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An Audible Demonstration of the Speed of Sound in Bubbly Liquids Preston S

An Audible Demonstration of the Speed of Sound in Bubbly Liquids Preston S

An audible demonstration of the of in bubbly Preston S. Wilson and Ronald A. Roy

Citation: American Journal of 76, 975 (2008); doi: 10.1119/1.2907773 View online: http://dx.doi.org/10.1119/1.2907773 View Table of Contents: http://scitation.aip.org/content/aapt/journal/ajp/76/10?ver=pdfcov Published by the American Association of Physics Teachers

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This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.83.205.78 On: Mon, 11 May 2015 21:48:30 APPARATUS AND DEMONSTRATION NOTES

Frank L. H. Wolfs, Editor Department of Physics and Astronomy, University of Rochester, Rochester, New York 14627

This department welcomes brief communications reporting new demonstrations, laboratory equip- ment, techniques, or materials of interest to teachers of physics. Notes on new applications of older apparatus, measurements supplementing data supplied by manufacturers, information which, while not new, is not generally known, procurement information, and news about apparatus under development may be suitable for publication in this section. Neither the American Journal of Physics nor the Editors assume responsibility for the correctness of the information presented. Manuscripts should be submitted using the web-based system that can be accessed via the American Journal of Physics home page, http://www.kzoo.edu/ajp/ and will be forwarded to the ADN editor for consideration.

An audible demonstration of the speed of sound in bubbly liquids ͒ Preston S. Wilsona Mechanical Engineering Department and Applied Research Laboratories, The University of Texas at Austin, Austin, Texas 78712-0292 Ronald A. Roy Department of Aerospace and Mechanical Engineering, Boston University, Boston, Massachusetts 02215 ͑Received 5 January 2007; accepted 18 March 2008͒ The speed of sound in a bubbly liquid is strongly dependent upon the volume fraction of the phase, the bubble size distribution, and the of the acoustic excitation. At sufficiently low , the speed of sound depends primarily on the gas volume fraction. This effect can be audibly demonstrated using a one-dimensional acoustic waveguide, in which the flow rate of air bubbles injected into a -filled tube is varied by the user. The normal modes of the waveguide are excited by the sound of the bubbles being injected into the tube. As the flow rate is varied, the speed of sound varies as well, and hence, the resonance frequencies shift. This can be clearly heard through the use of an amplified hydrophone and the user can create aesthetically pleasing and even musical . In addition, the apparatus can be used to verify a simple mathematical model known as Wood’s equation that relates the speed of sound of a bubbly liquid to its void fraction. © 2008 American Association of Physics Teachers. ͓DOI: 10.1119/1.2907773͔

I. INTRODUCTION a drinking glass filled with a “frothy liquid” produces a dull sound. Perhaps inspired by his former laboratory assistant,2 The interaction of sound and bubbles has been studied in Rayleigh published a paper in 1917 that described the sound 3 modern science for almost 100 years and has been part of produced by water boiling in a tea kettle. In 1933, Minnaert human consciousness for many hundreds of years, perhaps studied the sounds produced by running water, such as the 4 longer. It is a scientifically interesting and complex phenom- flowing stream and the crashing of ocean . The latter enon with a long and diverse history, yet our understanding two studies both introduced relationships between the size of of the phenomenon is still incomplete. Part of the intrigue a bubble and its period of oscillation with water as the host comes from the fact that minute parameter changes in a bub- medium, but Rayleigh considered steam bubbles while Min- bly fluid can have large and unexpected effects. For example, naert considered air bubbles. consider a column of water contained within a rigid tube. The of bubbly liquids now has a number of im- The addition of just a small number of bubbles, representing portant practical applications. These include a variety of in- as little as 0.1% by volume, can transform the once simple dustrial processes utilizing multiphase flow,5 microfluidic medium into a highly dispersive and nonlinear one, with a manipulation,6 and the design of nuclear reactor containment speed of sound less than 20% of the speed of sound in the systems.7 In the ocean, various acoustic techniques are used host medium. That corresponds to a 4 parts in 5 change in in conjunction with oceanic bubble populations to make the speed of sound for a 1 part in 10 000 change in the measurements relevant to meteorology, chemical oceanogra- . It is the added of the gas phase that phy, marine fluid dynamics, and marine biology.8 By shroud- causes the reduction of the speed of sound. ing propellers and hulls with man-made bubble screens, un- The early scientific work on the acoustics of bubbles was wanted noise produced by ships and submarines is reduced.9 motivated primarily by the desire to understand certain man- Bubbles are used in medical applications to enhance ultra- made and natural curiosities. In 1910, a paper was published sound imaging,10 hemostasis,11 and therapy.12 by Mallock1 that sought to explain the fact that when struck, Both theory and measurement reveal that for acoustic ex-

975 Am. J. Phys. 76 ͑10͒, October 2008 http://aapt.org/ajp © 2008 American Association of Physics Teachers 975 This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.83.205.78 On: Mon, 11 May 2015 21:48:30 citation frequencies that are close to the individual bubble If the bubble radius is perturbed, equilibrium is lost and resonance frequency, the acoustic attenuation within a bub- oscillations occur. The alternating compression and expan- bly liquid is measured in tens of decibels per centimeter, and sion of the bubble is due to the transfer of energy between that the sound speed can vary from as little as 20 m/stoas the inertia provided by the surrounding liquid and the com- much as 10 000 m/s, which is supersonic relative to bubble- pressibility of the gas inside the bubble. For the frequency free water. In the littoral ocean environment, this behavior is range between 100 and 1200 Hz and a bubble size of about a dominant obstacle for effective operation because 1 mm, it is sufficient to assume lossless conditions within the bubbles and bubble clouds are produced in abundance by bubble, but dissipative effects are important in other 20 breaking waves. The acoustic contrast provided by bubbly regimes. At a hydrostatic Pϱ, the compressibility liquid causes sound to be scattered from bubble clouds, and of the gas inside the bubble to an effective 21 hence naval sonar and mine hunting operations are hindered. 12␲¯a␯Pϱ. The bubble behaves acoustically like a pulsating Bubbles are also a source of ambient noise in the ocean. sphere and radiates sound uniformly in all directions. Since Rain drops, spray from breaking waves, and even snow can the bubble size is always much smaller than the resulting entrain bubbles when they impact the ocean surface. The acoustic , the low frequency limiting value of the impact deforms the ocean surface and creates an air cavity. acoustic radiation reactance can be used to define an effec- 3 When this cavity closes, a bubble is formed and a tive mass, known as the radiation mass 4␲¯a ␳ᐉ, where ␳ᐉ is impact that occurs when the air cavity closes causes the the liquid density. The bubble’s resonance frequency is then 13–15 bubble to oscillate and radiate sound. Breaking waves given by the square root of the stiffness to mass ratio, or produce clouds composed of large numbers of bubbles that 1 3␯Pϱ oscillate collectively and radiate sound. These bubble clouds ␻ ͱ ͑ / ͒ ͑ ͒ 0 = radians sec . 2 are responsible for part of the ambient noise in the ocean ¯a ␳ᐉ 16 below about 1000 Hz. 4 The physics of bubble acoustics has been the subject of This relation was first derived by Minnaert in 1933, al- previous “Apparatus and Demonstration Notes” within this though via a different methodology. Minnaert considered journal,17–19 but the apparatus described here complements bubbles in the size range between 3 and 6 mm, and con- and expands upon the previous work. The apparatus consists cluded that the period of oscillation would be small com- of a transparent PVC pipe about 0.5 m in length. Air bubbles pared to the required for the heat of compression to be are injected at the bottom of the water-filled tube and a hy- conducted away. The bubble was considered to be undergo- ␯ drophone is used to detect the resulting sounds. The appara- ing an , and hence the polytropic index ␥ tus can be used to demonstrate the bubble sound production was taken to be the ratio of specific heats . mechanism and the bubbly liquid’s speed of sound depen- The assumption of adiabatic behavior made by Minnaert dence on the air/water volume fraction. The latter is observed was reasonable for the bubble sizes he was considering, but it is instructive to consider a more accurate description. indirectly via the pitch of the sound. If the sounds are re- 22 corded, the data can be used to verify Wood’s equation, Leighton considered bubbles undergoing natural oscillation which is a mathematical model that relates the speed of and those driven by an external acoustic field, and showed sound in a bubbly liquid to the fractional volume of air. that because of the high and specific This demonstration has been conducted for audiences of the surrounding liquid, the gas in direct con- ranging from professional acousticians to prospective under- tact with the bubble wall can be considered isothermal. The graduate students and their parents. It is always a hit and gas at the center of the bubble behaves adiabatically. There- usually generates questions and discussion. The basic theory fore, the effective polytropic index for the bubble as a whole governing bubble oscillation, sound propagation in bubbly can take on a range of intermediate values depending on the liquids, and the acoustics of one-dimensional waveguides is bubble’s size relative to the size of the thermal boundary presented in Sec. II. The demonstration apparatus is de- layer. The ratio of the bubble equilibrium radius to the length scribed in Sec. III. In Sec. IV the use of the apparatus is of thermal boundary layer lth is given by described. Finally, the technique used to verify Wood’s equa- ¯a ¯aͱ2␻ tion with our apparatus is discussed in Sec. V. = ͱ , ͑3͒ lth Dg ␻ II. THEORETICAL CONSIDERATIONS where Dg is the thermal diffusivity of the gas and is the frequency of excitation.22 At sufficiently high frequencies, Consider a spherical air bubble with equilibrium radius ¯a the bubble is much larger than lth and it behaves adiabati- suspended in a liquid at room conditions: atmospheric pres- cally. At sufficiently low frequencies, the bubble is much sure and 20 °C. For ¯aϾ10 ␮m, the surface tension and va- smaller than lth and it behaves isothermally. The value of Dg por pressure can be ignored and the hydrostatic pressure of for air at room is 2.08ϫ10−5 m2 /s, and there- the surrounding water is balanced by the gas pressure inside Ϸ fore ¯a 90lth for Minnaert’s bubbles, confirming the adia- the bubble. Assuming the air is an , it is governed by batic assumption. the relation PV␯ = constant, ͑1͒ A. Low frequency sound propagation in bubbly liquids where P is the pressure, V is the volume, and ␯ is the poly- Now consider propagation of sound through a collection tropic index. For air bubbles in water, the value of ␯ can of bubbles, where the incident acoustic pressure is the source range from ␯=␥, where ␥ is the ratio of specific heats of the of the perturbations. In 1910 Mallock1 applied mixture air, to ␯=1. These limits bound a continuum of behavior that theory to what he termed “frothy liquids,” under the assump- ranges from adiabatic to isothermal, depending on the bubble tion that the gas bubbles were of such number and separation size and the frequency of excitation. that the medium could be considered homogenous, and he

976 Am. J. Phys., Vol. 76, No. 10, October 2008 Apparatus and Demonstration Notes 976 This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.83.205.78 On: Mon, 11 May 2015 21:48:30 obtained an approximate expression for the speed of sound in the bubbly liquid. In 1930, Wood23 began with a thermody- namic argument and confirmed Mallock’s work, but cast the results in a more useful form, and specifically considered air bubbles in water. In this section, the equation governing the low-frequency speed of sound in bubbly liquids is devel- oped, based on the work of Wood.23 The following assumptions and definitions are used. The bubbly fluid is composed solely of spherical gas bubbles suspended in liquid under the influence of acoustic pressure fluctuations. The subscripts ᐉ and g refer to the liquid and gas phases, respectively. Effective quantities pertaining to the mixture are denoted with the subscript m. Mass is con- served such that the mixture mass is the sum of the liquid and gas masses, with no relative translational motion be- tween the two phases. The volume fraction of the gas phase, known as the void fraction ␹, is given by ␹ / ͑ ͒ = Vg Vm, 4 Fig. 1. The low-frequency speed of sound of sounds of bubbly liquid given where Vg is the volume of the gas phase and Vm is the total by Wood’s equation ͓Eq. ͑12͔͒ for air bubbles in water, with ␳ᐉ / 3 ␳ / 3 / / volume of the mixture, given by Vm =Vg +Vᐉ. The effective =998 kg m , g =1.21 kg m , cᐉ =1481 m s and cg =333 m s. At zero void mixture density is fraction, the speed of sound is that of the liquid alone. At unity void fraction, the speed of sound is that of the gas alone. The minimum speed of sound is ␳ ͑ ␹͒␳ ␹␳ ͑ ͒ / ␹ m = 1− ᐉ + g, 5 about 23 m s and occurs at =0.5. ␳ ␳ where ᐉ is the density of the liquid and g is the density of ␬ the gas. The mixture compressibility m is 1 1 ␬ ͑ ␹͒␬ ␹␬ ͑ ͒ ␬ ͑ ͒ m = 1− ᐉ + b, 6 g = = . 10 ␯Pϱ 1.3Pϱ ␬ ␬ where ᐉ is the compressibility of the liquid and b is the ͑ ͒ ͑ ͒ ͑ ͒ compressibility due to the gas in the bubbles. In general, the Substitution of Eqs. 5 and 6 into Eq. 7 results in an speed of sound c is defined as equation for the mixture sound speed, 1 1 = ͓͑1−␹͒␳ᐉ + ␹␳ ͔͓͑1−␹͒␬ᐉ + ␹␬ ͔, ͑11͒ c2 = . ͑7͒ 2 g g ␳␬ cm 23 ␬ which is equivalent to Wood’s 1930 equation. For bubbles The value of b and the way it is eventually included in Eq. ͑ ͒ well below resonance size where surface tension and dissi- 7 depends in part upon the conditions that prevail during ͑ ͒ ͑ ͒ the compression phase. For air bubbles in water at frequen- pative effects are negligible, Eqs. 7 and 10 can be used to cies that approach zero, the process can be considered iso- relate the previous result to the density and of sound thermal, as was shown rigorously by Hsieh and Plesset.24 of the two mixture components. After manipulation this re- The response of an individual bubble is dominated by the sults in the equivalent relation ␬ ͑ ␹͒2 ␹2 ␳2 2 ␳2 2 compressibility of the gas within the bubble, hence b 1 1− gcg + ᐉcᐉ ͑␬ ͒ ͑␯ ͒−1 ␯ = + + ␹͑1−␹͒ , ͑12͒ = g isothermal= Pϱ with =1. As the excitation fre- 2 2 2 2 2 c cᐉ c ␳ᐉ␳ cᐉc quency approaches the bubble resonance frequency, the dy- mlf g g g ␬ →␬ ͑ ͒␬ namics of the bubble become important and b b f g. where cmlf is the low-frequency speed of sound of the mix- Well above resonance, the bubbles behave adiabatically and ture. The behavior of Eq. ͑12͒ for air bubbles in distilled −1 ␬ =͑␬ ͒ =͑␯Pϱ͒ with ␯=␥=1.4 for air. In between, water is shown in Fig. 1. Note that as the mixture approaches b g adiabatic ␹→ → ␹→ the polytropic index that governs the compression and ex- pure liquid, 0 and cmlf cᐉ. Alternatively, as 1, → ␹ pansion of the gas inside the bubbles is frequency dependent, cmlf cg. Upon approach to =1, a change to the adiabatic as shown by Prosperetti,25 and is given by ␯=Re͓⌽͔/3, speed of sound in air would be required. The speed of sound ͑ ͒ ͑ ͒ where in bubbly liquid, given by Eq. 11 or 12 , is known as Wood’s low-frequency speed of sound or the Wood limit of 3␥ the speed of sound. ⌽ = ͑8͒ 1−3͑␥ −1͒iX͓͑i/X͒1/2coth͑i/X͒1/2 −1͔ A minimum occurs at ␹=0.5, the speed of sound is lower than the intrinsic speeds of sound in air or water across much and of the void fraction range. This remarkable result can be X = D /␻a2. ͑9͒ qualitatively explained: bubbles greatly increase the mixture g compressibility yet effect the density much less. High com- In this work, the excitation frequency is well below the pressibility and high density yields a low speed of sound. ␬ ␬ bubble resonance, therefore b = g. In the frequency range Another interesting result is that in this regime, the bubble between 200 and 1000 Hz and a bubble size of about size does not play a role. The primary environmental variable 1.0 mm, Eq. ͑8͒ yields 1.26Ͻ␯Ͻ1.33. It is sufficient for our is the void fraction. purposes to take the average value of ␯ and set the compress- Because of the great difference in density and speed of ibility of the gas to be sound between air and water, Eq. ͑12͒ can be approximated

977 Am. J. Phys., Vol. 76, No. 10, October 2008 Apparatus and Demonstration Notes 977 This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.83.205.78 On: Mon, 11 May 2015 21:48:30 by a much simpler expression for a limited range of ␹. For air bubbles in water, in the range 0.002Ͻ␹Ͻ0.94, the first two terms on the right-hand side of Eq. ͑12͒ are much smaller than the third term. Neglecting them and noting that ␳2 2 ␳2 2 gcg ᐉcᐉ results in 1 c2 Ӎ . ͑13͒ mlf ␳ ␬ ␹͑ ␹͒ ᐉ g 1− Substitution of Eq. ͑10͒ into Eq. ͑13͒ results in a simple approximate expression for the low-frequency speed of sound of the mixture ␯P Ӎ ͱ ϱ ͑ ͒ cmlf , 14 ␳ᐉ␹͑1−␹͒ and differs less than 1.2% from Eq. ͑12͒. Extensions to Eq. ͑14͒ accounting for relative motion between the phases, i.e., bubbly fluid flows, have been provided by Crespo26 and Ruggles et al.27

B. Mode shapes and eigenfrequencies of a one-dimensional waveguide The compressibility of a bubbly liquid is large in compari- son to the pipe wall material; hence, the waveguide walls appear effectively rigid. The top of the tube is a water-air interface, hence it is approximated as a pressure release boundary, with p=0. Again, because of the high compress- ibility of the bubbly liquid inside the tube, the water just below the opening of the needles appears as a rigid surface, x=0. Consider a rigid tube of length L and circularץ/pץ with cross section with radius b, filled with a fluid possessing a Fig. 2. A schematic of the demonstration apparatus. speed of sound c0. The cutoff frequency of the lowest order transverse mode is given by28 ␣Ј c fc = 11 0 , ͑15͒ 11 2␲b The acoustic waveguide was constructed from transparent schedule 40 PVC pipe that is available from plumbing sup- ␣Ј Ј͑ ͒ where 11=1.841 and is the first root of J1 x , the Bessel ply houses. The inside diameter was 5.2 cm and the length function of the first kind of order one, and the prime indi- was 0.6 m. The pipe is typically filled with distilled water to cates the derivative with respect to the function’s argument. a level of Ϸ0.5 m. A bubble injection manifold was fabri- c 29 At frequencies below f11, only plane waves can propagate cated with 12 22-gauge needles embedded in an epoxy within the tube. Solving the equation within this tube filled PVC pipe fitting that attached to the bottom of the pipe. for the given boundary conditions yields the eigenfunctions The needles were positioned in a vertical orientation and therefore released air bubbles directly upward. A small me- ␲͑n +1/2͒x ͑ ͒ ͑ ͒ chanical float was positioned at the top of the pipe, such that pn x = an cos , 16 L it floated on the upper surface of the water, as shown in inset ͑a͒ of Fig. 2. The float was attached to a pointer that indi- and eigenfrequencies cated the height of the water surface on a centimeter scale. c The overall length L of the water column in the tube could be f = 0 ͑n +1/2͒, n = 0,1,2, ... . ͑17͒ n 2L measured to an accuracy of about 1 mm. The void fraction is

The resonance frequency fn is a linear function of the mode ␹ = ⌬ ᐉ /͑ᐉ + ⌬ ᐉ ͒ = ⌬ ᐉ /L, ͑18͒ number n with the sound speed c0 as a parameter. where ᐉ is the length of the water column in the absence of III. DESCRIPTION OF DEMONSTRATION bubbles and ⌬ᐉ is the change in length when bubbles are APPARATUS present. Any convenient source of air can be used, as long as the The apparatus, shown in Fig. 2, consists of five basic flow rate can be controlled by the user. In our implementa- parts: an acoustic waveguide that is filled with bubbly water, tion, a canister of medical air was used. The air was directed an air injection system that creates the bubbles, a hydro- through a flow meter with a needle valve to control the flow phone that senses the acoustic pressure within the wave- rate, although a measurement of the flow rate is not required. guide, a data acquisition system that records and displays the If available, one can use a video camera with a macro ͑clo- hydrophone signal, and an amplified loudspeaker system that seup͒ lens to observe the bubbles and hence determine the broadcasts the hydrophone signal to the audience. bubble size. A typical bubble is shown in Fig. 3.

978 Am. J. Phys., Vol. 76, No. 10, October 2008 Apparatus and Demonstration Notes 978 This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.83.205.78 On: Mon, 11 May 2015 21:48:30 is not important at this stage.31 The demonstration begins with no air flow from the needles, and hence, no bubbles in the waveguide. The VSA and the PA system are turned on and a flat spectrum of low amplitude is displayed, which gives an indication of the background noise in the room. The needle valve is then slowly opened to allow single bubbles to breakoff the needles one at a time. The individual breakoff events are audible through the PA system. Although not de- scribed here, the time series of the hydrophone signal could be presented using an oscilloscope. A bubble breakoff event would appear as a transient signal that approximates a damped sinusoid. The flow rate is then slowly increased and a larger number of bubbles is created. The acoustic excitation becomes effec- tively continuous and standing waves develop in the wave- guide. The excitation is sufficiently broadband to excite the first several normal modes of the waveguide. One can then continue to increase the flow rate and the number of bubbles further increases. This reduces the speed of sound in the waveguide, and hence reduces the frequency of the normal modes. As the flow rate continues to increase, the pitch of the sound produced continues to drop. Changing the void frac- tion via flow rate from 0.1% to 2% over a few re- sults in the speed of sound in the tube changing from ap- proximately 300 m/s to 100 m/s over the same time scale, Fig. 3. A photograph of a typical bubble produced by one of the needles. and hence the pitch of the sound is changed from about 150 Hz to 50 Hz, which is an octave and a half. This is easily detected by ear, but because the frequencies are relatively The acoustic energy is supplied by the bubbles them- low, a large speaker is required to reproduce the sound. The selves. When each individual bubble breaks off its needle, a overtones can still be heard if only a small speaker is avail- transient acoustic pulse is produced. Continuous bubble pro- able, but the demonstration is more effective with a large duction excites an acoustic field inside the speaker. waveguide that is observed with a hydrophone. A Brüel and As the flow rate is increased, the height of the water col- Kjær model 8103 miniature hydrophone was used in this umn also increases. This also plays a role in establishing the work, but a small lapel can also be used.30 frequencies of the normal modes, but it is small compared to The hydrophone signal was amplified and bandpass fil- the variation due to the void fraction induced change in the tered between 10 Hz and 10 kHz by a Brüel and Kjær model speed of sound. In addition to ramping the flow rate up and 2692 charge amplifier. The high-pass characteristic of the filter suppresses very low-frequency ͑Ͻ50 Hz͒ flow noise, then down, which produces decreasing and then increasing pitch, the user can vary the flow rate in any manner desired, which can in some cases be dominant. Suppressing the flow noise allows the desired part of the signal ͑the eigenmodes of which results in corresponding pitch changes. The user can the waveguide͒ to be more easily heard by the audience. If also include a return to very low flow rates, which produces the public address system has its own filter function, such as individual bubble breakoff events and their corresponding a graphic equalizer, a separate high-pass filter is not required. sounds. With some practice, one can achieve at humor- The low-pass filter was used for antialiasing the signal prior ous, and at other times aesthetically pleasing and even mu- to digitization. If the apparatus is used solely for an audio sical sounds. demonstration, the low-pass filter is not required. During the demonstration, the VSA displays the spectrum The hydrophone signal was directed to both a signal ana- of the hydrophone signal in real time. Near the beginning of lyzer and a public address system. An Agilent 89410A vector the demonstration, we establish a flow rate sufficiently high signal analyzer ͑VSA͒ was used to digitize the conditioned to yield continuous acoustic excitation, such as that shown in hydrophone signal and display and record the spectrum of Fig. 4͑a͒. The demonstrator can point out that the peaks in the signal via fast Fourier transform calculations performed the spectra correspond to the eigenfrequencies of the wave- onboard the VSA. Alternatively, a consumer-grade sound guide and that these frequencies combine to produce the card can perform the same functions on many personal com- acoustic sound we perceive during the experiment. As the puters. Finally, the hydrophone signal must be amplified and frequencies of these peaks increase, the pitch is also per- directed to the loudspeaker system. We used a musical in- ceived by a listener to increase. The spectra of these sounds strument amplifier designed for bass guitar, since the lowest can be seen in real time on the signal analyzer, with the eigenfrequencies can be below 100 Hz. peaks shifting up and down with the flow rate. Ideally, a waterfall plot can be displayed showing a brief history, like ͑ ͒ IV. DEMONSTRATION PROCEDURES that shown in Fig. 4 b . This demonstration has been per- formed by the authors many times to the approval of observ- The PVC waveguide is filled with water and the hydro- ers ranging from elementary school students to deans of phone is placed midway down the tube. The exact placement large research universities.

979 Am. J. Phys., Vol. 76, No. 10, October 2008 Apparatus and Demonstration Notes 979 This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.83.205.78 On: Mon, 11 May 2015 21:48:30 Fig. 5. Linear fits to the measured resonance frequencies as function of mode number for three void fractions: the lowest, the highest, and an inter- mediate void fraction.

Fig. 4. ͑a͒ A single spectrum from the 0.4% void fraction case for which the individual modes and their eigenfrequencies fn have been identified. The ͑ ͒ ␮ ͓ / ͔ bubbles are fewer in number and less homogeneously spaced units are decibels dB referenced to 1 Pa, where dB=20 log10 p pref , p is ␮ ͑ ͒ throughout the tube. This results in a speed of sound that is the RMS acoustic pressure measured in the tube, and pref =1 Pa. b A waterfall plot of all the experimental spectra. inhomogeneous and resonance frequencies that are not per- fectly harmonic, and hence less well-represented by a linear fit. V. VERIFICATION OF WOOD’S EQUATION The speeds of sound thus inferred for the entire dataset are plotted versus void fraction in Fig. 6. Vertical error bars rep- The bubbles produced in the apparatus are typically about resent the uncertainty in the speed of sound due to uncertain- 1 mm in radius, hence they have resonance frequencies near ties in the tube length, L, Ϯ1 mm, and the finite bandwidth 3 kHz, according to Eq. ͑2͒. As long as the eigenfrequencies resolution of the spectra, which was 2.5 Hz. The line is of the waveguide are well below the bubble resonance fre- Eq. ͑12͒ plotted using the following physical parameters: quency, the bubbly liquid-filled waveguide will exhibit a speed of sound nearly independent of frequency and bubble size and the mixture will behave as an effective medium with a speed of sound given by Wood’s equation ͓Eq. ͑12͔͒. To experimentally verify Eq. ͑12͒, spectra like the one shown in Fig. 4͑a͒ are obtained for a range of air flow rates. At each flow rate, the height of the column L= ᐉ +⌬ᐉ is measured using the float at the top of the waveguide and the void fraction ␹ is determined from Eq. ͑18͒. Due to motion of the water surface, the uncertainty in ⌬ᐉ is about Ϯ0.5 mm, which leads to an uncertainty in the void fraction ␹ of Ϯ0.001. The bubble-free column height was ᐉ =0.485 mϮ0.005 m. A collection of spectra at void frac- tions ranging up to ␹=2.5% is shown in Fig. 4͑b͒. As indicated by Eq. ͑17͒, a plot of the modal frequencies versus mode number ideally yields a straight line with a slope directly proportional to the speed of sound in the bub- bly liquid. Examples for three void fractions are shown in Fig. 5. A least squares linear fit of the measured resonance frequencies yields the slope from which the speed of sound is obtained for each void fraction. At higher void fractions, ␹ Fig. 6. Experimental verification of Wood’s equation. The solid line is the =2.46% and 0.6%, the bubbles are approximately homoge- ͑ ͒ neously spaced along the length of the tube, which yields an prediction of Eq. 12 and the open circles are the measured values. Vertical error bars represent the uncertainties in the measurement of the speed of effective speed of sound that is constant throughout the tube. sound due to uncertainties in the column length L and the finite bandwidth The resulting resonance frequencies are nearly harmonic. At resolution of the spectra. Horizontal error bars represent uncertainties in the the lower void fractions, exemplified by ␹=0.13%, the void fractions due to the uncertainty in the measured values of ⌬ᐉ.

980 Am. J. Phys., Vol. 76, No. 10, October 2008 Apparatus and Demonstration Notes 980 This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.83.205.78 On: Mon, 11 May 2015 21:48:30 ␳ / 3 ␳ / 3 / of Illinois Institute of Technology ͑1958͒. ᐉ =998 kg m , g =1.21 kg m , cᐉ =1494 m s, and cg 8 H. Medwin and C. S. Clay, Fundamentals of Acoustical Oceanography =334 m/s. The speed of sound in the liquid, cᐉ, was deter- ͑Academic Press, Boston, 1998͒. mined from the water temperature, 24 °C, and Eqs. ͑5͒, ͑6͒, 9 ͑ ͒ Underwater Ship Husbandry Manual, Naval Sea Systems Command, and 8 of Ref. 21. The speed of sound in the gas, cg, was 1995, NAVSEA Publication S0600-AA-PRP-050. determined from Eqs. ͑7͒ and ͑10͒ for a hydrostatic pressure 10 F. Calliada, R. Campani, O. Bottinelli, A. Bozzini, and M. G. Som- due to the standard atmospheric pressure, 101.3 kPa, and one maruga, “Ultrasound contrast agents basic principles,” Eur. J. Radiol. 27, half the column height of water, 0.24 m. Agreement within S157–S160 ͑1998͒. the measurement uncertainty is found between the measured 11 S. Vaezy, R. Martin, P. Mourad, and L. A. Crum, “Hemostasis using high intensity focused ultrasound,” Eur. J. Ultrasound 9, 79–87 ͑1999͒. and predicted speeds of sound throughout the range of void 12 ␹ X. Yang, R. A. Roy, and R. G. Holt, “Bubble dynamics and size distri- fractions. Below =0.3%, the agreement is somewhat less butions during focused ultrasound insonation,” J. Acoust. Soc. Am. than at higher void fractions. This is due to the inhomoge- 116͑6͒, 3423–3431 ͑2004͒. neous conditions inside the waveguide discussed before. The 13 H. C. Pumphrey and L. A. Crum, “Free oscillations of near-surface effective medium theory breaks down under such conditions bubbles as a source of the underwater noise of rain,” J. Acoust. Soc. Am. and can no longer be used to describe the system. To inves- 87͑1͒, 142–148 ͑1990͒. 14 tigate lower void fractions, a bubble injection scheme ca- J. H. Wilson, “Low-frequency wind-generated noise produced by the im- ͑ ͒ pable of evenly spacing bubbles at low flow rates would be pact of spray with the ocean’s surface,” J. Acoust. Soc. Am. 68 3 ,952– 956 ͑1980͒. required. 15 L. A. Crum, H. C. Pumphrey, R. A. Roy, and A. Prosperetti, “The un- derwater sounds produced by impacting snowflakes,” J. Acoust. Soc. Am. VI. CONCLUSIONS 106͑4͒, 1765–1700 ͑1999͒. 16 W. M. Carey and J. W. Fitzgerald, “Low Frequency Noise from Breaking A simple laboratory apparatus is described that provides Waves,” in Natural Physical Sources of Underwater Sound, edited by B. an audible demonstration of several aspects of the acoustics R. Kerman ͑Kluwer, Boston, 1993͒, pp. 277–304. of bubbly liquids. The speed of sound in a bubbly liquid 17 F. S. Crawford, “The hot chocolate effect,” Am. J. Phys. 50͑5͒, 398–404 ͑1982͒. contained within a PVC pipe is altered by varying the injec- 18 tion rate of air bubbles. The bubble injection process itself S. Aljishi and J. Tatarkiewicz, “Why does heating water in a kettle pro- duce sound?,” Am. J. Phys. 59͑7͒, 628–632 ͑1991͒. produces broadband noise that excites one-dimensional 19 V. Leroy, M. Devaud, and J.-C. Bacri, “The air bubble: Experiments on standing waves within the apparatus. The variation of the an unusual harmonic oscillator,” Am. J. Phys. 70͑10͒, 1012–1019 ͑2002͒. speed of sound as a function of the void fraction is apparent 20 K. W. Commander and A. Prosperetti, “Linear pressure waves in bubbly in the associated change of the pitch of the sounds produced. liquids: Comparison between theory and experiments,” J. Acoust. Soc. Am. 85͑2͒, 732–746 ͑1989͒. The apparatus can also be used to verify an effective medium 21 model of sound propagation in bubbly liquid, which relates L. E. Kinsler, A. R. Frey, A. B. Coppens, and J. V. Sanders, Fundamen- tals of Acoustics, 4th ed. ͑Wiley, New York, 2000͒. the void fraction to the speed of sound. 22 T. G. Leighton, The Acoustic Bubble ͑Academic Press, London, 1994͒. 23 A. B. Wood, A Textbook of Sound, 1st ed. ͑MacMillan, New York, 1930͒. ACKNOWLEDGMENT 24 D.-Y. Hsieh and M. S. Plesset, “On the propagation of sound in a liquid containing gas bubbles,” Phys. Fluids 4͑8͒, 970–975 ͑1961͒. This work was supported by the U.S. Navy Office of Na- 25 A. Prosperetti, “The thermal behavior of oscillating gas bubbles,” J. Fluid val Research Ocean Acoustics Program. Mech. 222, 587–616 ͑1991͒. 26 A. Crespo, “Sound and shock waves in liquids containing bubbles,” Phys. Fluids 12͑11͒, 2274–2282 ͑1969͒. ͒ a Electronic mail: [email protected] 27 A. E. Ruggles, H. A. Scarton, and R. T. Lahey Jr., “An Investigation of 1 A. Mallock, “The damping of sound by frothy liquids,” Proc. Royal Soc. the Propagation of Pressure Perturbations in Bubbly Air/Water flows,” in 84͑A 572͒, 391–395 ͑1910͒. First International Multiphase Fluid Transients Symposium, edited by H. 2 Arnulph Mallock worked for Lord Rayleigh in the 1870s and conducted H. Safwat, J. Braun, and U. S. Rohatgi ͑American Society of Mechanical experiments in the hydraulic laboratory at Terling, Lord Rayleigh’s estate, Engineers, New York, 1986͒, pp. 1–9. as discussed in R. J. Strutt, Life of John William Strutt, Third Baron 28 D. T. Blackstock, Fundamentals of ͑Wiley, New York, Rayleigh, O. M., F. R. S ͑University of Wisconsin Press, Madison, 1968͒. 2000͒. 3 L. Rayleigh, “On the pressure developed in a liquid during the collapse of 29 Hamilton Company; 4970 Energy Way; Reno, Nevada 89502 USA. a spherical cavity,” Philos. Mag. 34,94–98͑1917͒. http://www.hamiltoncompany.com/ 4 M. Minnaert, “On musical air-bubbles and the sounds of running water,” 30 The lapel mic must be made water proof. This can be done by coating the London, Edinburgh Dublin Philos. Mag. J. Sci. 16, 235–248 ͑1933͒. microphone with rubber, such as Plasti Dip, available at hardware stores. 5 C. Crowe, M. Sommerfeld, and Y. Tsuji, Multiphase Flows with Droplets http://www.plastidip.com/home_solutions/Plasti_Dip and Particles ͑CRC Press, Boca Raton, 1998͒. 31 The field inside the waveguide is spatially dependent according to Eq. 6 L. E. Faidley and J. A. I. Mann, “Development of a model for acoustic ͑16͒. One can accentuate or suppress a mode by placing the hydrophone liquid manipulation created by a phased array,” J. Acoust. Soc. Am. near an antinode or a , respectively. Since the excitation is broad- 109͑5͒, 2363 ͑2001͒. band, a number of modes are excited, but the relative amplitude of a 7 H. B. Karplus, “The velocity of sound in a liquid containing gas particular mode received on the hydrophone depends on the hydrophone’s bubbles,” Technical Report No. C00-248, Armour Research Foundation location inside the waveguide.

981 Am. J. Phys., Vol. 76, No. 10, October 2008 Apparatus and Demonstration Notes 981 This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.83.205.78 On: Mon, 11 May 2015 21:48:30

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