V. Vidal et al- Acoustic signal associated with the bursting of a soap film which initially closes an overpressurized cavity: Experiment and theory

8/3/2019 V. Vidal et al- Acoustic signal associated with the bursting of a soap film which initially closes an overpressurized cavity: Experiment and theory

1/19

Eur. Phys. J. B (2006)DOI: 10.1140/epjb/e2006-00450-0 THE EUROPEAN

PHYSICAL JOURNAL B

Acoustic signal associated with the bursting of a soap film whichinitially closes an overpressurized cavity

Experiment and theory

V. Vidal1,a, J.-C. Geminard2,b, T. Divoux1, and F. Melo1

1 Laboratorio de Fsica No Lineal and Center for Advanced Interdisciplinary Research in Materials (CIMAT),Departamento de Fsica, Universidad de Santiago de Chile (USACH), Avenida Ecuador 3493, Santiago, Chile

2 Laboratoire de Physique, Ecole Normale Superieure de Lyon, 46 Allee dItalie, 69364 Lyon Cedex 07, France

Received 3 May 2006 / Received in final form 10 November 2006Published online 20 December 2006 c EDP Sciences, Societa Italiana di Fisica, Springer-Verlag 2006

Abstract. 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 ofthe phenomenon, which can be very useful in numerous practical cases. For instance, in the nature, thevolcanologists observe the bursting of large, elongated, gas-bubbles at the surface of lava lakes and recordthe associated sound emission. One can wonder which pieces of information they can get from such acousticmeasurements. For a didactic purpose, we provide also the reader with all the theoretical backgroundnecessary for the understanding of the physical processes that govern the various characteristics of theacoustic 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 thefilm-rupture more than about the energy initially loaded in the cavity.

PACS. 68.15.+e Liquid thin films 43.20.Mv Waveguides, wave propagation in tubes and ducts 43.20.KsStanding waves, resonance, normal modes 91.40.Yt Remote sensing of volcanoes

1 Introduction

The sound produced by the quick opening of an under- oroverpressurized cavity is a common phenomenon. It can

be encountered in many situations, from everyday life tolarge-scale natural systems.One of the most well-known examples is certainly the

characteristic pop sound when opening a champagne bot-tle: the pressure inside the bottle, initially larger thanthe atmospheric pressure, drops down when the cork isreleased. Excited by the sudden change in pressure, thefree volume close to the bottleneck resonates. In addi-tion, the initial overpressure is due to a large amount ofgas dissolved in the liquid, which, once the pressure hasdecreased, escapes the system as the famous champagne

a Permanent address: Laboratoire de Physique, Ecole Nor-male Superieure de Lyon, 46 Allee dItalie, 69364 Lyon Cedex07, France.b e-mail: [email protected]

bubbles: by listening carefully to a glass full of champagne(or whatever bubbling liquid), one can notice the softpopping sound generated by each bubble-bursting at theliquid surface. In this latter case, the resonators are thebubbles themselves.

To date, the studies of the sound emission have limitedto the bursting of bubbles at the free surface of a newto-nian fluid [1] in which, because of the rounded shape of thebubble reaching the free surface, the resonator consists ofa liquid lens, the depth comparing with the bubble diam-eter [2]. By contrast, the body of a bubble rising in a non-newtonian fluid is likely to be strongly elongated alongthe vertical because of specific fluid rheological proper-ties [3]. One can wonder about the characteristics of thesound emission in this latter case, which is relevant in vari-ous practical situations. For instance, the bursting of elon-gated bubbles can be observed in the kitchen, when cook-ing sauces [4]. Volcanoes are large-scale natural systemswhich exhibit very similar phenomena; a large amount ofdissolved gas is released while lava rises up the magmatic

ensl00124832,

version1

16Jan2007

Author manuscript, published in "European Physical Journal. B, Condensed Matter and Complex Systems. 3, 54 (2006) 321-339"DOI : 10.1140/epjb/e2006-00450-0
http://dx.doi.org/10.1140/epjb/e2006-00450-0http://hal.archives-ouvertes.fr/http://prunel.ccsd.cnrs.fr/ensl-00124832/fr/http://dx.doi.org/10.1140/epjb/e2006-00450-0


2/19

2 The European Physical Journal B

conduit [5]; bursting of large gas bubbles is sometimes ob-served at the surface of lava lakes or at vents. The char-acteristics of the acoustic signal beeing likely to provideuseful pieces of information on the system geometry or

dynamics, geophysicists have long been recording soundproduced by volcanoes [6,7]. Because of the gas release,lava at the surface forms a foam having non-newtonianrheological properties. We can guess that one observes,at the surface of lava lakes, the bursting of bubbles thatare elongated along the vertical.

One can then wonder what pieces of information canbe extracted from acoustic measurements. We thus per-formed preliminary experiments in polymeric solutionsand observed that, in some cases, the bursting of the thinfilm that initially closes the bubble at the free surface ex-cites the elongated bubble body, which does not deformsignificantly during the sound emission and resonates likea tube [8]. We also observed a puzzling dependence ofthe acoustic energy on the bubble volume as well as, inthe same experimental conditions, a large scatter of theacoustic energy. In order to account for the observations,it is fundamental to deeply understand the sound emit-ted as the consequence of the bursting of a well-controlledthin-film which initially closes an overpressurized cavityof well-defined geometry. Thus, we consider a resonatingcavity consisting of a simple cylindrical-tube, closed at thebottom, which is excited by the bursting of a soap film atthe top (Fig. 1). Previous studies of soap film burstingshow that the typical film rupture is very fast [9], usuallyfaster than all the physical processes inside and outsidethe cavity (acoustic resonance, radiation and sound wave

propagation). Therefore, immediately after bursting, thecharacteristics of the sound waves, recorded inside andoutside the resonating tube, depend generally on the tubegeometry only. With this very simple experiment, we wantto analyze the different physical mechanisms responsiblefor the temporal and spectral characteristics of the signal.We pay special attention in describing carefully the ex-perimental procedures and propose in appendices all thetheoretical tools and references necessary for the under-standing of the phenomenon. We intend thus to providethe reader with useful and didactic material which, wehope, depending on his own interest, will help him in theinterpretation of more complex systems, from the crack-

ling of the champagne bubbles to the bursting of giantelongated bubbles on volcanoes.

2 Experimental setup and procedure

The experimental setup consists of a vertical tube (diame-ter = 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 lo-cated d = 5 cm away from the upper-output plane andoriented at 45 from the system axis. At bottom,the system is either closed by means of a solid rod or bya second microphone (microphone 377A10 + preamplifier426B03 + amplifier 482A16, PCB Piezotronics Inc.). Both

Fig. 1. Sketch of the experimental setup. The experimentalsetup consists of a tube (inner diameter ) drilled in a plex-iglass block. The length L of the cavity is tuned by changingthe position of the microphone 2 (or rod) at bottom. The pres-sure difference P across the film is controlled by means of asyringe pump and a pressure sensor. After the film bursting,the sound emitted outside the cavity is recorded with the helpof the microphone 1 whereas the pressure at the bottom, whichresults from the standing waves inside the cavity, is monitoredby using the microphone 2.

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

At the top, an initially-flat soap-film is produced bystretching a droplet of a water-soap mixture (Quix Limon)with the help of a razor blade. Then, a capillary con-nected to the main vertical tube makes it possible to in-troduce a chosen volume of air from a syringe pump intothe cavity limited by the lower wall and the film. A differ-ential pressure transducer (PX277-01D5V, OMEGA En-gineering Inc.) is used to measure the resulting pressure-difference, P, across the soap film which then takes theshape of a part of sphere. In addition, the system can beimaged either from top or side with the help of a fast-camera (HiSIS 2002, KSV Instruments Ltd).

Subjected to gravity, the film thins (drainage, [10])and, after a few tens of second, spontaneously bursts [11],which produces a characteristic sound. A digital oscillo-scope (54602B, Hewlett Packard) is used to monitor anddigitize the signals from the microphones (Fig. 2) and totransfer the raw data to a computer through the HPIBinterface. Subsequent analysis is performed numerically(Matlab, The MathWorks, Inc. and Igor Pro, Wavemet-rics, Inc.).

Thus, the experimental setup makes possible the anal-ysis of the sound emission associated with the bursting ofa curved soap film, which initially ends an almost cylin-drical cavity of length L and diameter filled with air

ensl00124832,

version1

16Jan2007


3/19

V. Vidal et al.: Acoustic signal associated with the bursting of a soap film 3

-0.1

0.0

0.1

-10

0

10

403020100

Time (ms)

Inneran

dou

terpressures

(Pa

)

Inside

Outside

Film bursting

2

1

0

100

06000400020000

Frequency (Hz)

Acous

ticenergy

(a.u.)

Inside

Outside

Fig. 2. Left: signals P(t) from the microphones. Right: corresponding energy spectra. The typical signals from the twomicrophones exhibit, immediately after bursting of the film (t > 0), damped oscillations (left-hand-side). Note that the higherharmonics are damped faster in the signal from microphone 1. One can notice that their amplitude, relative to that of thefundamental, 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(P = 13.1 Pa, = 10 mm, L = 8 cm, microphone 2 at the bottom, sampling rate 20 kHz).

at the overpressure P above the pressure of the outsideatmosphere. We shall report our experimental results andinterpretations in the following Section 3 and draw conclu-sions in Section 4. The theoretical background necessaryfor the interpretation of our observations is presented inAppendices A to D.

3 Experimental results and interpretation

In the present section, we report our experimental obser-vations, starting with the spectral analysis of the sound

emitted after the bursting of the film (Sect. 3.1). Wethen discuss the temporal evolution of the amplitude(Sect. 3.2), focusing on the physical processes causing thetemporal damping of the signal. In Section 3.3, we showthat the analysis of the initial amplitude of the harmonicsprovides us with interesting pieces of information aboutthe film rupture. Finally, the last Section 3.4 is dedicatedto the energy balance.

3.1 Resonant frequencies n

3.1.1 The fundamental frequency 0

From the experimental signals P(t) provided by the mi-crophone 1, we measure easily the fundamental frequency

0 of the acoustic wave in air, which corresponds to thelowest-frequency peak in the power spectrum (for in-stance, 0 1 kHz in Fig. 2). In order to analyze theeffect of the cavity geometry, we shall report the resultsin terms of the associated wavelength 0, defined to be0 c/0. Note that the velocity of sound in air, c, de-pends on the temperature T according to [12]:

c(T) =

RT

M(1)

where = 1.4, R = 8.1 4 J K1 mol1, and M =

29 g mol1. In our experimental conditions, the room tem-perature is about T 298 K and c 346 m s1.

Let us now tune the length L by changing the positionof the microphone 2 (or rod) and determine the corre-sponding wavelength 0 of the fundamental. We observethat 0 depends linearly on L according to:

0 = (L + L) (2)

where = 4.000.05 (Fig. 3) whatever the tube diameter in our experimental range (6 to 10 mm). On the otherhand, the length correction, L, is observed to dependon the tube diameter : for instance, we measure L =(4.2

0.1) mm in the case = 10 mm.

The tube, open at one end, constitutes a resonatorwhich is excited by the pressure drop associated with the

ensl00124832,

version1

16Jan2007


4/19


100

80

60

40

20

0

Wavelength

0(cm)

2520151050

Tube length (cm)

0

= 4 L = 1.7 cm

Fig. 3. Fundamental wavelength 0 vs. tube length L. Thefundamental frequency 0 exhibited by the system after break-ing of the soap film is associated with the wavelength 0, whichis found to depend linearly on the tube length L according to

equation (2) with L 0.42 cm (P = 10 Pa, = 10 mm,rigid wall at the bottom, sampling rate 20 kHz).

sudden breaking of the soap film (Sect. A.5). What one canhear in air is the acoustic wave radiated outside the cavity(Appendix B) that selects the frequency (Appendix A). Ina first approximation (Sect. A.1), we expect the pressurein the output plane to continuously equal the outside pres-sure (thus, a pressure node). At the bottom, we expect avelocity node and, accordingly, a pressure antinode. In thiscase, the cavity length equals one fourth of the fundamen-

tal wavelength and 0 = 4L. The experimental slope is in fairly good agreement with this theoretical expecta-tion (Fig. 3). However, the correction length L is finite,as the pressure node does not exactly locate in the out-put plane. Indeed, because of the radiation of the acousticwave in the output plane, one must consider the effectivetube length L = L + L where one expects L = 4/3in the case of a flanged aperture [13,14] (Eq. (A.11)). Oneobtains a similar correction L = 0.3 , which is onlyslightly smaller than the latter, in the case of an unflangedaperture [13,15]. Experimentally, the correction length isfound to be in fairly good agreement with the correctionexpected for a flanged aperture (we measure L 0.42 cmfor = 10 mm).

3.1.2 Higher-order harmonics n (n = 0)

We point out that, in the theory (Sect. A.2.2), the cor-rection L is expected not to depend on the wavelength, so that the larger resonant frequencies n are harmon-ics of the fundamental 0, n = (2n + 1)0 associatedwith n = 4L/(2n + 1), where L = L + L. The pre-diction is well satisfied experimentally. Indeed, we observeclearly that the higher harmonics have frequencies thatare odd multiples of the fundamental frequency: 30, 50,70, . . . (Fig. 4).

10-5

10-4

10-3

10-2

10-1

100

Acousticenergy(a.u.)

80006000400020000

Frequency (Hz)

5078 Hz3046 Hz1015 Hz 7110 Hz

Fig. 4. Energy spectrum (microphone 1). We report, in logscale, the data from microphone 1 (Fig. 2). The vertical dashedlines clearly point out that the harmonics frequencies satisfyn = (2n + 1)0 (P = 13.1 Pa, = 10 mm, L = 8 cm,microphone 2 at the bottom, sampling rate 20 kHz).

3.1.3 Conclusion on the resonant frequencies

The characteristic frequencies of the sound wave emittedafter the film bursting depend on the geometry of theresonant cavity only. Indeed, the signal recorded by themicrophone 1 mainly contains the several harmonics ofthe fundamental 0, which is associated with the effectivelength L = L + L: L is the length of the cavity, fromthe bottom to the aperture plane, and L a length correc-tion, due to the radiation at the open end, which mainly

depends on the tube diameter . In the next Section 3.2,we discuss the temporal envelope of the signal and, morespecifically, how the amplitude of each of the harmonicsvanishes.

3.2 Amplitude-damping characteristic-times n

3.2.1 Qualitative observation

As can be seen on the signals P(t), especially from micro-phone 1 (Fig. 2), the amplitude of harmonics associated

with larger frequencies decreases faster. Indeed, the signalclearly contains several harmonics at short time (typicallyt < 8 ms) and only the fundamental afterwards (typicallyt > 8 ms). Measurements performed with a long enoughcavity make it possible to determine with accuracy thecharacteristic time n, over which the amplitude of theharmonic n vanishes, as a function of the correspondingfrequency n.

3.2.2 Analysis procedure

The analysis procedure is as follows: from the position ofthe peaks in the power spectrum (Fig. 4), we determinethe fundamental resonant frequency 0. We then write the

ensl00124832,

version1

16Jan2007


5/19


signal from microphone 1 for t > 0:

P(t) =

n=0An sin(nt + n)exp(t/n) (3)

where n = 2(2n + 1)0. The interpolation of the ex-perimental data with equation (3) provides us with thecharacteristic damping-time, n, of each of the harmonicsthat exhibits a significant amplitude (thus generally upto n = 2). The agreement between equation (3) and theexperimental data is excellent as shown, as an example,in Figure 5.

3.2.3 Experimental results

Each of the n gives the experimental value of the charac-

teristic damping time, d, at the corresponding frequency = (2n + 1)0. Collecting the whole set of experimentalmeasurements obtained for the various hamonics and dif-ferent tube lengths L, we obtain the experimental d as afunction of the frequency . We checked that the exper-imental data reported in Figure 6 do not depend on theinitial overpressure P.

3.2.4 Discussion

The radiation at the open end alone would lead to d 3 (Sect. A.2.3), which is not observed experimentally,

especially for small frequencies. We have evaluated otherpossible sources of dissipation [among them, the partialreflection at the closed end (Sect. A.3) and the thermaland viscous dissipation in the boundary layer at the ver-tical wall of the cavity (Sect. A.4)]. We estimated that,in the case of a rigid wall at bottom (rod), the dampingof the acoustic wave due to the partial reflection at theclosed end is negligible (this contribution must be takeninto account when microphone 2 is placed at the bottom).Thus, taking into account only the radiation at the openend and the viscous and thermal dissipation in the bound-ary layer at the vertical wall of the cavity, we estimate thecharacteristic damping-time to be:

1

d= 2

[1 + ( 1)P 12r ]

1/2 + 22

2

c23

2n + 1(4)

where = 1.5 105 m2 s1 denotes the kinematicviscosity of air, Pr 0.7, the Prandtl number, and = 1.4, the specific-heat ratio. Equation (4) gives, with-out any adjustable parameter, the characteristic damping-time which we find to be in rather good agreement withour experimental measurements (Fig. 6).

The damping of the standing wave inside the cavity ismainly governed by both the viscous and thermal dissi-pation at the wall, which dominates at small frequencies,and the radiation at the open end, which dominates atlarge frequencies. Whatever the dominant dissipation pro-cess, the harmonics having larger frequencies are damped

-0.1

0.0

0.1

Pressure(Pa)

654321

Time (ms)

Fig. 5. Signal P(t) from the microphone 1 vs. time t. Thickgrey line: signal from the microphone. Thin black line: inter-polation with equation (3). The experimental data are fromFigure 2 (P = 13.1 Pa, = 10 mm, L = 8 cm, microphone 2at the bottom, sampling rate 20 kHz).

0.1

2

4

6

1

2

4

6

10

2

Dampingtimed

(10

-3s)

121086420

Frequency (kHz)

and n=2

and n=1

and n=0 ( =0)

Fig. 6. Characteristic time d vs. frequency . The data areobtained from the fundamental and its harmonics for variousvalues of the cavity-length L, the diameter = 8 mm beingkept constant (squares: fundamental n = 0; diamonds: firstharmonic n = 1; open circles: second harmonic n = 2). Thecharacteristic time d decreases when the frequency increasesas expected from equation (4) [the error bars are estimatedfrom the interpolation of the experimental data with Eq. (3)].The lines are obtained from equation (4) without any ad-

justable parameter (solid line: n = 0; long-dashed line: n = 1,short dashed line: n = 2). The thin continuous line points outthe contribution of the radiation alone obtained from equa-tion (4) with = 0 (P = 18 Pa, = 8 mm, rigid wall at the

bottom).

faster. This is the reason why the sound, recorded outside,contains several harmonics right after the bursting of thefilm whereas only the fundamental is sensed later.

At this point, we know that the frequencies recordedby the microphone correspond to the resonant frequen-cies of the cavity and how the amplitude of the variousharmonics vanishes when time increases. In the followingSection 3.3, we discuss how the resonator is excited bythe pressure drop associated with the film bursting andwe show that a detailed analysis of the acoustic signalmakes it possible to determine accurately the character-istic time of the resulting pressure drop in the apertureplane.

ensl00124832,

version1

16Jan2007


6/19


-0.1

0.0

0.1

-10

0

10

151050

Time (ms)

-0.1

0.0

0.1

-10

0

10

151050

Time (ms)

Inneran

dou

terpressures

(Pa)

InsideShort rupture-time Long rupture-time

Inside

Outside Outside

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 P to P 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 doesnot drop to P immediately after bursting. Accordingly, the amplitude of the acoustic signal outside is smaller (P 13 Pa, = 10 mm, L = 8 cm, microphone 2 at the bottom, sampling rate 20 kHz).

3.3 Spectral analysis of the pressure drop associatedwith the film bursting

In this section, we obtain, from the detailed analysis ofthe coefficients An and n which appear in equation (3),values of the Laplace transform of the pressure drop inthe output plane and, consequently, the associated char-acteristic time, .

3.3.1 Qualitative observations

For identical initial conditions (given L, and P), val-ues of the initial amplitude of the acoustic signal can bevery scattered. Even if the resonant frequencies are not af-fected, the acoustic signal varies significantly both in am-

plitude and shape, inside and outside the cavity (Fig. 7).This feature points out the fundamental role of the filmbreaking process in the excitation of the acoustic waves.

Using the fast camera, we observe the dynamics of thesoap-film breaking, when not too fast, and we monitor theacoustic signal. We report in Figure 7, the acoustic signalsobtained for two very different characteristic times of thefilm rupture. If the film rupture time is short, short enoughfor the soap film to disappear in the time between two im-ages (1/4000 s), the amplitude of the signal inside the cav-ity initially drops down from +P to P (Fig. 7, left).We note that this case corresponds to the rupture of avery thin film, which has reached the minimum accessiblethickness [common black film (CBF, [10]) in our experi-mental conditions] before bursting. If the film rupture timeis longer (in the same experimental conditions but, here,

the drainage is not complete when the film breaks), theamplitude of the acoustic signal inside the cavity, rightafter bursting, does not drop down to

P anymore. Ac-

cordingly, the amplitude of the acoustic signal outside issmaller (Fig. 7, right).

In order to point out the variability of the amplitude ofthe acoustic signal from one experiment to another if oneperforms the experiment without controlling the thicknessof the soap film when it breaks, we report measurementsof the initial amplitude of the acoustic signal inside thecavity for two sets of experiments performed in the sameexperimental conditions (Fig. 8). One observes that theamplitude of the first oscillation is often much smaller thanP, especially for the shorter cavity.

In the next Section 3.3.2, we report experimental re-sults obtained for the bursting of CBFs. In this latter case,

the measurements are reproducible and we report a thor-ough analysis of the signal and draw conclusions.

3.3.2 Spectral analysis

The theoretical analysis (Sect. A.5.4) demonstrates thatthe amplitude and phase of the acoustic signal inside thecavity are related to the Laplace transform of the pressuredrop P(t) in the output plane which results from the film

bursting. In what follows, we define f(s) L[P(t)/P1]where L denotes the Laplace transform. As the acousticwave outside the cavity results from the radiation at theopen end of the acoustic wave inside (Sect. B), we canestimate that the amplitude An and the phase n of the

ensl00124832,

version1

16Jan2007


7/19


20

15

10

5

0

Initialamplitude(Pa)

20151050

Initial over ressure P Pa)

Fig. 8. Initial amplitude at the closed end vs. P. We reportthe amplitude of the first oscillation of the pressure signal atthe closed end of the cavity as a function of the overpressureP 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, islargest for the shortest cavity length ( = 10 mm, microphone2 at the bottom).

10-7

10-6

10-5

|f(j

)|(a

.u.)

6 7 8 9

104

2 3 4 5 6 7

rad/s

^

1/

Fig. 9. Amplitude |f(jw)| as a function of . The experi-mental data are from the fundamental (full squares), the first(open squares), second (full diamonds) and third (open dia-monds) harmonics. The amplitude is constant below and de-creases like 5 above 1

(vertical solid line). From the cut-off

frequency, we obtain 40 s (P = 18 Pa, = 8 mm).

signal from the microphone 1, located outside, satisfy:

An

nf(jn) (5)n = arg[f(jn)] n

t0 +

d

c

(6)

where d denotes the distance between the microphone andthe output plane [we introduce the time delay t0 as, ex-perimentally, the origin of time is taken when the signalexceeds a given threshold value and not when the filmbreaks. We thus expect t0 to be of the order of magnitudeof the signal period 2/0].

From the interpolation of the acoustic signal outside,we measure An and n and, then, report the amplitude(Fig. 9) and the phase (Fig. 10) of f(jw) as a function ofthe frequency . At small frequency, both the amplitudeand the phase are constant, whereas the amplitude de-creases significantly (like k with k 5) above a typical

-6

-4

-2

0

(j

)(rad)

6 7 8 9

104

2 3 4 5 6 7

rad/s

1/

Fig. 10. Phase (j) arg[f(jn)] as a function of . Theexperimental data set is the same as in Figure 9. In order toget a constant phase for < 1

, we take t0 = 2.3 10

4 s(P = 18 Pa, = 8 mm).

value 1/ of the frequency . We observe a decrease in the

phase arg[f(jn)] around the value 1/ but the scat-ter of the experimental data at large frequency avoids anaccurate determination of the limit reached when .Nevertheless, the experimental results clearly exhibit acharacteristic time associated with the pressure dropat the open end. We estimate 40 s.

At this point, one could be tempted to recover thecharacteristic time in the signal recorded at the closedend of a long cavity. However, because of the dissipationat the side walls (Sect. A.4) which leads to a dispersivepropagation (the harmonics have different velocity), the

width of the initial pressure front, generated by the burst-ing of the soap film, increases during the propagationtoward the closed end. As a consequence, the temporalbehavior of the pressure from microphone 2 does not im-age what happened at the open end. As an experimentalproof, we report results obtained for two different cavity-lengths (8 and 23 cm) and different viscosities of the soapfilm (Fig. 11).

The viscous soap films are obtained by mixing water,soap, sugar and glycerine. We observe with the fast cam-era that the characteristic rupture time is, at least, 2 to 3times larger in this case. However, the first pressure dropmeasured at the closed end of the cavity looks very similar

in the cases of the regular and viscous soap films, in thecase of the longest cavity (L = 23 cm): the initial pres-sure drop associated with the film bursting is hidden bythe widening of the pressure front during the propagationtowards the closed end. Note also that the initial ampli-tude of the signal does not depend on the characteristictime of the pressure drop as, because of the choice ofa long cavity, remains smaller than 1/0. Making useof a different cavity length (L = 8 cm), we estimate fromthe data reported in Figure 11 the widening rate to be 15 s/cm.

Even if the pressure drop at the bottom does not ex-actly image the pressure drop at the open end, we can getan estimate of the rupture time in the case of a shortcavity. For each signal, we determine the time drop nec-essary for the pressure at the bottom to reach the first

ensl00124832,

version1

16Jan2007


8/19


-10

-5

0

5

10

Pressure(Pa)

2.52.01.51.00.50.0

Time ms

Fig. 11. Pressure P(t) from microphone 2. Black solid line:regular soap film and L = 23 cm; thick grey line: viscous soapfilm and L = 23 cm; dashed line: viscous soap film and L =8 cm. We report the first pressure drop (the origin of timeis arbitrary). Even if the characteristic rupture time is 2 to3 times larger for the viscous soap film (observation with thefast camera), one observes that the initial pressure drop at theclosed end is almost the same for the regular and viscous soapfilms. By contrast, for the same characteristic rupture time,the initial pressure drop at the bottom drastically depends onthe tube length L (P = 10 Pa, = 10 mm).

1.0

0.8

0.6

0.4

0.2

0.0Initial

relative-amplitude

5004003002001000

Estimated (s)

Fig. 12. Initial relative-pressure amplitude vs. estimated rup-ture time . The initial acoustic-signal amplitude is normalizedto P. The rupture time is estimated from the pressure dropat the bottom. In this experiment, 1/0 43 s (L = 2 cm, = 10 mm).

minimum. We estimate to be about (drop

L)/2. Wereport in Figure 12 the initial amplitude of the pressuresignal, normalized to P, as a function of the estimated .We observe, whatever the initial P, a drastic decreaseof the amplitude when the rupture time increases, whichproves that the scattering of the data observed in Figure 8is due to the variability of the rupture time.

Finally, we can compare the characteristic time mea-sured in the experiment to an estimate of the typical time,typ, expected for the opening of a film stretched over anaperture of radius /2. The opening of the film is drivenby the surface tension and is limited either by the viscousdissipation [16] or by the inertia, this latter case beingrelevant in our experimental conditions. As a simple de-termination of the opening velocity appeared to be rathersubtle [11], we detail here the model [17]. Let us now con-

Fig. 13. Sketch of the opening pore.

sider a pore of radius r which grows with the velocity vfrom the center of a frame (diameter /2, Fig. 13). The liq-uid, initially located in the central disk of radius r, formsa rim of mass m = r2e, which grows with time (e standsfor the film thickness and for the density of the liquid).

One could be tempted, at this point, to write the energyconservation by simply balancing the gain of surface freeenergy 2 r2 ( 30 2 mN/m stands for the surfacefree energy of the liquid-air interface, the surface energyof the film is twice larger) with the kinetic energy 1/2mv2.However, if one considers the problem in the frame of ref-erence of the rim (Fig. 13, inset), one can see that kineticenergy is continuously injected in inner flows (these flowsare likely to be damped by the viscous dissipation but it isnot nessecary to solve the hydrodynamical problem insideto go farther). We can easily see that the amount of ki-netic energy which is introduced per unit time in the rimwrites 2erv1/2v2 (2erv corresponds to the volume ofliquid, having the kinetic energy 1/2v

2

per unit volume,which enters the rim per unit time). As a consequence,the energy balance must be written:

2 2rv = ddt

12

mv2

+ 2erv 12

v2. (7)

Taking into account the dependence of m on r, we obtainthe constant velocity of the rim v =

2/e. Note that

one half of the energy is dissipated and that an estimateof the velocity obtained without introducing the dissipa-tion would be wrong by a factor

2 [11]. Assuming that

typ corresponds to the time the torus needs to reach the

edge of the aperture, we get typ =

2

e/2. We canmake use of typ = = 40 s to determine an estimate ofthe unknown film thickness. We get e = 5.7 nm, which issmall but of the order of magnitude of the CBF thicknessreported in the literature. We do not expect a quantita-tive agreement between typ and as, first, there is noexperimental proof that the bursting of the soap film cor-responds to the opening of a pore at the center; second, asthe relation between the dynamics of the pore opening andthe associated pressure drop can be rather complex; and,finally, because the initial curvature of the soap film is nottaken into account. Nevertheless, we note that typ and are, at least, of the same order of magnitude, which isan indication that the acoustic measurements do providea realistic estimate of the characteristic time associatedwith the pressure drop at the open end.

ensl00124832,

version1

16Jan2007


9/19


3.4 Energy transferred to the acoustic wave

The sudden breaking of the soap film produces a pres-sure drop inside the cavity. The total potential energy Ep

released can be evaluated by considering the adiabatic ex-pansion of the volume V = (/2)2L of gas when the in-

ner pressure drops down by P. One obtains Ep =12

V P2

c2 .

In the present section, we report how the energy Ep istransferred to the acoustic signal after the bursting of thesoap film. Before reporting our experimental results, wediscuss in Section 3.4.1 what the microphone 1 senses. Thebehavior of the total energy is analyzed in Section 3.4.2.

3.4.1 Preliminary remarks

The microphone 1 senses the acoustic signal P(t) at a

given point outside the cavity. As the acoustic wave out-side the cavity is produced by a source of finite spatialextension (the surface area of the open end), the acousticintensity I is likely to depend on the position of the mi-crophone (distance and orientation) in a rather complexway.

Experimentally, the microphone is located at a finitedistance d = 5 cm from the open end. One could worrythe distance between the open end and the microphoneto alter the amplitude of the various harmonics relativeto each other. However, in our experimental conditions,the wavelength associated with the resonant frequenciesis large compared to the cavity diameter and, as shown in

Section B.1, the far fieldapproximation applies. Thus, theamplitude associated with each of the modes decreases like1/d where d is the distance to the open end, independent ofthe frequency. Moving the microphone toward or outwardthe tube end only rescales the power spectrum (as 1/d2)and does not modify the shape of the envelope.

On the other hand, the microphone is oriented atabout 45 degrees from the cavity axis and we could alsoworry the spatial structure of the acoustic wave to al-ter the measurements. We show in Section B.2, that theangular dependence of the acoustic wave outside the cav-ity leads to a slight dependence of the acoustic intensityon the frequency for a given orientation of the micro-

phone [Eq. (B.5)]. However, the correction is less than10% in all our experimental conditions ( < 10 mm, < 10 kHz). In addition, we estimate from the first zero ofthe form factor that the modes having frequencies higherthan about 40 kHz will not be sensed by the microphone.

The microphone is located at a finite distance from thetube aperture. One must, in principle, consider the spher-ical nature of the acoustic wave outside when estimatingthe acoustic energy from the pressure signal P(t) providedby the microphone 1. However, we show in the appendix(Sect. A.5.7) that the total acoustic energy measured out-side the cavity is given, to within better than 2.5% in allour experimental conditions, when neglecting the curva-ture of the wave front.

Finally, the typical lateral size of the experimentalsetup 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 ra-dius d. As a consequence, we estimate the total acousticenergy outside EoutT to be simply given by

EoutT 4d2

c

t=0

P2(t)dt, (8)

where P(t) is the pressure signal from the microphone 1.We expect the main source of error to be the integrationover the sphere which leads to the prefactor 4. Even ifthe potential error is there difficult to estimate, we canguess that EoutT cannot overestimate the acoustic energyby more than 15% as we checked experimentally that theamplitude of the acoustic signal does not depend signif-icantly on the angle the microphone makes with thecavity axis, up to 135.

3.4.2 Measurements and discussion of the energy balance

We already noticed that the soap film is likely to burstbefore having reached its minimum accessible thicknesswhich leads to a scatter in the initial amplitude of theacoustic signals. We measured the acoustic energy EoutT fordifferent initial P and report the data obtained when thefilm has reached its minimum accessible thickness (CBF)before bursting. In this case, the energy is maximum andthe measurements are reproducible (Fig. 14).

We observe that EoutT P2 for small P in agree-ment with Ep

P2. However, we note a departure from

the quadratic law for larger P. This feature is qualita-tively explained by a change in the geometry of the film.Indeed, the film is almost flat at small P and its burst-ing results in an almost-planar pressure-front propagatingtoward the cavity. In this case, the geometry of the frontmatches that of the planar modes inside the cavity andthe energy transfer is efficient. By contrast, at large P,the film bends significantly and its bursting results in acurved pressure-front propagating toward the cavity. Thegeometry of the front does not match that of the planarmodes inside the cavity anymore and the efficiency of theenergy transfer drops down.

More precisely, in the appendix A.5.7, we determine

the input transmission coefficient for the energy:

Ti =

1

x

6j + x

1 cos 0

1 ejx tan 02

2

(9)

where 0 is defined by P = (8/)sin 0 and x = /.The dependence of Ti on P accounts for the non-trivialbehavior of EoutT . Indeed, from the dynamics of the pres-sure drop at the open end and from Ep, the total en-ergy transferred to the resonant modes inside the cavity,EinT , can be obtained by taking into account the trans-mission of each of the harmonics. In addition, we expectEout

T Ein

T

, provided that only P is varied. We observethat the dependence of EoutT on P, predicted from Ti,compares nicely with the experimental observations (lines

ensl00124832,

version1

16Jan2007


10/19


30

20

10

0

m

(%)

3020100

e(%)

5

4

3

2

1

0

AcousticenergyE

ou

t (10

-10J)

403020100

Pressure P (Pa)

T

= 10 mm = 8 mm = 6 mm

Fig. 14. Total acoustic energy outside EoutT vs. initial over-pressure P. Whatever the cavity diameter , the acoustic en-ergy scales like P2 for small P whereas a departure fromthe quadratic law is observed for large P. The bending of the

film accounts nicely (lines) for the experimental observations(L = 5.5 cm). Black arrows indicate the maximum P = 8/for each of the diameters. In inset is plotted the percentage ofenergy we would measure outside (from the fit of the experi-mental points in this figure), as a function of the percentageof the energy expected to be measured, when excluding thegeometrical effects. Note the good agreement between theoryand the experimental observations.

in Fig. 14). The agreement proves that the geometry ofthe film is responsible for non-trivial dependence of theacoustic energy on the initial overpressure.

To go farther, we deduce from the interpolation of the

experimental data the fraction = (1/Ti)EoutT /Ep of thetotal energy Ep, that would be transferred to the acousticmodes outside the cavity in absence of the geometricaleffects due to the bending of the film. Measured valuesm are reported in Table 1.

We can compare m to another estimate ofobtained,in addition, by considering the viscous dissipation and ra-diation only. On the one hand, assuming that the burst-ing of the film results in an exponential decrease of thepressure in the output plane with the characteristic time, one can estimate the amount of energy initially trans-ferred to the acoustic modes inside the cavity (Sect. A.5.3)in absence of geometrical effects:

EinT =

1 20

tanh

20

Ep Ep. (10)

Note that the fraction depends on the rupture time ,which explains the scatter of the experimental data ob-tained when the film thickness is not controlled. In thecase of a common black film, we estimated 40 s for = 8 mm, which leads to 0.75. As we expect todepend linearly on , we can also report estimates of for = 6 and 10 mm in Table 1.

On the other hand, the total acoustic energy recoveredoutside, EoutT , corresponds to the part of the energy E

inT

which is radiated and not dissipated at the walls. One canshow that the fraction Zn /(Zn + vn ) of the energy initiallytransferred to the mode n inside the cavity is dissipated

Table 1. Percentages . Data in bold font are experimentalvalues, whereas other data are estimates (see text).

6 mm 8 mm 10 mm

81% 75% 68%v 16% 30% 46%

e 13% 22% 31%

m 14% 23% 26%

at the cavity wall. We remind that vn and Zn are the

characteristic times associated with the dissipation at thewall and with the radiation at the open end, respectively.Defining v = EoutT /E

inT and summing the contribution of

all the harmonics, we obtain the expected value of v for = 8 mm. Taking into account the scaling behaviors of

vn and Zn on , we get the estimates reported in Table 1for = 6 and 10 mm.

Thus, from the dissipation at the wall and from thefinite rupture time , we expect, in absence of the geomet-rical effects due to the bending of the film, EoutT eEp,with e = v, which compares nicely to m (see in-set Fig. 14). However, note that the agreement must beconsidered with caution, considering the uncertainty inmeasuring the characteristic time and in the surface ofintegration in equation (8).

As a conclusion of the energy study, we would like tounderline again the strong influence of the film rupturetime on the energy conversion. Hence, direct measure-

ments of the acoustic energy do not provide any infor-mation on the energy initially loaded inside the cavity ifthe film rupture is unknown. Any attempt to do so wouldlead to false interpretations.

4 Conclusion and perspectives

We reported an experimental study of the acoustic signalassociated with the bursting of a thin liquid film at theopen end of a cylindrical cavity.

Let us first explain qualitatively why the signals frommicrophones 1 and 2 look so different. The radiation at the

open end and the propagation inside the cavity are respon-sible for the drastic differences in the signals recorded atthe bottom of the cavity and outside. Indeed, the trans-mission coefficient associated with the radiation is propor-tional to the frequency n so that, at a given time t, theamplitude of the harmonic n, relative to that of the fun-damental, is (2n + 1) times larger outside than inside thecavity. Moreover, the propagation along the cavity lengthmakes the relative phase of two successive harmonics ro-tate by the angle , so that, while in phase at the bottom,they have opposite phase in the output plane. The har-monics are thus clearly visible in the signal outside thecavity while they are hardly distinguishable in the signalrecorded at the bottom. In addition, the harmonics areonly observed at short times as they are damped fastermainly because of the radiation.

ensl00124832,

version1

16Jan2007


11/19


Our experimental findings can be summarized as fol-lows:

The acoustic signal exhibits a well defined fundamen-tal frequency 0 which is mainly governed by the cav-

ity length L. However, because of the radiation at theopen end, the resonant length L, associated with thefundamental frequency 0, does not exactly equal thecavity length L but L + L where L is a fraction ofthe tube diameter .

For long cavities, we observe a banded spectrum, withfrequency peaks n = (2n + 1)0, where 0 is the fun-damental frequency.

Because of the radiation at the open end and of thethermal and viscous dissipation at the cavity walls,the acoustic signals we record inside and outside thecavity are damped in time. The associated character-istic times, we measure experimentally, are in fairly

good agreement with the theoretical predictions with-out any adjustable parameter. The spectral content of the acoustic signals depends on

how the cavity is opened. As a consequence, the care-ful analysis of the acoustic signal recorded outside thecavity makes it possible to determine accurately thecharacteristic time associated with the initial pressuredrop at the open end of the cavity.

The total amount of energy recovered in the acousticsignals depends drastically on the film curvature andon the dynamics of the film bursting.

We would like to emphasize once again the didactic goalsof the work we reported herein. We were only aiming at the

deep understanding of all the physical processes that gov-ern the acoustic signals associated with the sudden open-ing of an under- or overpressurized cavity.

Focusing on one experimental situation, we can con-clude that we can make use of the resonant cavity forperforming the spectral analysis of the pressure drop asso-ciated with the film bursting. We are taught, for instance,that the bursting of a common black film, stretched overan 8 mm-in-diameter aperture, is associated with a char-acteristic time of about 40 s.

Generalizing to other application fields, we can makeuse of our findings to better understand acoustic signalsrecorded in the nature. For instance, our study provides

useful tools for the interpretation of the acoustic signalsrecorded on field by the volcanologists. First, the spec-tral content (frequencies) gives direct access to pieces ofinformation about the geometry of the system: the res-onant length relates to the bubble length whereas, fromthe surface, only the diameter of the bubble head or ofthe conduit can be observed. In the case of large bubblesbursting at the surface of a lava lake, the aspect ratio ofthe bubble depends on the rheological properties of thelava so that listening to the volcano makes it possible toaccess some rheological properties of the fluid. We pointout that our experimental geometry is also relevant forinterpreting the acoustic signal from the bursting of a gi-ant elongated bubble at the top of a volcanic vent [7].In this case the resonant length informs us about the con-duit length and the damping characteristic-time about the

processes that lead to the energy loss in the system, whichgives constraints for modeling the resonant cavity [viscos-ity 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 theacoustic methods: as they drastically depend on the film-rupture characteristic-time, which is neither a controllednor a measured quantity in the field, the acoustic-signalamplitude and/or energy cannot be used to measure thetotal energy released during the bursting of a single bub-ble. Any attempts to do so would lead to misinterpretationof the field data.

The authors are very grateful to S. Vergniolle, M. Ripepe andD. Legrand for having provided us with exciting and usefulpieces of information about the acoustics of volcanoes. We also

highly thank S. Job, S. Douady, D. Constantin and B. Castaingfor 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. Geminard wouldlike to thank the Centre National de la Recherche Scientifique(CNRS, France) and T. Divoux the Ecole Normale Superieure(ENS, Paris) for supporting the research of their members andstudents in foreign laboratories.

Appendix A: Theoretical background

A.1 The perfect resonant cavity

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

P 1c2

2P

t2= 0. (A.1)

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

v

t

=

P. (A.2)

Let us consider the cylindrical cavity [length L (x [L, 0]), diameter ] initially (t < 0) filled with air atthe pressure Pa + P. We are aiming at the description ofthe evolution of the pressure field inside the cavity result-ing from the opening of the cavity at one end (x = 0). Welimit our study to the planar waves, P(r, t) = P(x, t), thatcan propagate in the cavity and ignore the higher-order,non planar, modes that could exist in the cavity but thatare not relevant for our purpose [18].

Denoting P(x, s) the Laplace transform of P(x, t), de-fined to be [19]

P(x, s) =

0

P(x, t)estdt, (A.3)

ensl00124832,

version1

16Jan2007


12/19


we obtain from the general equation (A.1),

P(x, s) s2

c2P(x, s) = s

c2P(x, 0) 1

c2P

t(x, 0). (A.4)

Taking into account the initial condition P(x, 0) = P,x [L, 0[, the general solution to equation (A.4) satis-fies

P(x, s) = A+esxc + Ae

sxc +

P

s(A.5)

where the constants A+ and A, which are functions of s,are given by the boundary conditions at both ends.

Assuming that the reflection of the sound wave at theclosed end (plane x = L) occurs without any loss, theparticle velocity along the x-axis, vx(L, t) = 0 (t). Fromthe Laplace transform of equation (A.2),

svx(x, s) vx(x, 0) = 1

P(x, s), (A.6)

and from the initial condition, vx(x, 0) = 0 (x [L, 0[),we get P(L, s) = 0 (s). As a consequence, A+ = Aand, thus,

P(x, s) = A(s)coshs(x + L)

c

+

P

s(A.7)

where the amplitude A, which is a function of s, is givenby the boundary condition at the open end.

In a first approximation, once the cavity is opened, theoverpressure at x = 0 instantaneously vanishes so that

P(0, t) = P (t < 0) and P(0, t) = 0 (t > 0) (we thusconsider in this section that the output impedance at theopen end is zero). As the consequence, the Laplace trans-

form of the pressure at the open end P(0, s) = 0 and, fromequation (A.7),

P(x, s) =P

s

cosh

sc L cosh sc (L + x)cosh

sc 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,sn = j(2n + 1)

c2L

(n integer ranging from

to +

),

which cancel the denominator cosh

sc L

in equation (A.8).Thus, the perfectcavity (reflection without any loss at theclosed end and zero output impedance) exhibits a series ofresonant frequencies n (2n+1) c2L which are associatedwith the wavelengths n cn = 4L2n+1 .

Once the resonator has been excited due to the openingof the cavity at one end, one could expect to hear the res-onant frequencies n outside the cavity. However, as thefrequencies n correspond to perfectly resonant modes, bydefinition, the amplitudes of these modes do not decreasewith time and, accordingly, no acoustic wave can existoutside the cavity. This conclusion, which contradicts theexperimental observations, is the consequence of the spe-cific boundary condition at the open end that we chose inthis section: we considered that the pressure in the output

plane is zero for t > 0 (zero output impedance). As dis-cussed in Appendix B, this assumption is not experimen-tally satisfied and the resonant modes can indeed escapethe cavity, producing the acoustic wave that we can hear

outside. In the next Section A.2, we will consider a morerealistic boundary condition at the open end and analyzethe 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,Zr, is finite (Appendix B) and that there exists acousticwaves inside and outside the cavity. Writing the continuityof the pressure and velocity fields in the output plane (x =

0), we get:

P(x, s) =P

s

1 cosh

sc (L + x)

cosh

sc L

+ Zr(s)c sinh

sc L

(A.9)

which reduces to equation (A.8) in the limit Zr(s) 0.

A.2.2 Resonant frequencies

The resonant modes are again obtained by determiningthe poles of P(x, s). Using the expression (B.7) of the

acoustic impedance Zr determined in the case of a flangedaperture, we get the poles sn which satisfy

1

tanh

snc L

= Zr(sn)c

= 43c

sn +2

8c2s2n. (A.10)

Neglecting the second order term in equation (A.10), weobtain the corresponding wavelengths

n =4

2n + 1

L +

4

3

(A.11)

where n is an integer ranging from 1 to .Thus, to the first order in /L, the radiation at the

open end leads to a slight increase in the wavelength nand, accordingly, to a slight decrease in the frequencyn = 2c/n. In the case of a flanged aperture [20], theeffective resonant length of the cavity is L = L + 4

3

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

A.2.3 Damping of the resonant modes

Taking into account the second order term in equation(A.10) and defining Zn by sn jn 1/Zn , we get

Zn =8cL

22n= 4

c2

22n + 1

3n. (A.12)

ensl00124832,

version1

16Jan2007


13/19


One can notice that Zn independently depends on thecavity 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 n, with a characteristic time Z

n ,which scales like 3n . Equation (A.12) holds true in thecase of a flanged aperture. In the case of an unflangedaperture [13], the characteristic time associated with theradiation is expected to be twice longer, 2Zn [the real partof 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 so-lution for the pressure field inside the cavity:

P(x, t) =4P

n=0

(1)n2n + 1

cosn

c(L + x)

cos(nt)exp

t

Zn

(A.13)

in which we neglect terms having amplitudes smallerthan or of the order of (/L)2P. Note that, from equa-tion (A.13), P(x, 0) = P, x [L, 0[.

Equation (A.13) describes the inner pressure field inthe case of an instantaneous opening of the cavity. In-

deed, we consider the initial condition of constant pres-sure inside the cavity for t < 0 (the output impedanceZ is thus infinite for t < 0) and assume that the out-put impedance is Zr for t > 0. Thus, we assume that theoutput impedance drops down from the infinity to Zr inan infinitely short time. We shall discuss the effect of thecharacteristic time of the cavity opening in Section A.5.However, at this point, we do prefer to analyze first twoadditional physical processes which lead to the damping ofthe resonant modes inside the cavity: the partial reflectionat the closed end (Sect. A.3) and the viscous dissipationat the cavity walls (Sect. A.4).

A.3 Effect of partial reflection at the closed end


Up to now, we considered that the reflection at the closedend of the cavity occured without energy loss. However,because of finite acoustic impedance of the material theframe is made of or, especially, when using the micro-phone 2 at the bottom, part of the acoustic energy is lostin the plane x = L.

Let us assume that a planar wave propagating towardthe closed end is reflected partially as a planar wave prop-agating toward the open end with the reflection coefficientr0. Assuming zero output impedance at the open end, we

get:

P(x, s) =P

s

1 cosh

sc (L + x)

+ 1+r0

1r0 sinh

sc (L + x)

1 cosh

sc L

+ 1+r01r0

sinh

sc L .

(A.14)

Note that equation (A.14) reduces to equation (A.8) inthe limit r0 1.


Defining r0n such that sn jn 1/r0n we get, from thereal part of the poles, nr0n =

21r01+r0

(n ). Thus,the harmonics amplitude decreases exponentially with thecharacteristic time

r0n =1 r01 + r0

1

4n. (A.15)

Note that, assuming that r0 is real, the reflection at theclosed 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 theacoustic wave due to viscous dissipation can usually be

neglected. Indeed, the characteristic damping length foran acoustic wave in air is of about 300 m. However, theplanar waves propagating inside the cavity are not com-patible with the condition of zero velocity along the x-axisat the side walls and dissipation occurs in the boundarylayer, in which air is subjected to important shear.

A.4.1 Physical background

The dynamical equation including the kinematic viscos-ity writes

v

t

=

1

P + v (A.16)

which reduces to equation (A.2) in the limit = 0. Incylindrical coordinates, let us consider that the velocityalong the x-axis (r = 0) oscillates at the frequency withthe amplitude v. Assuming that the thickness of theboundary layer, Lbl, is much smaller than the cavity di-ameter , one can neglect the curvature of the walls andwrite the velocity along the x-axis

vx(r) = v

1 exp

(1 +j)(

2 r)

Lbl

(A.17)

where Lbl = 2/ ( = 1.5 105 m2 s1 stands forthe kinematic viscosity of air). From equation (A.17), onecan estimate the viscous stress applied by the side walls to

ensl00124832,

version1

16Jan2007


14/19


the fluid and write the force balance in the cross sectionof the cavity:

vxt

= Px

vx (A.18)

where = 1/8(1 + j) and vx stands for the meanparticle velocity along the x-axis in the cross section of thecylindrical cavity (note that Eq. (A.18) holds true only inthe limit Lbl and in the harmonic regime).

A.4.2 Modified wave equation and pressure field

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

t+

(vx)

x= 0, (A.19)

the wave equation

2P

t2 c2

2P

x2= P

t(A.20)

which takes into account the viscous dissipation in theboundary layers at the side walls. Accordingly, thepressure field inside the cavity, assuming zero outputimpedance and reflection without loss at the closed end,can be written

P(x, s) =P

scoshKL coshK(L + x)coshKL (A.21)

where c2K2 s2 + s. Note that, in the limit 0,K = s/c and equation (A.21) reduces to equation (A.8).


Seeking for the damped oscillatory-modes, we write s =j 1/ and determine the resonant frequencies n andthe corresponding viscous-damping characteristic-timesvn cancelling the denominator in equation (A.21).

Thus, assuming that the reflection at the closed endoccurs without energy loss and zero output impedance,we estimate that the dissipation within the boundary layerslightly affects the resonant frequencies and, in addition,leads to the damping of the mode n inside the cavity overthe characteristic time

vn

2n. (A.22)

The shear in the boundary layer induces temperaturegradients. The associated thermal losses at the wallsare accounted for by replacing in equation (A.22) by

[1 + ( 1)P1

2

r ] where Pr 0.7 stands for the Prandtlnumber and = 1.4, the specific-heat ratio [20].

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

A.5.1 Introduction

In this section, we discuss how the energy initially loadedin the cavity is transferred, immediately after bursting ofthe soap film, to the resonant modes. In order to under-stand the behavior of the system, before generalization, wedetermine the solution in the simple case of an exponentialdecay of the pressure at the open end with the character-istic time . We assume P(0, t) = P e

t (t > 0).

The Laplace transform of the pressure at the open endis given by P(0, s) = P

1+sso that, from equation (A.7),

P(x, s) =P

s

1 cosh

sc (L + x)

(1 + s)cosh sc L

. (A.23)


The solution is obtained from the poles of equation (A.23),which are s = 1/ and sk = j(2k + 1) c2L (k integerranging from to +). After some algebra, we get:

P(x, t)

P=

cosh( L+xc )

cosh( Lc)e

t

+

n=0an cos

nc

(L + x)

sin

nt + n

(A.24)

where we define the frequency n, amplitude an and phasen of the mode n:

n (2n + 1) c2L

, (A.25)

an 4

0n

(1)n1 + (n)2

, (A.26)

tan(n) 1n

, n [0, 2

[. (A.27)

In the limit 0, we recover equation (A.13) excludingthe radiation at the open end (i.e. Zn ).

The first term in equation (A.24) corresponds to anexponential decrease of the pressure, at each position x inthe cavity, with the characteristic time . This contribu-tion does not correspond to a resonant mode but to themean gas flow which escapes the cavity. By constrast, theterms in the sum correspond to the resonant modes. Theamplitude an decreases when n increases and depends onthe characteristic time : if n 1, an 1n and, ifn 1, an (2n)1. The change in regime occurs atthe characteristic frequency c = 1/.

A.5.3 Energy injected in the resonant modes

The total amount of acoustic energy inside the cavity onceopened, EinT , corresponds to the contribution of the terms

ensl00124832,

version1

16Jan2007


15/19


in the sum in equation (A.24). We remind that the den-

sity of acoustic energy equals 12

(v2 + P2

2c2). After simple

algebra, taking into account the relation (A.26) and inte-grating over the inner volume V, we get

EinT =1

2

V P2

c2

1 20

tanh

20

. (A.28)

In the limit 0, EinT = 12 V P2

c2. Note that EinT corre-

sponds to the work necessary for increasing adiabaticallythe pressure inside the cavity by P (ac2 = 1, wherea is the adiabatic compressibility of air [13]). Thus, ifthe typical duration of the pressure drop is short com-pared to the period 2/0 associated with the fundamen-tal frequency 0, the total amount of potential energy ini-tially loaded in the cavity is transferred to the acousticmodes. To the contrary, when the typical opening time

increases, the energy transferred to the acoustic modesdecreases according to equation (A.28) in the case of anexponential decay of the pressure in the output plane.

A.5.4 Generalization

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

Let us first define the functions g(x, t) P(x, t)/P1and f(t) g(0, t). From equation (A.7), we get

g(x, s) = f(s)cosh[ s

c(L + x)]

cosh[ sc L]. (A.29)

One can determine the solution g(x, t) by calculating theinverse Laplace transform of g(x, s). Aiming solely at thedescription of the resonant modes, we consider the poles

of 1/cosh[s/cL] and not the poles of f(s) that correspondto a non-resonant part of the pressure field. Limiting thusour study to the oscillatory part of the pressure field, wewrite

g(x, t) =

n=0

an cosn

c(L + x)

sin(nt + n) (A.30)

together with

an =40

(1)n

f(jn) (A.31)n = arg[f(jn)]. (A.32)

Note that equation (A.30) reduces to the sum in equa-tion (A.24) for f(t) = exp( t) 1.

The amplitude and phase of the various harmonics aregoverned by the pressure drop at the open end: measuringthe amplitude and phase of several harmonics makes itpossible, in principle, to determine the Laplace transformP(0, s) of the pressure drop at the open end and, then,the pressure in the ouput plane P(0, t) as a function oftime. Finally, note that, if we denote EinT any value of the

initial energy in the acoustic modes inside the cavity att = 0, we obtain that f(s) satifies the condition

n=0

f(jn)2 = 2c2

4V 20

1

P2EinT . (A.33)

A.5.5 Radiation and damping of the resonant modes

The radiation at the open end (Sect. A.2), the partialreflection at the closed end (Sect. A.3) and the viscousdissipation at the cavity walls (Sect. A.4) lead the damp-ing of the resonant mode in time. The partial reflection atthe closed end and the viscous dissipation are not affectedby the opening of the cavity for t 0. To the contrary,the bursting of the soap film affects the evolution of thepressure at the open end with time and, thus, governs the

characteristic time (the output impedance drops downfrom the infinity to Zr during the opening of the film).The phenomenon can hardly be described in the theoryand, seeking for simplicity, we will assume that the out-put impedance is Zr for t 0 and that the opening ofthe film only governs the characteristic time . We expectsuch an approximation to be valid as long as is smallerthan the time needed by the acoustic wave to go back andforth from the open end to the closed end. This is to sayas long as < 2L/c or 0 < , which is generally thecase in our experimental conditions.

In this framework, we suggest to write from equa-tion (A.30), the pressure field associated with the acoustic

modes inside the cavity

P(x, t)

P=

n=0 an cosn

c(L + x)

(A.34)

sin(nt + n)exp t

dn

where an and n are given by equations (A.31) and (A.32).The frequencies n are associated with the modified wave-lengths n which takes into account the effect of the ra-diation [Eq. (A.11), we neglect here the slight shift of theresonant frequency due to the viscous dissipation at thecavity walls]. The damping characteristic-time dn , which

appears in equation (A.34), is given by1

dn=

1

Zn+

1

r0n+

1

vn(A.35)

where the various characteristic times n are from Sec-tions A.2 to A.4.

A.5.6 Acoustic wave outside the cavity

From the solution inside the cavity [Eq. (A.34)], we cancalculate the particle velocity in the output plane andthus, from equation (B.1), the pressure field, Pout(r, t),outside the cavity (Sect. B). We first assume that themicrophone is located far from the output plane, at a

ensl00124832,

version1

16Jan2007


16/19


distance d large enough (d ) for the far field ap-proximation to be valid (Sect. B.1). Second, we assumethat the wavelength n, associated with the harmonicswe consider, is large compared to the cavity diameter

so that the pressure field outside has the spherical ge-ometry (Sect. B.2). Finally, the damping of the mode nis supposed to be slow enough for considering that theamplitude is constant over one period (i.e. n

dn 1). In

this framework, equation (B.1) can be used (the imaginaryparts of and k are small and can be neglected) and theintegration over the output plane reduces to a multiplica-tion by the surface area of the cross section. Calculatingthe velocity vx from the solution inside the cavity, we get

Pout(r, t)

P=

2

4Lr

n=0n

f(jn)

sin

n

t r

c

+ n

exp

t

dn

.(A.36)

We remind that n = arg[f(jn)].Thus, listening to the sound outside the cavity makes

it possible to determine the amplitude and phase of theLaplace transform of the pressure drop in the output planeand, then, the pressure drop itself. Finally, we point outthat the conclusion holds true only for the harmonicswhich satisfy the condition n

dn 1. Considering the

radiation only (dn = Zn ), we can rewrite this latter condi-

tion (2n + 1) 1/(4L/)2. The equality leads to n = 10for the shortest and widest cavity we use in the experi-ments whereas we only sense harmonics up to n = 3. Wecan thus consider that equation (A.36) gives a fairly goodestimate of the pressure field outside the cavity in ourexperimental conditions.

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

Assuming that the pressure and, thus, velocity fields out-side the cavity are spherical, we can write the general ex-pression of the total acoustic energy that crossed a sphereof radius r from t = 0 to

EoutT = 4r

2

t=0

Idt (A.37)

= 4r2

t=0

Pout(r, t)vout(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 wave-length of the fundamental (we point out here that this isnot experimentally the case as the microphone 1 is not lo-cated far enough from the output plane). In this case, theacoustic wave is locally planar so that the acoustic inten-

sity reduces toI= Pout2/(c). Calculating the integral inequation (A.37), one obtains the general expression of the

total energy in the acoustic wave outside as a function off(s)

Eout

T

=4

16L2P2

n=0

d

n

2

nf(jn)2. (A.38)

The result must be independant of the distance r andEoutT is the total amount of acoustic energy that crossesany surface surrounding the cavity aperture.

Equation (A.38) applies whatever the physical origin ofdn . If the radiation is the only source of damping,

dn =

Zn

and equation (A.38) reduces to

EoutT =2

4LcP2

n=0

f(jn)2 = EinT (A.39)where the second equality is obtained by using the condi-tion (A.33). Thus, if the acoustic waves are only radiatedat the open end, without any source of dissipation, thetotal amount of energy in the acoustic wave outside thecavity EoutT equals the total amount of acoustic energyEinT initially loaded inside the cavity at t = 0. We recoverhere the energy conservation.

At last, we already noticed that the microphone 1 is lo-cated at a distance d from the open end which is not largein comparison to the wavelength 0 of the fundamental.As the microphone only provides us with the local pres-sure and not with the particle velocity, let us define anestimate of the total acoustic energy by

EapproxT 4d2

c

t=0

Pout(d, t)

2

dt (A.40)

from equation (A.37) and assuming that the acousticimpedance is that of a planar wave. One can show thatEapproxT = E

outT [1 + O(22/64Ld)]. Thus, we estimate

that EapproxT differs from EoutT by less than 2.5% in all

our experimental conditions ( 1 cm, L 2 cm andd = 5 cm).

Appendix B: Radiation at the open end

Let us consider a planar wave which propagates inside thecavity toward the open end. When crossing the outputplane x = 0, the sound wave can invade the whole half-space x > 0. Thus, the planar wave is diffracted at theopen end. The phenomenon has been widely studied [14],and we will summarize below the main results, obtainedin the piston approximation, that we need for discussingour experimental findings.

Assuming the pressure field depends on time like ejt ,we can decompose, at the open-end, the incident planar-wave into an infinite number of infinitesimal source ele-ments of the form

dP(r) = jvx2

ejk|rr|

|r r| dS (B.1)

ensl00124832,

version1

16Jan2007


17/19


where vx is the axial component of the velocity field as-sociated with the planar wave inside the cavity, r theposition of the source element in the aperture plane anddS an infinitesimal surface element around the source. In-

tegration of equation (B.1) over the whole surface of theaperture 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 byintegration the pressure amplitude

|P(r)|y=0z=0

= 2cvx

sin12

kx

1 +

2x

2 1

.(B.2)

Thus, along the x-axis, the amplitude of the pressure waveoscillates when the open end is approached from large val-ues of x. Moving toward the cavity, one encounters thefirst local minimum in the pressure amplitude at x0 whichsatisfies

4x0

=

. (B.3)

The distance x0 is a convenient demarcation between thenear field region, in which the behavior of the pressurefield is complicated, and the far field region, in which thepressure amplitude decreases monotically along the x-axis.Note that x0 < 0 for > , which means that, in this case,there is no near field region. In our experimental condi-tions, the length L of the cavity is always much larger thanthe diameter . As a consequence there is no near field re-gion, at least for the fundamental and several harmonicshaving low enough frequency.

B.2 Spatial structure of the radiated wave

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

|P(r)

||r|= jc

8

vx

rk ejkr

H() (B.4)

where r = |r| and H() denotes the dependence of thepressure amplitude on the angle (cos x r) given by:

H() = 2 J1(k2 sin )

k2 sin

. (B.5)

In equation (B.5), J1 denotes the first-order Bessel func-tion of the first kind [19]. Thus, even in the far field ap-proximation, the pressure field outside the cavity stronglydepends on the angle . However, in our experimental con-ditions

and, as discussed in the text (Sect. 3.4.1),

our measurements do not depend on the angular positionof the microphone 1.

B.3 Output impedance

We can make use of the piston approximation for eval-uating the acoustic impedance at x = 0 of the acoustic

wave outside the tube. To do so, we can calculate thepressure field in the plane of the aperture by integrationof equation (B.1). A second integration makes it possi-ble to estimate in the case of a flanged pipe, as doneby Rayleigh [14,21], the mean value

P

, in the aperture

plane, of the pressure P associated with the acoustic wave

P

= 2jcvx

0

J1(k

2

)2

2 1 d. (B.6)

In the limit , we get the acoustic impedance asso-ciated with the radiation at the open end in the case of aflanged pipe

Zrc

=

P

cvx 1

8(k)2 +

4

3jk. (B.7)

Similar calculations by Levine and Schwinger [15] in thecase of an unflanged pipe lead to a similar, but numericallydifferent, expression

Zrc

=

P

cvx 1

16(k)2 + 0.3 jk. (B.8)

Because of the radiation, the acoustic impedance at theopen end is non-zero.

B.4 Transmission coefficient

From the continuity of the pressure and velocity fields inthe aperture plane, we can estimate energy transmissioncoefficient at the open end [13]. One obtains

T =1

2

k22

[1 + 18

k22]2 + [ 43

k]2, (B.9)

in the case of a flanged aperture. In the limit k 1,the transmission coefficient reduces to T

1/2k22 (the

approximation is better than 10% up to k = 0.5). Notethat T is an increasing function of the frequency as longas > .

Appendix C: Input impedance

C.1 Qualitative analysis

The rather complex behavior of the total energy ET asa function of the overpressure P can be understood byconsidering the geometrical structure of the initial config-uration (film shape). Indeed, before bursting, the soap filmis spherical-shaped (radius R) and the pressure differenceis P = 4/R where denotes the surface free-energy

ensl00124832,

version1

16Jan2007


18/19


associated with the liquid-air interface (the surface free-energy of the soap film is 2 whereas the total curvatureof the film is 2/R).

Qualitatively, for small P, the film is almost flat and

its bursting results in a pressure drop along an interfacewhich has the geometry of the planar standing-waves in-side the cavity; as a consequence, the planar front associ-ated with the pressure drop penetrates easily the cavityand the energy ET is proportional to P2 as one wouldexpect (limit P 0 in Fig. 14). When P is increased,the center of curvature of the soap film approaches theaperture plane. The spherical geometry of the front dif-fers from that of the planar modes inside the cavity; asa consequence, the spherical modes associated with thepressure drop hardly enter the cavity and the total energytransferred to the cavity ET decreases.

C.2 Input impedance

In order to account for the whole behavior of ET as a func-tion of P, consider first the initial geometry sketched inFigure C.1. We will work in spherical coordinates centeredin O, the center of the spherical film. The initial pressuredistribution, which writes P(r) = P for r < R andP(r) = 0 for r > R, can be written as a sum of spheri-cal modes centered in O. In the sake of simplification, wefurther assume that the film bursting leads to an homoge-neous change in the pressure or velocity along the surfaceof the film. One can thus first consider one pressure modepropagating toward the cavity,

P(r, s) = P0(s)R

resc (rR), (C.1)

and the corresponding velocity-mode:

v(r, s) = P0(s)

c

1 c

sr

Rr

esc(rR). (C.2)

Note that P0(s) denotes the Laplace transform of the pres-sure drop associated with the film bursting. From equa-tions (C.1) and (C.2), we can calculate, in the plane of theopen end, the mean pressure:

P

(s) = 8P0(s)

Rc

s2

1 e sRc (1cos 0)

(C.3)

and velocity along the x-axis:

v (s) = 8 P0(s)

c

R

s2

cos 0 e sc R(1cos 0)

. (C.4)

where R sin 0 = /2. We deduce from (C.3) and (C.4),the input impedance Zi, which is associated with the ho-mogeneous variations of the pressure and velocity in theplane of the open end:

Zi(s)c

= 1 es

c R(1cos 0)

cos 0 e scR(1cos 0). (C.5)

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

C.3 Consequence Transmission coefficient

Let us now consider that a fraction of the incident waveis transferred to a planar wave propagating toward theclosed end (acoustic impedance c) while the remainingpart is diffracted leading to a wave propagating away fromthe cavity [acoustic impedance in the plane of the openend, Zr (Eq. B.7)]. Doing so, we implicitely assume thatthe characteristic time associated with the pressure drop isshort compared with the time needed for the acoustic waveto go back and forth along the cavity length. Denoting Pcthe amplitude of the wave transferred to the cavity andwriting the continuity of the pressure- and velocity-fieldsin the plane of the open end, we get after some simplealgebra:

Pc(jw) = P0(jw)

1 x

6j + x

1 cos 01 ejx tan 02

(C.6)

where x 2c

= . We remind that P =4R =

8 sin 0

so that the transmission coefficient:

Ti Pc(jw)P0(jw)

2

=

1 x6j + x 1 cos 01 ejx tan 022

(C.7)

is a function of the overpressure P through 0(P).

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()cos(t) et (D.1)

Let us denote tk k/s the time samples with s the sam-pling frequency. By definition, the fast Fourier transformof the signal writes:

F F T[V](n) 1N2

N1k=0

V(tk) e2j knN . (D.2)

In what follows, we assume that the number of samplesN is large (i.e. N s). We define the value of theenergy spectrum (n) at the frequency n 2 nNsby (n) 2|F F T[V](n)|2. The spectrum , associatedwith the voltage given in equation (D.1), exhibits a max-imum at:

n =

1 + 422

122

. (D.3)

ensl00124832,

version1

16Jan2007


19/19


Thus, in the limit , n = . In the presence ofdamping, in the limit 1, the maximum is slightlyshifted to a higher value n such that

n 122 .

The corresponding maximum value of the power spectrum

is then given by:

(n) = V2

()2s

2

4N2

1 +

1 +

4

22

12

2s 2

N2V2

() ( 1). (D.4)

Note that, from its definition, depends on the acquisi-tion time N/s. In the limit , the acquisition timeis smaller than and one must take = N/s in order to

recover the expected value = 1/2V2

(). In our exper-imental conditions, we choose the acquisition time to belarge in comparison to and equation (D.4) applies.

References

1. D.E. Spiel, J. Geophys. Res. 97, 11443 (1992)2. H.M. Princen, Surface and colloid science, edited

Matijevic, (Wiley-Interscience, NY, pp. 184 1969); Vol. 2,V. Bergeron, Forces and structure in surfactant-laden thin-liquid films, Thesis, University of California at Berkeley(1993)

3. A.A. Kulkarni, J.B. Joshi, Ind. Eng. Chem. Res. 44, 5873(2005)

4. M. Ichihara, T. Yanagisawa, Y. Yamagishi, H. Ichikawa, K.Kurita, Japan Earth and Planetary Science Joint Meeting,Abstract, (CD-ROM), A111-P004 (2005)

5. D. Dingwell, Magma degassing and fragmentation: re-cent experimental advances, in From Magma to Tephra- Modelling physical processes of explosive volcanic erup-

tions, Developments in Volcanology 4, edited by A.Freundt, M. Rosi (Elsevier, 1998)

6. M. Ripepe, P. Poggi, T. Braun, E. Gordeev, Geophys. Res.Lett. 23, 181 (1996)

7. S. Vergniolle, G. Brandeis, J. Geophys. Res. 101, 20433(1996); S. Vergniolle, G. Brandeis, J. Geophys. Res. 101,20449 (1996)

8. T. Divoux, Listening to singing bubbles at the free surfaceof a non-newtonian fluid, Stage de recherche de deuxiemeannee du MIP, ENS, Paris (2004); T. Divoux, V.

Documents

V. Vidal et al- Acoustic signal associated with the bursting of a soap film which initially closes an overpressurized cavity: Experiment and theory