the energy study, we would like to underline again the strong influence of the film rupture time on the energy conversion. Hence, direct measurements of the acoustic energy do not provide any information on the energy initially loaded inside the cavity if the film rupture is unknown. Any attempt to do so would lead to false interpretations.

Conclusion and perspectives

We reported an experimental study of the acoustic signal associated with the bursting of a thin liquid film at the open end of a cylindrical cavity.

Let us first explain qualitatively why the signals from microphones 1 and 2 look so different. The radiation at the open end and the propagation inside the cavity are responsible for the drastic differences in the signals recorded at the bottom of the cavity and outside. Indeed, the transmission coefficient associated with the radiation is proportional to the frequency ν n so that, at a given time t, the amplitude of the harmonic n, relative to that of the fundamental, is (2n + 1) times larger outside than inside the cavity. Moreover, the propagation along the cavity length makes the relative phase of two successive harmonics rotate by the angle π, so that, while in phase at the bottom, they have opposite phase in the output plane. The harmonics are thus clearly visible in the signal outside the cavity while they are hardly distinguishable in the signal recorded at the bottom. In addition, the harmonics are only observed at short times as they are damped faster mainly because of the radiation.

Our experimental findings can be summarized as follows:

• The acoustic signal exhibits a well defined fundamental frequency ν 0 which is mainly governed by the cavity length L. However, because of the radiation at the open end, the resonant length L , associated with the fundamental frequency ν 0 , does not exactly equal the cavity length L but L + δL where δL is a fraction of the tube diameter Φ. • For long cavities, we observe a banded spectrum, with frequency peaks ν n = (2n + 1)ν 0 , where ν 0 is the fundamental frequency. We would like to emphasize once again the didactic goals of the work we reported herein. We were only aiming at the deep understanding of all the physical processes that govern the acoustic signals associated with the sudden opening of an under-or overpressurized cavity. Focusing on one experimental situation, we can conclude that we can make use of the resonant cavity for performing the spectral analysis of the pressure drop associated with the film bursting. We are taught, for instance, that the bursting of a common black film, stretched over an 8 mm-in-diameter aperture, is associated with a characteristic time of about 40 µs.

Generalizing to other application fields, we can make use of our findings to better understand acoustic signals recorded in the nature. For instance, our study provides useful tools for the interpretation of the acoustic signals recorded on field by the volcanologists. First, the spectral content (frequencies) gives direct access to pieces of information about the geometry of the system: the resonant length relates to the bubble length whereas, from the surface, only the diameter of the bubble head or of the conduit can be observed. In the case of large bubbles bursting at the surface of a lava lake, the aspect ratio of the bubble depends on the rheological properties of the lava so that listening to the volcano makes it possible to access some rheological properties of the fluid. We point out that our experimental geometry is also relevant for interpreting the acoustic signal from the bursting of a giant elongated bubble at the top of a volcanic vent [7]. In this case the resonant length informs us about the conduit length and the damping characteristic-time about the processes that lead to the energy loss in the system, which gives constraints for modeling the resonant cavity [viscosity of the medium in which the acoustic wave propagates (gas or magma), material present at the closed end (either magma or solid rock from the reflection coefficient), etc.]. On the other hand, we point out some limitations of the acoustic methods: as they drastically depend on the filmrupture characteristic-time, which is neither a controlled nor a measured quantity in the field, the acoustic-signal amplitude and/or energy cannot be used to measure the total energy released during the bursting of a single bubble. Any attempts to do so would lead to misinterpretation of the field data.

we obtain from the general equation (A.1), where the constants A + and A -, which are functions of s, are given by the boundary conditions at both ends.

∆ P (x, s) - s 2 c 2 P (x, s) = - s c 2 P (x, 0) - 1 c 2 ∂P ∂t (x,
Assuming that the reflection of the sound wave at the closed end (plane x = -L) occurs without any loss, the particle velocity along the x-axis, v x (-L, t) = 0 (∀t). From the Laplace transform of equation (A.2),

sv x (x, s) -v x (x, 0) = - 1 ρ ∇ P (x, s), (A.6)
and from the initial condition, v x (x, 0) = 0 (∀x ∈ [-L, 0[), we get ∇ P (-L, s) = 0 (∀s). As a consequence, A + = A - and, thus,

P (x, s) = A(s) cosh s(x + L) c + δP s (A.7)
where the amplitude A, which is a function of s, is given by the boundary condition at the open end. In a first approximation, once the cavity is opened, the overpressure at x = 0 instantaneously vanishes so that P (0, t) = δP (t < 0) and P (0, t) = 0 (t > 0) (we thus consider in this section that the output impedance at the open end is zero). As the consequence, the Laplace transform of the pressure at the open end P (0, s) = 0 and, from equation (A.7),

P (x, s) = δP s cosh s c L -cosh s c (L + x) cosh s c L . (A.8)
The solution for the pressure field P (x, t) is obtained by calculating the inverse Laplace transform of P (x, s).

The resonant frequencies are associated with the poles, s n = j(2n + 1) πc 2L (n integer ranging from -∞ to +∞), which cancel the denominator cosh s c L in equation (A.8). Thus, the perfect cavity (reflection without any loss at the closed end and zero output impedance) exhibits a series of resonant frequencies ω n ≡ (2n+1) πc 2L which are associated with the wavelengths λ n ≡ c νn = 4L 2n+1 . Once the resonator has been excited due to the opening of the cavity at one end, one could expect to hear the resonant frequencies ω n outside the cavity. However, as the frequencies ω n correspond to perfectly resonant modes, by definition, the amplitudes of these modes do not decrease with time and, accordingly, no acoustic wave can exist outside the cavity. This conclusion, which contradicts the experimental observations, is the consequence of the specific boundary condition at the open end that we chose in this section: we considered that the pressure in the output plane is zero for t > 0 (zero output impedance). As discussed in Appendix B, this assumption is not experimentally satisfied and the resonant modes can indeed escape the cavity, producing the acoustic wave that we can hear outside. In the next Section A.2, we will consider a more realistic boundary condition at the open end and analyze the consequences on the resonant modes inside the cavity.

A.2 Effect of the radiation at the open end

A.2.1 Introduction

Let us now assume that the output acoustic impedance, Z r , is finite (Appendix B) and that there exists acoustic waves inside and outside the cavity. Writing the continuity of the pressure and velocity fields in the output plane (x = 0), we get:

P (x, s) = δP s × 1 - cosh s c (L + x) cosh s c L + Zr(s) ρc sinh s c L (A.9)
which reduces to equation (A.8) in the limit Z r (s) → 0.

A.2.2 Resonant frequencies

The resonant modes are again obtained by determining the poles of P (x, s). Using the expression (B.7) of the acoustic impedance Z r determined in the case of a flanged aperture, we get the poles s n which satisfy

1 tanh sn c L = - Z r (s n ) ρc = - 4Φ 3πc s n + Φ 2 8c 2 s 2 n . (A.10)
Neglecting the second order term in equation (A.10), we obtain the corresponding wavelengths

λ n = 4 2n + 1 L + 4 3π Φ (A.11)
where n is an integer ranging from 1 to ∞. Thus, to the first order in Φ/L, the radiation at the open end leads to a slight increase in the wavelength λ n and, accordingly, to a slight decrease in the frequency ω n = 2πc/λ n . In the case of a flanged aperture [START_REF] Pierce | Acoustics -An introduction to its physical principles and applications[END_REF], the effective resonant length of the cavity is

L = L + 4 3π Φ [Eq. (B.

A.2.3 Damping of the resonant modes

Taking into account the second order term in equation (A.10) and defining τ Z n by s n ≡ jω n -1/τ Z n , we get

τ Z n = 8cL Φ 2 ω 2 n = 4π c 2 Φ 2 2n + 1 ω 3 n . (A.12)
One can notice that τ Z n independently depends on the cavity length L and on the harmonic in consideration (i.e. on n).

The radiation leads to the damping of the harmonic n, having the frequency ω n , with a characteristic time τ Z n , which scales like ω -3 n . Equation (A.12) holds true in the case of a flanged aperture. In the case of an unflanged aperture [START_REF] Kinsler | Fundamentals of acoustics[END_REF], the characteristic time associated with the radiation is expected to be twice longer, 2τ Z n [the real part of the output impedance is twice smaller, Eq. (B.8)].

A.2.4 Pressure field

By calculating the inverse Laplace transform of P (x, s) given in equation (A.9), we can write an approximate solution for the pressure field inside the cavity:

P (x, t) = 4δP π × ∞ n=0 (-1) n 2n + 1 cos ω n c (L + x) cos(ω n t) exp - t τ Z n (A.13)
in which we neglect terms having amplitudes smaller than or of the order of (φ/L) 2 δP . Note that, from equation (A.13), P (x, 0) = δP, ∀x ∈ [-L, 0[. Equation (A.13) describes the inner pressure field in the case of an instantaneous opening of the cavity. Indeed, we consider the initial condition of constant pressure inside the cavity for t < 0 (the output impedance Z is thus infinite for t < 0) and assume that the output impedance is Z r for t > 0. Thus, we assume that the output impedance drops down from the infinity to Z r in an infinitely short time. We shall discuss the effect of the characteristic time of the cavity opening in Section A.5. However, at this point, we do prefer to analyze first two additional physical processes which lead to the damping of the resonant modes inside the cavity: the partial reflection at the closed end (Sect. A.3) and the viscous dissipation at the cavity walls (Sect. A.4).

A.3 Effect of partial reflection at the closed end

A.3.1 Pressure field

Up to now, we considered that the reflection at the closed end of the cavity occured without energy loss. However, because of finite acoustic impedance of the material the frame is made of or, especially, when using the microphone 2 at the bottom, part of the acoustic energy is lost in the plane x = -L.

Let us assume that a planar wave propagating toward the closed end is reflected partially as a planar wave propagating toward the open end with the reflection coefficient r 0 . Assuming zero output impedance at the open end, we get:

P (x, s) = δP s × 1 - cosh s c (L + x) + 1+r0 1-r0 sinh s c (L + x) 1 cosh s c L + 1+r0 1-r0 sinh s c L . (A.14)
Note that equation (A.14) reduces to equation (A.8) in the limit r 0 → -1.

A.3.2 Damping of the resonant modes

Defining τ r0 n such that s n ≡ jω n -1/τ r0 n we get, from the real part of the poles,

ω n τ r0 n = π 2 1-r0 1+r0 (λ n Φ).
Thus, the harmonics amplitude decreases exponentially with the characteristic time

τ r0 n = 1 -r 0 1 + r 0 1 4ν n . (A.15)
Note that, assuming that r 0 is real, the reflection at the closed end does not affect the resonant frequencies.

A.4 Effect of dissipation at the cavity walls

The shear viscosity of air is small and the damping of the acoustic wave due to viscous dissipation can usually be neglected. Indeed, the characteristic damping length for an acoustic wave in air is of about 300 m. However, the planar waves propagating inside the cavity are not compatible with the condition of zero velocity along the x-axis at the side walls and dissipation occurs in the boundary layer, in which air is subjected to important shear.

A.4.1 Physical background

The dynamical equation including the kinematic viscosity η writes

∂v ∂t = - 1 ρ ∇P + η∆v (A.16)
which reduces to equation (A.2) in the limit η = 0. In cylindrical coordinates, let us consider that the velocity along the x-axis (r = 0) oscillates at the frequency ω with the amplitude v ∞ . Assuming that the thickness of the boundary layer, L bl , is much smaller than the cavity diameter Φ, one can neglect the curvature of the walls and write the velocity along the x-axis

v x (r) = v ∞ 1 -exp - (1 + j)( Φ 2 -r) L bl (A.17)
where L bl = 2η/ω (η = 1.5 × 10 -5 m 2 s -1 stands for the kinematic viscosity of air). From equation (A.17), one can estimate the viscous stress applied by the side walls to the fluid and write the force balance in the cross section of the cavity:

ρ ∂v x ∂t = - ∂P ∂x -ρβv x (A.18)
where β = 1/Φ √ 8ωη(1 + j) and v x stands for the mean particle velocity along the x-axis in the cross section of the cylindrical cavity (note that Eq. (A.18) holds true only in the limit L bl Φ and in the harmonic regime).

A.4.2 Modified wave equation and pressure field

We can rewrite, from the dynamical equation (A.18) and from the mass conservation,

∂ρ ∂t + ∂(ρv x ) ∂x = 0, (A.19)
the wave equation

∂ 2 P ∂t 2 -c 2 ∂ 2 P ∂x 2 = -β ∂P ∂t (A.20)
which takes into account the viscous dissipation in the boundary layers at the side walls. Accordingly, the pressure field inside the cavity, assuming zero output impedance and reflection without loss at the closed end, can be written

P (x, s) = δP s cosh KL -cosh K(L + x) cosh KL (A.21)
where c 2 K 2 ≡ s 2 + βs. Note that, in the limit β → 0, K = s/c and equation (A.21) reduces to equation (A.8).

A.4.3 Damping of the resonant modes

Seeking for the damped oscillatory-modes, we write s = jω -1/τ and determine the resonant frequencies ω n and the corresponding viscous-damping characteristic-times τ v n cancelling the denominator in equation (A.21). Thus, assuming that the reflection at the closed end occurs without energy loss and zero output impedance, we estimate that the dissipation within the boundary layer slightly affects the resonant frequencies and, in addition, leads to the damping of the mode n inside the cavity over the characteristic time

τ v n Φ √ 2ω n η . (A.22)
The shear in the boundary layer induces temperature gradients. The associated thermal losses at the walls are accounted for by replacing η in equation (A.22) by

η[1 + (γ -1)P -1 2
r ] where P r 0.7 stands for the Prandtl number and γ = 1.4, the specific-heat ratio [START_REF] Pierce | Acoustics -An introduction to its physical principles and applications[END_REF].

A.5 Response of the resonant cavity to a pressure jump at the open end

A.5.1 Introduction

In this section, we discuss how the energy initially loaded in the cavity is transferred, immediately after bursting of the soap film, to the resonant modes. In order to understand the behavior of the system, before generalization, we determine the solution in the simple case of an exponential decay of the pressure at the open end with the characteristic time τ . We assume P (0, t) = δP e -t τ (∀t > 0). The Laplace transform of the pressure at the open end is given by P (0, s) = τ δP 1+sτ so that, from equation (A.7),

P (x, s) = δP s 1 - cosh s c (L + x) (1 + sτ ) cosh s c L . (A.23) A.5.2 Pressure field
The solution is obtained from the poles of equation (A.23), which are s = -1/τ and s k = j(2k + 1) πc 2L (k integer ranging from -∞ to +∞). After some algebra, we get:

P (x, t) δP = cosh( L+x cτ ) cosh( L cτ ) e -t τ + ∞ n=0 a n cos ω n c (L + x) sin ω n t + θ n (A.24)
where we define the frequency ω n , amplitude a n and phase φ n of the mode n:

ω n ≡ (2n + 1) πc 2L , (A.25) a n ≡ 4 π ω 0 ω n (-1) n 1 + (ω n τ ) 2 , (A.26) tan(θ n ) ≡ 1 ω n τ , φ n ∈ [0, π 2 [. (A.27)
In the limit τ → 0, we recover equation (A.13) excluding the radiation at the open end (i.e. τ Z n → ∞). The first term in equation (A.24) corresponds to an exponential decrease of the pressure, at each position x in the cavity, with the characteristic time τ . This contribution does not correspond to a resonant mode but to the mean gas flow which escapes the cavity. By constrast, the terms in the sum correspond to the resonant modes. The amplitude a n decreases when n increases and depends on the characteristic time τ : if

ω n τ 1, a n ∝ ω -1 n and, if ω n τ 1, a n ∝ (ω 2 n τ ) -1 .
The change in regime occurs at the characteristic frequency ω c = 1/τ .

A.5.3 Energy injected in the resonant modes

The total amount of acoustic energy inside the cavity once opened, E in T , corresponds to the contribution of the terms in the sum in equation (A.24). We remind that the density of acoustic energy equals 1 2 ρ(v 2 + P 2 ρ 2 c 2 ). After simple algebra, taking into account the relation (A.26) and integrating over the inner volume V , we get

E in T = 1 2 V δP 2 ρc 2 1 - 2ω 0 τ π tanh π 2ω 0 τ . (A.28) In the limit τ → 0, E in T = 1 2 V δP 2 ρc 2 .
Note that E in T corresponds to the work necessary for increasing adiabatically the pressure inside the cavity by δP (χ a ρc 2 = 1, where χ a is the adiabatic compressibility of air [START_REF] Kinsler | Fundamentals of acoustics[END_REF]). Thus, if the typical duration of the pressure drop τ is short compared to the period 2π/ω 0 associated with the fundamental frequency ω 0 , the total amount of potential energy initially loaded in the cavity is transferred to the acoustic modes. To the contrary, when the typical opening time τ increases, the energy transferred to the acoustic modes decreases according to equation (A.28) in the case of an exponential decay of the pressure in the output plane.

A.5.4 Generalization

The result reported in Section A.5.2 can easily be generalized to any behavior P (0, t) of the pressure at the open end if one considers only the resonant modes.

Let us first define the functions g(x, t) ≡ P (x, t)/δP -1 and f (t) ≡ g(0, t). From equation (A.7), we get

ĝ(x, s) = f (s) cosh[ s c (L + x)] cosh[ s c L] . (A.29)
One can determine the solution g(x, t) by calculating the inverse Laplace transform of ĝ(x, s). Aiming solely at the description of the resonant modes, we consider the poles of 1/cosh[s/cL] and not the poles of f (s) that correspond to a non-resonant part of the pressure field. Limiting thus our study to the oscillatory part of the pressure field, we write

g(x, t) = ∞ n=0 a n cos ω n c (L + x) sin(ω n t + θ n ) (A.30)
together with

a n = 4ω 0 π (-1) n f (jω n ) (A.31) θ n = arg[ f (jω n )]. (A.32)
Note that equation (A.30) reduces to the sum in equation (A.24) for f (t) = exp(-t τ ) -1. The amplitude and phase of the various harmonics are governed by the pressure drop at the open end: measuring the amplitude and phase of several harmonics makes it possible, in principle, to determine the Laplace transform P (0, s) of the pressure drop at the open end and, then, the pressure in the ouput plane P (0, t) as a function of time. Finally, note that, if we denote E in T any value of the initial energy in the acoustic modes inside the cavity at t = 0, we obtain that f (s) satifies the condition The phenomenon can hardly be described in the theory and, seeking for simplicity, we will assume that the output impedance is Z r for t ≥ 0 and that the opening of the film only governs the characteristic time τ . We expect such an approximation to be valid as long as τ is smaller than the time needed by the acoustic wave to go back and forth from the open end to the closed end. This is to say as long as τ < 2L/c or ω 0 τ < π, which is generally the case in our experimental conditions. In this framework, we suggest to write from equation (A.30), the pressure field associated with the acoustic modes inside the cavity

∞ n=0 f (jω n ) 2 = π 2 ρc 2 4V ω 2 0 1 δP 2 E in T . (A.
P (x, t) δP = ∞ n=0 a n cos ω n c (L + x) (A.34) × sin (ω n t + θ n ) exp - t τ d n
where a n and φ n are given by equations (A.31) and (A.32).

The frequencies ω n are associated with the modified wavelengths λ n which takes into account the effect of the radiation [Eq. (A.11), we neglect here the slight shift of the resonant frequency due to the viscous dissipation at the cavity walls]. The damping characteristic-time τ d n , which appears in equation (A.34), is given by

1 τ d n = 1 τ Z n + 1 τ r0 n + 1 τ v n (A.35)
where the various characteristic times τ n are from Sections A.2 to A.4.

A.5.6 Acoustic wave outside the cavity

From the solution inside the cavity [Eq. (A.34)], we can calculate the particle velocity in the output plane and thus, from equation (B.1), the pressure field, P out (r, t), outside the cavity (Sect. B). We first assume that the microphone is located far from the output plane, at a distance d large enough (d Φ) for the far field approximation to be valid (Sect. B.1). Second, we assume that the wavelength λ n , associated with the harmonics we consider, is large compared to the cavity diameter Φ so that the pressure field outside has the spherical geometry (Sect. B.2). Finally, the damping of the mode n is supposed to be slow enough for considering that the amplitude is constant over one period (i.e. ω n τ d n 1). In this framework, equation (B.1) can be used (the imaginary parts of ω and k are small and can be neglected) and the integration over the output plane reduces to a multiplication by the surface area of the cross section. Calculating the velocity v x from the solution inside the cavity, we get

P out (r, t) δP = Φ 2 4Lr × ∞ n=0 ω n f (jω n ) sin ω n t - r c + θ n exp - t τ d n . (A.36) We remind that θ n = arg[ f (jω n )].
Thus, listening to the sound outside the cavity makes it possible to determine the amplitude and phase of the Laplace transform of the pressure drop in the output plane and, then, the pressure drop itself. Finally, we point out that the conclusion holds true only for the harmonics which satisfy the condition ω n τ d n 1. Considering the radiation only (τ d n = τ Z n ), we can rewrite this latter condition (2n + 1)

1/π(4L/Φ) 2 . The equality leads to n = 10 for the shortest and widest cavity we use in the experiments whereas we only sense harmonics up to n = 3. We can thus consider that equation (A.36) gives a fairly good estimate of the pressure field outside the cavity in our experimental conditions.

A.5.7 Total energy in the acoustic wave outside the cavity

Assuming that the pressure and, thus, velocity fields outside the cavity are spherical, we can write the general expression of the total acoustic energy that crossed a sphere of radius r from t = 0 to ∞

E out T = 4πr 2 ∞ t=0 Idt (A.37) = 4πr 2 ∞ t=0 P out (r, t)v out (r, t)dt
where I stands for the acoustic intensity. By principle, this integral must be independant of r. Consider then first that r is much larger than the wavelength of the fundamental (we point out here that this is not experimentally the case as the microphone 1 is not located far enough from the output plane). In this case, the acoustic wave is locally planar so that the acoustic intensity reduces to I = P out 2 /(ρc). Calculating the integral in equation (A.37), one obtains the general expression of the total energy in the acoustic wave outside as a function of f (s)

E out T = πΦ 4 16L 2 δP 2 ∞ n=0 τ d n ω 2 n f (jω n ) 2 . (A.38)
The result must be independant of the distance r and E out T is the total amount of acoustic energy that crosses any surface surrounding the cavity aperture.

Equation (A.38) applies whatever the physical origin of τ d n . If the radiation is the only source of damping, τ d n = τ Z n and equation (A.38) reduces to

E out T = πΦ 2 4L cδP 2 ∞ n=0 f (jω n ) 2 = E in T (A.39)
where the second equality is obtained by using the condition (A.33). Thus, if the acoustic waves are only radiated at the open end, without any source of dissipation, the total amount of energy in the acoustic wave outside the cavity E out T equals the total amount of acoustic energy E in T initially loaded inside the cavity at t = 0. We recover here the energy conservation.

At last, we already noticed that the microphone 1 is located at a distance d from the open end which is not large in comparison to the wavelength λ 0 of the fundamental. As the microphone only provides us with the local pressure and not with the particle velocity, let us define an estimate of the total acoustic energy by

E approx T ≡ 4πd 2 ρc ∞ t=0 P out (d,

Appendix B: Radiation at the open end

Let us consider a planar wave which propagates inside the cavity toward the open end. When crossing the output plane x = 0, the sound wave can invade the whole halfspace x > 0. Thus, the planar wave is diffracted at the open end. The phenomenon has been widely studied [START_REF] Kemp | Theoretical and experimental study of wave propagation in brass musical instruments[END_REF], and we will summarize below the main results, obtained in the "piston" approximation, that we need for discussing our experimental findings.

Assuming the pressure field depends on time like e jωt , we can decompose, at the open-end, the incident planarwave into an infinite number of infinitesimal source elements of the form

dP (r) = jωρ v x 2π e -jk|r-r | |r -r | dS (B.1)
where v x is the axial component of the velocity field associated with the planar wave inside the cavity, r the position of the source element in the aperture plane and dS an infinitesimal surface element around the source. Integration of equation (B.1) over the whole surface of the aperture leads to the pressure field outside the cavity.

B.1 Near field -far field regions

Consider first a point located on the x-axis. We get by integration the pressure amplitude

|P (r)| y=0 z=0 = 2ρcv x sin 1 2 kx 1 + Φ 2x 2 -1 . (B.
2) Thus, along the x-axis, the amplitude of the pressure wave oscillates when the open end is approached from large values of x. Moving toward the cavity, one encounters the first local minimum in the pressure amplitude at x 0 which satisfies 4x 0

Φ = Φ λ - λ Φ . (B.3)
The distance x 0 is a convenient demarcation between the near field region, in which the behavior of the pressure field is complicated, and the far field region, in which the pressure amplitude decreases monotically along the x-axis. Note that x 0 < 0 for λ > φ, which means that, in this case, there is no near field region. In our experimental conditions, the length L of the cavity is always much larger than the diameter φ. As a consequence there is no near field region, at least for the fundamental and several harmonics having low enough frequency.

B.2 Spatial structure of the radiated wave

Consider now a point located far away from the aperture (|r| Φ). We get by integration, to the leading order in Φ/r,

|P (r)| |r| Φ = j ρc 8 v x Φ r kΦ e -jkr H(θ) (B.4)
where r = |r| and H(θ) denotes the dependence of the pressure amplitude on the angle θ (cos θ ≡ x • r) given by:

H(θ) = 2 J 1 ( kΦ 2 sin θ) kΦ 2 sin θ . (B.5)
In equation (B.5), J 1 denotes the first-order Bessel function of the first kind [START_REF] Arfken | Mathematical method for physicists[END_REF]. Thus, even in the far field approximation, the pressure field outside the cavity strongly depends on the angle θ. However, in our experimental conditions λ Φ and, as discussed in the text (Sect. 3.4.1), our measurements do not depend on the angular position of the microphone 1.

B.3 Output impedance

We can make use of the "piston" approximation for evaluating the acoustic impedance at x = 0 of the acoustic wave outside the tube. To do so, we can calculate the pressure field in the plane of the aperture by integration of equation (B.1). A second integration makes it possible to estimate in the case of a flanged pipe, as done by Rayleigh [START_REF] Kemp | Theoretical and experimental study of wave propagation in brass musical instruments[END_REF][START_REF]Lord Rayleigh in Theory of sound[END_REF], the mean value P , in the aperture plane, of the pressure P associated with the acoustic wave

P = 2jρcv x ∞ 0 J 1 (ξk Φ 2 ) 2 ξ ξ 2 -

B.4 Transmission coefficient

From the continuity of the pressure and velocity fields in the aperture plane, we can estimate energy transmission coefficient at the open end [START_REF] Kinsler | Fundamentals of acoustics[END_REF]. One obtains

T = 1 2 k 2 Φ 2 [1 + 1 8 k 2 Φ 2 ] 2 + [ 4 3π kΦ] 2 , (B.9)
in the case of a flanged aperture. In the limit kΦ 1, the transmission coefficient reduces to T 1/2k 2 Φ 2 (the approximation is better than 10% up to kΦ = 0.5). Note that T is an increasing function of the frequency as long as λ > Φ. associated with the liquid-air interface (the surface freeenergy of the soap film is 2Γ whereas the total curvature of the film is 2/R).

Qualitatively, for small δP , the film is almost flat and its bursting results in a pressure drop along an interface which has the geometry of the planar standing-waves inside the cavity; as a consequence, the planar front associated with the pressure drop penetrates easily the cavity and the energy E T is proportional to δP 2 as one would expect (limit δP → 0 in Fig. 14). When δP is increased, the center of curvature of the soap film approaches the aperture plane. The spherical geometry of the front differs from that of the planar modes inside the cavity; as a consequence, the spherical modes associated with the pressure drop hardly enter the cavity and the total energy transferred to the cavity E T decreases.

C.2 Input impedance

In order to account for the whole behavior of E T as a function of δP , consider first the initial geometry sketched in Figure C.1. We will work in spherical coordinates centered in O, the center of the spherical film. The initial pressure distribution, which writes δP (r) = δP for r < R and δP (r) = 0 for r > R, can be written as a sum of spherical modes centered in O. In the sake of simplification, we further assume that the film bursting leads to an homogeneous change in the pressure or velocity along the surface of the film. One can thus first consider one pressure mode propagating toward the cavity,

Appendix D: Energy spectrum

Let us now consider one damped harmonics which gives, directly from the microphone, the voltage V (t) of the form:

V (t) = V (ω) cos (ωt) e -t τ (D.1)
Let us denote t k ≡ k/ν s the time samples with ν s the sampling frequency. By definition, the fast Fourier transform of the signal writes:

F F T [V ](n) ≡ 1 N 2 N -1 k=0 V (t k ) e 2πj kn N . (D.2)
In what follows, we assume that the number of samples N is large (i.e. N ν s τ ). We define the value of the energy spectrum Π(ω n ) at the frequency ω n ≡ 2π n N ν s by Π(ω n ) ≡ 2|F F T [V ](n)| 2 . The spectrum Π, associated with the voltage given in equation (D.1), exhibits a maximum at:

ω n = ω 1 + 4 ω 2 τ 2 - 1 ω 2 τ 2 .
(D.3) Thus, in the limit τ → ∞, ω n = ω. In the presence of damping, in the limit ωτ 1, the maximum is slightly shifted to a higher value ω n such that δω ω ≡ ωn-ω ω 1 ω 2 τ 2 . The corresponding maximum value of the power spectrum is then given by:

Π(ω n ) = V 2 (ω) ν 2 s τ 2 4N 2 1 + 1 + 4 ω 2 τ 2 ∝ 1 2 ν 2 s τ 2 N 2 V 2 (ω) (ωτ 1). (D.4)
Note that, from its definition, Π depends on the acquisition time N/ν s . In the limit τ → ∞, the acquisition time is smaller than τ and one must take τ = N/ν s in order to recover the expected value Π = 1/2V 2 (ω). In our experimental conditions, we choose the acquisition time to be large in comparison to τ and equation (D.4) applies.

Fig. 2 .

 2 Fig.2. Left: signals P (t) from the microphones. Right: corresponding energy spectra. The typical signals from the two microphones exhibit, immediately after bursting of the film (t > 0), damped oscillations (left-hand-side). Note that the higher harmonics are damped faster in the signal from microphone 1. One can notice that their amplitude, relative to that of the fundamental, is larger outside (microphone 1) than inside (microphone 2) the cavity: even noticeable on the raw signals, this feature is clearly demonstrated by the energy spectra (see Appendix D for definition) displayed on the right-hand-side (δP = 13.1 Pa, Φ = 10 mm, L = 8 cm, microphone 2 at the bottom, sampling rate 20 kHz).

Fig. 3 .

 3 Fig.3. Fundamental wavelength λ0 vs. tube length L. The fundamental frequency ν0 exhibited by the system after breaking of the soap film is associated with the wavelength λ0, which is found to depend linearly on the tube length L according to equation (2) with δL 0.42 cm (∆P = 10 Pa, Φ = 10 mm, rigid wall at the bottom, sampling rate 20 kHz).

Fig. 5 .

 5 Fig. 5. Signal P (t) from the microphone 1 vs. time t. Thick grey line: signal from the microphone. Thin black line: interpolation with equation (3). The experimental data are from Figure 2 (δP = 13.1 Pa, Φ = 10 mm, L = 8 cm, microphone 2 at the bottom, sampling rate 20 kHz).

Fig. 6 .

 6 Fig. 6. Characteristic time τ d vs. frequency ν. The data are obtained from the fundamental and its harmonics for various values of the cavity-length L, the diameter Φ = 8 mm being kept constant (squares: fundamental n = 0; diamonds: first harmonic n = 1; open circles: second harmonic n = 2). The characteristic time τ d decreases when the frequency ν increases as expected from equation (4) [the error bars are estimated from the interpolation of the experimental data with Eq. (3)]. The lines are obtained from equation (4) without any adjustable parameter (solid line: n = 0; long-dashed line: n = 1, short dashed line: n = 2). The thin continuous line points out the contribution of the radiation alone obtained from equation (4) with η = 0 (δP = 18 Pa, Φ = 8 mm, rigid wall at the bottom).

Fig. 8 .

 8 Fig.8. Initial amplitude at the closed end vs. δP . We report the amplitude of the first oscillation of the pressure signal at the closed end of the cavity as a function of the overpressure δP for two different cavity lengths (full squares: L = 2 cm; open circles: L = 8 cm). Note that the scattering of the data, which is due to the variability of the bursting-film thickness, is largest for the shortest cavity length (Φ = 10 mm, microphone 2 at the bottom).

Fig. 9 .

 9 Fig. 9. Amplitude | f (jw)| as a function of ω. The experimental data are from the fundamental (full squares), the first (open squares), second (full diamonds) and third (open diamonds) harmonics. The amplitude is constant below and decreases like ω -5 above 1 τ (vertical solid line). From the cut-off frequency, we obtain τ 40 µs (δP = 18 Pa, Φ = 8 mm).

Fig. 10 .

 10 Fig. 10. Phase θ(jω) ≡ arg[ f (jωn)] as a function of ω. The experimental data set is the same as in Figure 9. In order to get a constant phase θ for ω < 1 τ , we take t0 = 2.3 × 10 -4 s (δP = 18 Pa, Φ = 8 mm).

Fig. 11 .

 11 Fig. 11. Pressure P (t) from microphone 2. Black solid line: regular soap film and L = 23 cm; thick grey line: viscous soap film and L = 23 cm; dashed line: viscous soap film and L = 8 cm. We report the first pressure drop (the origin of time is arbitrary). Even if the characteristic rupture time is 2 to 3 times larger for the viscous soap film (observation with the fast camera), one observes that the initial pressure drop at the closed end is almost the same for the regular and viscous soap films. By contrast, for the same characteristic rupture time, the initial pressure drop at the bottom drastically depends on the tube length L (δP = 10 Pa, Φ = 10 mm).

Fig. 12 .

 12 Fig.12. Initial relative-pressure amplitude vs. estimated rupture time τ . The initial acoustic-signal amplitude is normalized to δP . The rupture time τ is estimated from the pressure drop at the bottom. In this experiment, 1/ω0 ∼ 43 µs (L = 2 cm, Φ = 10 mm).

Fig. 13 .

 13 Fig. 13. Sketch of the opening pore.

Fig. 14 .

 14 Fig. 14. Total acoustic energy outside E out T

  0). (A.4) Taking into account the initial condition P (x, 0) = δP , ∀x ∈ [-L, 0[, the general solution to equation (A.4) satisfies P (x, s) = A + e sx c + A -e -sx c + δP s (A.5)

  7)]. In the same way, we obtain L = L + 0.3 Φ for an unflanged aperture [Eq. (B.8)].

33 )A. 5 . 5

 3355 Radiation and damping of the resonant modesThe radiation at the open end (Sect. A.2), the partial reflection at the closed end (Sect. A.3) and the viscous dissipation at the cavity walls (Sect. A.4) lead the damping of the resonant mode in time. The partial reflection at the closed end and the viscous dissipation are not affected by the opening of the cavity for t ≥ 0. To the contrary, the bursting of the soap film affects the evolution of the pressure at the open end with time and, thus, governs the characteristic time τ (the output impedance drops down from the infinity to Z r during the opening of the film).

δ 2 )

 2 P (r, s) = δ P 0 (s) Note that P 0 (s) denotes the Laplace transform of the pressure drop associated with the film bursting. From equations (C.1) and (C.2), we can calculate, in the plane of the open end, the mean pressure:∆ P (s) = 8δ P 0 (s) Rc sΦ 2 1e -sR c (1-cos θ0) (C.3)and velocity along the x-axis:v (s) = -8 δ P 0 (s) ρc R sΦ 2 cos θ 0e -s c R(1-cos θ0) . (C.4)where R sin θ 0 = Φ/2. We deduce from (C.3) and (C.4), the input impedance Z i , which is associated with the homogeneous variations of the pressure and velocity in the plane of the open end:Z i (s) ρc = -1e -s c R(1-cos θ0)cos θ 0e -s c R(1-cos θ0) .(C.5)

Fig. C. 1 .+ x 1 -cos θ 0 1 -e jx tan θ 0 2 (C. 6 ) 1 -cos θ 0 1 -e jx tan θ 0 2 2(C. 7 )

 111261127 Fig. C.1. Sketch of the initial geometry.

Table 1 .

 1 Percentages β. Data in bold font are experimental values, whereas other data are estimates (see text).

	Φ	6 mm	8 mm	10 mm
	β τ	81%	75%	68%
	β v	16%	30%	46%
	β e	13%	22%	31%
	β m	14%	23%	26%

•

  Because of the radiation at the open end and of the thermal and viscous dissipation at the cavity walls, the acoustic signals we record inside and outside the cavity are damped in time. The associated characteristic times, we measure experimentally, are in fairly good agreement with the theoretical predictions without any adjustable parameter. • The spectral content of the acoustic signals depends on how the cavity is opened. As a consequence, the careful analysis of the acoustic signal recorded outside the cavity makes it possible to determine accurately the characteristic time associated with the initial pressure drop at the open end of the cavity. • The total amount of energy recovered in the acoustic signals depends drastically on the film curvature and on the dynamics of the film bursting.

  t)

	2
	dt	(A.40)
	from equation (A.37) and assuming that the acoustic
	impedance is that of a planar wave. One can show that E approx T = E out T [1 + O(π 2 Φ 2 /64Ld)]. Thus, we estimate that E approx T differs from E out by less than 2.5% in all T our experimental conditions (Φ ≤ 1 cm, L ≥ 2 cm and
	d = 5 cm).

  we get the acoustic impedance associated with the radiation at the open end in the case of a flanged pipe Because of the radiation, the acoustic impedance at the open end is non-zero.

							1	dξ.	(B.6)
	In the limit λ Φ, Z r ρc = ρcv x P	1 8	(kΦ) 2 +	4 3π	jkΦ.	(B.7)
	Similar calculations by Levine and Schwinger [15] in the
	case of an unflanged pipe lead to a similar, but numerically
	different, expression
	Z r ρc	=	P ρcv x	1 16	(kΦ) 2 + 0.3 jkΦ.	(B.8)

The authors are very grateful to S. Vergniolle, M. Ripepe and D. Legrand for having provided us with exciting and useful pieces of information about the acoustics of volcanoes. We also highly thank S. Job, S. Douady, D. Constantin and B. Castaing for fruitful discussions. This work was supported by CONI-CYT under FONDAP Program No. 11980002 and FONDE-CYT Project No. 3040018. V. Vidal and J.-C. Géminard would like to thank the Centre National de la Recherche Scientifique (CNRS, France) and T. Divoux the École Normale Supérieure (ENS, Paris) for supporting the research of their members and students in foreign laboratories.

Appendix A: Theoretical background A.1 The perfect resonant cavity

Let us denote P (r, t) the pressure field associated with the acoustic wave (P does not include the constant pressure P a of the atmosphere). Denoting c the velocity of sound in air, we write the general equation governing an acoustic wave:

The particle velocity v, associated with the pressure field P (r, t), obeys the relation:

Let us consider the cylindrical cavity [length L (x ∈ [-L, 0]), diameter Φ] initially (t < 0) filled with air at the pressure P a + δP . We are aiming at the description of the evolution of the pressure field inside the cavity resulting from the opening of the cavity at one end (x = 0). We limit our study to the planar waves, P (r, t) = P (x, t), that can propagate in the cavity and ignore the higher-order, non planar, modes that could exist in the cavity but that are not relevant for our purpose [18]. Denoting P (x, s) the Laplace transform of P (x, t), defined to be [START_REF] Arfken | Mathematical method for physicists[END_REF] P (x, s) = ∞ 0 P (x, t)e -st dt, (A.3)

Appendix C: Input impedance C.1 Qualitative analysis

The rather complex behavior of the total energy E T as a function of the overpressure δP can be understood by considering the geometrical structure of the initial configuration (film shape). Indeed, before bursting, the soap film is spherical-shaped (radius R) and the pressure difference is δP = 4Γ/R where Γ denotes the surface free-energy


Absorption spectrum of very low pressure atomic hydrogen.

Jacques Moret-Bailly * 25 décembre 2015

Résumé

Les spectres des quasars résultent essentiellement d'interactions de la lumière naturelle avec de l'hydrogène atomique. Une absorption visible d'une raie spectrale fine et saturée dans un gaz requiert une basse pression, donc un long parcours sans perturbation comme un rougissement cosmologique. Burbidge et Karlsson ont observé que les rougissements des spectres des quasars résultent de rougissements fondamentaux écrits 3K et 4K, qui amènent les raies d'absorption Lyman beta ou gamma de l'atome H sur la raie alpha. Le spectre est donc rougi tant qu'une raie absorbée ne se superpose pas à la raie Lyman alpha du gaz : le rougissement se produit donc seulement si une absorption alpha pompe des atomes dans l'état 2P. L'espace est structuré en coquilles sphériques centrées sur le quasar contenant ou non des atomes excités. Dans les coquilles excitées la densité d'atomes 2P croît jusqu'à ce que l'amplification cohérente d'un mode ayant un long parcours dans la coquille, donc perpendiculaire au rayon observé, soit assez grande pour qu'un flash superradiant à la fréquence alpha éclate, absorbant l'énergie des atomes et du rayon observé, à la fréquence Lyman alpha locale. Ces oscillations de relaxation inscrivent de nombreuses raies d'absorption. Les rougissements des spectres des quasars sont dus à des interactions paramétriques composées d'effets Raman impulsionnels stimulés (ISRS) : Des atomes d'hydrogène excités catalysent des échanges d'énergie entre un rayon observé et des rayons froids du fond thermique, en accord avec la thermodynamique.

Abstract. Spectra of quasars result primarily from interactions of natural light with atomic hydrogen. A visible absorption of a sharp and saturated spectral line in a gas requires a low pressure, so a long path without blushing as a cosmological redshift. Burbidge and Karlsson observed that redshifts of quasars result from fundamental redshifts, written 3K and 4K, that cause a shift of absorbed beta and gamma lines of H to alpha gas line. Thus absorbed spectrum is shifted until an absorbed line overlaps with Lyman alpha line of gas : redshift only occurs if an alpha absorption pumps atoms to 2P state. Thus, space is divided into spherical shells centered on the quasar, containing or not 2P atoms. Neglecting collisional de-excitations in absorbing shells, more and more atoms are excited until amplification of a beam having a long path in a shell, thus perpendicular to the observed ray, is large enough for a superradiant flash at alpha frequency. Energy is provided by atoms and observed ray, absorbing a line at local Lyman alpha frequency. Redshifts of quasars spectra are due to parametric interactions composed of Impulsive Stimulated Raman Scatterings (ISRS) : Excited hydrogen atoms catalyze energy exchanges between observed light rays and background cold rays, in agreement with thermodynamics. 1 Introduction.

Study of spectra of quasars is difficult because usual spectroscopy which uses transitions between energy levels does not work.

Burbidge [START_REF] Burbidge | The Distribution of Redshifts in Quasi-Stellar Objects, N-Systems and Some Radio and Compact Galaxies[END_REF], then Karlsson [2] found an empirical formula on redshifts of selected quasars, formula which is transformed in section 2.2, so that its empirical parameter (Karlsson's constant) appears connected to Lyman spectrum of H atom. Subsection 2.3 applies transformed formula to spectral frequencies, correlating them. It appears that light is redshifted when Lyman α line is absorbed, while visible absorption of gas lines requires a stop of redshift which occurs if there remains no energy at Lyman α frequency.

Remarkable redshifts 3K or 4K (where K is Karlsson's constant 0.061) show that redshifts stop if absorbed lines Ly β or Ly γ reach ν α frequency. Thus we suppose that atoms in state 2P produce a redshift, so that a lack of these atoms allows an absorption of intense, sharp lines.

In 2.3, these remarks are used to build a set of spectral lines.

In 2.3.4, space is divided into spherical shells around star, absorbing (thus redshifting) or not light at local Lyman α frequency.

In section 3, gas in a redshifting shell may lose, during a short time, its excitation by emission of a superradiant beam, so that a sharp absorbed line appears.

Section 4, shows how impulsive stimulated Raman scatterings (ISRS) redshifts light.

2 Absorption of a gas spectrum by lack of energy at Lyman ν α frequency of H atom.

Karlsson's and Burbidge's periodicities.

Burbidge [START_REF] Burbidge | The Distribution of Redshifts in Quasi-Stellar Objects, N-Systems and Some Radio and Compact Galaxies[END_REF], then Karlsson [START_REF] Karlsson | Quasar redshifts and nearby galaxies[END_REF] observed that relative frequency shifts of spectral lines of quasars z = [ν emit -ν obsv ]/ν obsv are often close to remarkable values z = 0.061 and z = 1.95. This result was generalized by Karlsson's formula :

z(n) = nK, ( 1
)
where n is integer 3, 4, 6,... and K=0.061. Many sharp, saturated lines making "Lyman forests" of quasar obey these rules ; many authors tried to apply similar formula to galaxies, but it failed.

As his formula does not work well for large n, Karlsson proposed a periodicity of ln(1+z), additions of which multiplies sets of frequencies by parameters, as Doppler redshifts do, without very good results.

An aim of this paper is extension of Karlsson's work to understand formation of "Lyman forests" of quasars. We use mainly conventions and results found in P. Petitjean's paper [START_REF] Petitjean | Le contenu baryonique de l'univers révélé par les raies d'absorption dans le spectre des quasars[END_REF], also Rauch's ideas [START_REF] Rauch | The Lyman alpha forest in the spectra of QSOs[END_REF].

Discussion of Karlsson's formula.

In Doppler frequency shifts, emitted and received frequencies verify equation which involves, as parameters, emission and reception speeds V emission and V reception :

ν reception = c-V reception c-V emission * ν emission (2)
We assume that frequency shifts at various light frequencies obey an equation similar to equation ( 2), so that a single parameter is needed to compute frequency shifts at all frequencies, for instance the ratio f 1 /f 0 of a particular shifted frequency over initial frequency :

ν shif ted = f 1 f 0 ν initial (3)
The terms of strange serie 3, 4, 6, ... of Karlsson's formula may be written 3p + 4q, where p and q are any non-negative integers. Thus redshifts 3K and 4K are remarkable, able to generate simply all Karlsson's redshifts.

Compute redshifts which put Lyman ν β and ν γ frequencies of H atom to ν α frequency :

Z (β,α) = (ν β -ν α )/ν α = [(1 -1/32 -(1 -1/22)]/(1 -1/22)] ≈ 5/27 ≈ 0.1852 ≈ 3 * 0.0617 ≈ 3 * K; (4) Z (γ,α) = (ν γ -ν α )/ν α =[(1-1/42 -(1-1/22)]/(1-1/22) ] = 1/4 = 0,25 = 4*0.0625 ≈ 4K. (5)
Assuming that use of 3 lowest Lyman frequencies of H atom is better than use of K, and that variation of shifts at various frequencies is obtained from a particular shift as in a Doppler redshift, Karlsson's formula becomes simpler : shifted frequencies ν(p, q) depend only on absolute frequency ν 0 , three well known frequencies and two any non-negative integers p and q : ν(p, q) = (ν α /ν β ) p * (ν α /ν γ ) q * ν 0 . (6) Karlsson's formula (1), applied, for instance, to Lyman beta line with n=3 does not generate exactly Lyman alpha frequency, while formula (6) does. Though resolution of spectra does not allow to distinguish a doublet from a superposition of lines, we will choose formula (6) because choice of possible values of p and q seems more natural than extraction of n from a less natural serie. Is it only aesthetics ?

Model of a quasar surrounded exclusively by hy-

drogen.

Possible hypothesis.

We assume that "Lyman forest" is built by absorption of a thermal emission of an extremely hot star by pure, relatively cold (2 000-50 000 K), low pressure, non-excited, atomic hydrogen.

Following properties of many lines of "Lyman forest" are directly deduced from Petitjean's paper [START_REF] Petitjean | Le contenu baryonique de l'univers révélé par les raies d'absorption dans le spectre des quasars[END_REF] :

-A-As lines are sharp, widening of lines by collisions must be negligible, pressure of gas must be very low.

-B-To obtain absorption of Lyman lines at several frequencies, a redshift process of electromagnetic waves is necessary. Here, we make only hypothesis of formula (3) about this redshift.

-C-A single, unshifted Ly β is observed ; no unshifted Ly γ appears because it does not remain absorbable energy at high frequencies after redshifts of thermal emission profile.

Absence of an absorption line may result from :

-a-Absence of emitted energy around frequency of line. -b-A permanent shift of light frequencies dilutes absorption, so that all absorbed (or emitted) lines have width of shift and lines are weak, not observable. Accordingly, absorption (or emission) of sharp lines requires a stop of frequency shifts.

-c-Accurate superposition of observed lines results from the choice of redshift equations : Absence of shifted β and γ lines while sharp α lines are observed, is due to superposition of these lines with α lines.

In standard theory, sharpness of saturated absorbed lines requires contradictory conditions :

-Column density of gas must be large for saturation ; -Absorbing gas must be thin to avoid a broadening of lines by frequency shift during absorption.

-Pressure of gas must be low to avoid collisional broadening of line. Thus gas must be in filaments which are only detected on paths from quasars.

Supposing that redshift is related to a physical property of gas, condition is :

-Light is redshifted except if an absorbed line is at Lyman α frequency. Thus redshift requires a lyman α absorption, that is generation of 2P atomic hydrogen.

Building of spectrum.

Figure 1 represents a canvas of atomic hydrogen spectrum for building any absorption spectrum by addition of lines, in particular able to play the role of Ly β or Ly γ lines.

Rules used to build spectrum taking into account only hydrogen atoms are simple :

-i -At start, close to star, we suppose that α, β and γ lines have been absorbed.

-ii -It appears a frequency shift until an absorbed, shifted line of initial frequency ν reaches ν α frequency. Thus all absorbed frequencies have been multiplied by ν α /ν, coefficent lower than 1. In pure atomic hydrogen ν may be ν β or ν γ , assuming that higher frequency lines are too weak.

-iii -During stop of frequency shift, the three main lines of H could be absorbed, but there remain no energy at Ly α frequency.

-iv -Assume that Ly β absorption produces a very weak redshift. During this negligible redshift, gas lines are visibly absorbed. If weak redshift is able to shift absorbed frequencies off Ly α absorption line before full absorption of light at Ly β frequency, fast redshift restarts, go to -ii -. Else there is no more redshift, absorbed Ly β line is visible, but Ly γ line is probably not because its frequency is probably larger than the shifted high frequency limit of emission of star.

In pure H, total redshift results from several relative shifts Z β,α and Z γ,α which correspond to multiplication of light frequencies by ν α /ν β or ν α /ν γ .

Computed lower frequencies written on figure are not very good. They can be corrected by multiplication by a unique dispersion function F (ν) equal to 1 at ν α frequency.

Introduction of other gas in spectrum building.

Lines found in figure 1 appear in real spectra, after a correction of their frequencies by the factor F (ν). But many lines have an other origin : various lines absorbed at frequencies larger than ν α may play the role of Ly β and Ly γ , multiplying density of lines.

Figures much more complex than fig. 1 can be drawn.

Structuring space.

Redshifts stop if frequency of an absorbed line becomes ν α and not if absorbed ν γ frequency reaches ν β frequency. This shows that redshift results on generation of 2P hydrogen atoms.

Up to now, we have only supposed that it exists somewhere (sometime for light) a redshift. Thus, assuming that quasar is far from other stars, space is divided into spherical shells where generated 2P atoms shift light frequencies, and shells in which there is no 2P atoms and no frequency shifts.

As quasars are not alone in space, perturbation of generation of these regions may result from pumpings of atoms by light of other stars. Thus region in which quantized redshifts appear must be small enough to avoid an important lighting by other stars : This explains Burbidge's selection of quasars.

To avoid different pumpings by light coming from different regions of surface of the star, redshift must appear only at a distance much larger than size of the star, that is in a region where pressure of gas is low.

These conditions are evidently not verified for a set of relatively close stars as a galaxy. As the star must produce very high light frequencies, if it is small it must be extremely hot. An hypothesis which seems valuable is an accreting neutron star, a type of stars which were never found in nebulae while they should be seen.

3 Absorption of a gas spectrum by lack of energy at Lyman ν α frequency of H atom : Dynamics of space structure.

In model described in 2.3, we obtained spherical shells of hydrogen atoms either excited by absorption of Lyman ν α frequency, or in ground state. But it is difficult to obtain the large number of lines observed for instance between Ly α and Ly β lines ( Ly α forest ).

In 2.3.4, we took only ito account the lack of absorption at ν α frequency to explain stops of redshifts. But, in spherical shells in which ν α frequency is absorbed, Einstein's B coefficient of amplification at ν α may become large because temperature of exciting light is large and there are almost no collisions to de-excite atoms. Thus, Lyman alpha coherent, superradiant flash emissions may burst, into directions almost orthogonal to exciting rays, for which path in a spherical shell is maximum. The flash pumps energy from atoms and light which excites them, so that an absorption line appears in spectrum. These transfers of energy are in agreement with thermodynamics because energy flows from hot beams comming from star to resulting beams stemming from dark infinity through flash, radiance of which is average. The result is a sharp absorption at current frequency .

It is difficult to discuss on stability of relaxation oscillations which generate these absorptions, thus on stability of spectra. However, the medium is probably much more homogeneous and stable than the medium which provides aurora borealis, similar to permanently superradiant media which provide punctuated or not circles around Strömgren spheres, possibly transformed by planets into hourglasses as in SNR1987A. A computation of this spectrum seems very difficult, requiring a lot of hypothesis.

Physical interpretation.

Comparison of optical properties of 1S and 2P hydrogen.

In 1S state, hydrogen has 1420 MHZ hyperfine frequency, (period T=0.7ns ; wavelength λ = 21 cm), quadrupolar resonance frequency.

In its first excited state, hydrogen atom has quadrupolar resonance frequencies : 178 MHz (T=5.6 ns, λ =1,7 m) in state 2S 1/2 , 59 Mhz (T=17 ns, λ = 5 m) in 2P 1/2 and 24 MHz (T=42 ns, λ = 12 m) in 2P 3/2 .

A qualitative difference between laboratory and space ISRS is that ordi-nary incoherent light is made of longer, around 1 nanosecond pulses : these pulses must be shorter than hyperfine periods in excited states, as required by conditions of space coherence of "Impulsive Stimulated Raman Scattering" (ISRS) : Length of pulses must be shorter than all involved time constants [START_REF] Lamb | Analytical description of ultra-short optical pulse propagation in a resonant medium[END_REF]. Space coherence of incident and scattered light in ISRS allows an interference of exciting and scattered light, which shifts frequency of incident light and preserves the geometry of light beams.

The other condition for ISRS is : Collisional time must be longer than 1ns, that is pressure must be very low, so that pressure broadening of lines is low.

These conditions make ISRS in space very weak compared with ISRS in labs of chemistry, which uses around 10 nanosecond laser pulses, around k = 10 5 times shorter than light pulses of temporally incoherent light :

-Division of pressure by k divides the shift by k.

-Division of quadrupolar frequency by k divides the shift by k twice : --Once mixing incident and Raman frequency shifted of hyperfine frequency ;

--Once by division of difference of populations of hyperfine levels, assuming thermal equilibrium.

Thus order of magnitude of path needed to observe ISRS is ≈ 10 15 times longer than in laboratory experiments : astronomical paths are needed for observation.

De-excitation of hyperfine levels.

As pressure is very low, collisions between atoms are negligible. A radiative process is necessary to de-excite hyperfine levels, thus obtain Raman frequency shift.

Happily, there are cold background electromagnetic waves. Thus, the real process is not a single ISRS, but a set of ISRS such that hyperfine levels of 2P atoms are not excited. Excited atoms H catalyze an exchange of energy between light beams, it is a parametric interaction. Variations of energy produce frequency shifts of light beams, so that their entropy is increased. This parametric interaction is usually named "Coherent Raman Effects between Incoherent Light beams" (CREIL).

Conclusion.

Study of Lyman forests of quasars is difficult because they mix spectra origins of which are different, although all visibly absorbed lines are sharp and saturated : -Karlsson's relative frequency shifts 3K or 4K transform, shift, with a good approximation, absorbed Lyman frequencies ν β or ν γ of H atom into ν α frequency. During corresponding frequency shifts, H atoms are pumped to 2P level by permanent, diluted, invisible Ly α absorption.

-Fast, superradiant de-excitation of 2P atoms by emission of a superradiant flash absorbs light at ν α frequency, during a relaxation process. Burbidge and Karlsson selected quasars in which few lines have this origin to obtain their results.

Coherently scattered Raman light interferes with incident light, shifting its frequency (As Rayleigh scattering shifts the phase, producing refraction).

Stop of redshift which allows saturated, visible absorption of lines is due to lack of 2P atoms. This lack may result from flash, coherent de-excitation of 2P levels, or to presence of an absorbed line at Ly α frequency. In this last case, redshift may remain very weak by Ly β absorption, until absorbed line is shifted out of Ly α frequency, so that fast redshift restarts... Additional absorptions by various atoms may also play the role of Lyman β and γ absorptions, complicating the spectrum. Comparison of theoretical and observed spectra shows a small shift increasing at lowest frequencies, because, as refraction, ISRS has a dispersion which was not taken into account.

Hubble's law does not apply to quasars which may be accreting neutron stars lying in close galaxies.

Figure 1 -

 1 Figure 1 -Generation of lines of Lyman forest of a quasar by coincidence of lines α with more shifted β or γ. During stop of redshift Ly β and Ly γ lines are absorbed. Lines of other local gas may be also absorbed and may later play the role of Ly β and Ly γ lines if their frequencies are larger than ν α . Written frequencies do not take into account dispersion of hyperfine polarizability of H atom.