Abstract

Measurements of sound attenuation by a warm air fog are presented for dimensionless frequencies, ωτ/Cm (where τt is the thermal relaxation time for the droplet, ω the circular acoustic frequency, and Cm the liquid mass fraction), near unity where the effects of mass transfer are dominant. The tests are made in a Wilson cloud chamber by measuring the rate of decay of the fundamental mode of acoustic oscillation which is excited during the operation of the chamber. Measurements of pressure and volume are made continuously during the expansion. Droplet size and concentration of the monodisperse fog are determined from time-resolved measurements of the optical transmission at two wavelengths of light. The range of ωτ/Cm from 0.5– 16 is scanned by varying droplet size and concentration through the controlled addition of condensation nuclei. This range includes the Napier frequency (frequency for which attenuation per unit wavelength is maximum), and the resulting data indicate that a maximum dimensionless attenuation occurs for a value of ωτ/Cm≈1, as predicted by the theory of Cole and Dobbins. The magnitude of the maximum dimensionless attenuation is approximately 100 times that which would be obtained for a corresponding nonvolatile aerosol.

202 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME28:Measurements of the Attenuation of Sound by a Warm Air Fog Jo~s E. CoLz, III,' A~D RXC~AI~D A. DoB~n~sDivision of Engineering, Brown University, Providence, R. I.(Manuscript received 22 May 1970, in revised form 14 November 1970)ABSTRACT Measurements of sound attenuation by a warm air fog are presented for dimensionless frequencies, ~ort/Cm(where rt is the thermal relaxation time for the droplet, ~o the circular acoustic frequency, and C,, the liquidmass fraction), near unity where the effects of mass transfer are dominant. The tests are made in a Wilsoncloud chamber by measuring the rate of decay of the fundamental mode of acoustic oscillation which isexcited during the operation of the chamber. Measurements of pressure and volume are made continuouslyduring the expansion. Droplet size and concentration of the monodisperse fog are determined from timeresolved measurements of the optical transmission at two wavelengths of light. The range of ~rdC~ from0.5-16 is scanned by varying droplet size and concentration through the controlled addition of condensationnuclei. This range includes the Napier frequency (frequency for which attenuation per unit wavelength ismaximum), and the resulting data indicate that a maximum dimensionless attenuation occurs for a value ofo~rt/C,,= 1, as predicted by the theory of Cole and Dobbins. The magnitude of the maximum dimensionlessattenuation is approximately 100 times that which would be obtained for a corresponding nonvolatile aerosol.1. Introduction Experimental investigations of the effect of a fog onthe attenuation of sound have been reported by severalauthors. Qualitative tests performed outdoors in naturalfogs by Derham (1708) and by Tyndall in 1873 (seeTyndall, 1964) yielded little information on the effectsof droplets on sound attenuation, although refractiveeffects due to wind and thermal gradients were foundby Tyndall to be important considerations in a naturalenvironment. A reverberation technique was used byKnudsen et at. (1948) to obtain quantitative data onsound attenuation due to the presence of water droplets.The fog was created by use of a paint sprayer, anddroplet size was determined by photographing a sampleof the fog collected on an oil<oared glass slide. Therange of droplet diameters reported in a typical fogwas from 5-50 ~m, and the lowest acoustic excitationfrequency was 500 Hz. Wei (1950) made measurementsof sound attenuation by spray-produced fogs at lowaudible frequencies using a standing-wave tube. In thisexperiment, droplet size was also determined by collection on an oil-coated glass slide which was at a fixedlocation in the tube. Wei also reported a spectrum ofdroplet sizes, but he did not discuss the spatial uniformity of this distribution along the length of thechamber. In a more recent attempt to obtain quantitative data on sound attenuation in an outdoor environment, Wiener (1961) conducted an investigation innatural fogs off the coast of Maine. Evaluation of the ~ Present affiliation: Department of Mechanical Engineering,Tufts University, Medford, Mass.attenuation by the fog was not possible from these testsbecause of the refractive effects of the winds which werepresent. The history of the theory of the propagation of soundthrough fog was recently reviewed by Cole and Dobbins(1970) who also provide a theory that is comprehensiveto the extent that consideration is given to the combinedeffects of the transfer of momentum, heat and massbetween the liquid and gas phases. A similar theory isgiven by Marble and WoOten (1970) in calculatingsound propagation in a volatile aerosol but they did notconsider in detail the case of atmospheric fog. In the analysis of Cole and Dobbins the conservationequations are written for the droplets as well as for thegaseous mixture of a noncondensing gas (i.e., air) andvapor. Source terms appear in the equations whichaccount for mass, momentum and heat transfer betweenthe gaseous mixture and the droplets. The equationsare linearized and non-dimensionalized, and a solutionis obtained for the attenuation and dispersion of aplanar acoustic wave. Relaxation times for the mass,momentum and heat transfer processes are defined fromthe parameters of the equations. For instance, therelaxation time for the heat transfer process~ isrt=Dapt/(12aepo). In the case of the nonvolatileparticles, r, is approximately the time for an impulsivelyapplied unit temperature difference between the gasand particles to decrease to e-x when heat is transferredby gaseous conduction. A maximum value of attenuation per unit wavelength owing to the heat ta'ansferSee Appendix for list of symbols.MAr~c~r1971 JOHN E. COLE, III, AND RICHARD A. DOBBINS 203t)rocess is found to occur for a value of wrt~ 1. A timeequal to rt/U~ (where C,~ is the liquid mass fraction)is found to be the relaxation time for the mass transferprocess in the case of volatile droplets, and amaxin~um attenuation per unit wavelength occurs for~------wrt/C,,,ml. Since Cm(10-a for natural fogs andclouds, the relaxation time for volatile droplets is muchlarger than the relaxation time for nonvolatile particles,and the dimensional frequency for which mass transferis important is in the infrasonic region. Mass transfereffects become insignificant for large values of ~, and theattenuation of sound by volatile droplets becomes thesame as that for nonvolatile partldes. Neither Knudsenet at. (1.948) nor Wei (1950) investigated the region near~= 1 where the effects of mass transfer are important.Most of the data of Knudsen et al. were taken for valuesof ~>/1000. Wei worked in the range of low audiblefrequencies, and in one of his tests - was as low as 7.9 ascMculated using the most probable droplet diameter ofhis fog. The object of the present experimental study is toexamine the attenuation of sound by a fog for values of~ 1. The tests are made using a Wilson cloud chamberin which a cloud of nearly uniform size droplets (i.e., amonodisperse cloud) is produced. Droplet size andmkmber density are determined from measurements ofoptical transmission at two wavelengths of light asfunctions of time. Since the cloud is monodisperse, theoptical transmission tests can be interpreted using Miescattering theory for single-size droplets. The ability toproduce a monodisperse doud and to measure ch-opletsize and number density allows the relaxation time(i.e., rt/C'~) to be clearly defined in the experiment andreadily determined, and the acoustic theory for singlesize droplets to be tested. The size and concentrationof droplets produced in the chamber can be varied overa wide range by controlling the addition of condensationnuclei. Very dense fogs are produced in this manner withmass fractions much greater than those found in atmospheric fogs; typical values obtained in our cloudchamber arc C,~=iO-~ or liquid water contentm6.3gmm 'a. With this capability, the region near - = 1 isinvestigated using an audible frequency of --~80Acoustic tests performed in the cloud chamber areconducted before the effects of heat transfer from thechamber walls become importa~nt, and the attenuationdue to the presence of the fog can be easily separatedfrom other sources of attenuation.2. Cloud chamber description and operating procedure The cloud chamber is a hmSzontally mounted,smooth-walled steel cylinder with an internal diameterof 15.2 cm (Fig. 1). The test volume of the chamber canbe expanded to ratios as high as 1.8 by the movement ofa planar piston. The initial length of the chamber is1.28 m. The piston is sealed against the walls of thechamber by two sets of rubber "O" rings. A test sectionis located 1.23 m from the fixed end of the chamber andcontains quartz windows, which are flush-mounted withthe inside of the chamber, and which allow opticaltransmission tests to be performed. A pressure transducer is also placed at the test section. Ports for fillingand evacuating the chamber and also for connecting themercury manometer and the microphone are locatedat the fixed end of the chamber. All lines that feed intothe chamber are fitted with valves near the chamber inorder to minimize external volumes. The operating procedure for the cloud chamberbegins with the evacuation of the test volume to a pressure < 1 torr. A measured amormt of water is then addedto the volume through a glass capillary tube located at afill-port halfway along the chamber. The water isallowed to evaporate 15-30 min until the vapor pressure,as measured by a mercury manometer, reaches a steadyvalue. Air (so-called zero grade, with constituents otherthan O~ and N~ amounting to less than 0.0240/o) is thenadmitted through ports located at the test section, atthe midpoint of the chamber, and at the fixed end of thechamber. This multi-port filling procedure is requiredto achieve spatial homogeneity of the mixture of airand water vapor. An externally-driven fan, which islocated at the fixed end of the chamber, is used foradditional mixing on some occasions. As the air isadmitted into the test volume, it may be passed througha continuous 60-Hz spark in order to add sites fornucleation. The concentration of condensation nucleiis controlled by the duration of the spark excitation.The control of the duration of' the spark permits theconcentration of the fog to be varied by a factor of 100and the droplet size correspondingly varied by a factorof 6. The test volume is filled to atmospheric pressureand the right-hand side of the chamber is evaculatedto a pressure <100 mm of mercury. During thisprocedure, the piston is retained by wedges held inposition by a movable fork. The piston is released byquickly lowering the fork, which allows the wedges to beexpelled. As the piston travels, it actuates a brake whichsqueezes the piston shaft and brings the piston to asmooth stop. There are no fixed locations for stopping204 JOURNAL OF THE ATMOSPHERIC SCIENCES Vo~.tmvz28 ~.~oPHOTOMU LTiPLiER DETECTOR NOCHROMATOR' r-TEST SECTION '.~ ~'- MONOCHROMATOR LAMP.~ PHOTOMULTIPLIER DETECTORFro. 2. Schematic diagram of ,the optical system used for transmission tests.the piston, and the exact final volume is not knownprior to making a run. The total time required for theexpansion is normally 0.20 sec but varies with the pres~sure ratio across the piston.3. Instrumentation The cloud chamber is instrumented to provide measurements of pressure, piston displacement, acousticpressure, and optical transmission at two wavelengthsof light as functions of time. Electrical signals from thevarious transducers are suitably amplified and fed intoa 12-channel oscillograph (Consolidated Electrodynamics Corporation model 5-124). The galvanometersin the oscillograph have a time response <0.005 secand provide ample time resolution. Pressure is measured with a transducer of the straingage type (Consolidated Electrodynamics Corporationtype 4-313-001, 0-50 psia) with a time response ~0.005sec. High accuracy in the recorded pressure signal of thelatter portion of the test is achieved by expanding thescale and recording only the latter portion of the pressure signal. The absolute calibration of this transducer ismade using an 800-mm mercury manometer. This calibration is checked after each run by determining thepressure in the chamber both with the transducer andthe manometer after thermal equilibrium at room temperature has been restored (this usually occurs a minuteafter cessation of piston motion). Piston displacement is continuously measured by aslide-wire type of transducer attached to the end of thepiston shaft. The calibration of this transducer is madeat the end of every run by measuring final piston positionwith a meter scale. In order to measure the droplet size and numberdensity of the fog, continuous spectral transmissiontests are performed at two optical wavelengths duringthe expansion of the chamber. The optical equipmentshown in Fig. 2 consists of a tungsten light source andlens system which produces a collimated beam passingthrough the quartz windows of the test section. Afteremerging from the test section, the beam is focused by areceiver lens, split by either a high-pass filter or a halfsilvered mirror, and focused on the apertures of twomonochromators. Light passing from the monochromators goes through narrow-band-pass filters and tophotomultiplier detectors. The spectral transmission curves; that. are recordedduring the formation and growth of the cloud displayan oscillatory behavior (Fig. 3). Examination of the Miescattering coefficient reveals that as the parameter~rD/X (where x is the wavelength o:i light:) is increased,tlhe scattering coefficient also exhibits an oscillatorybehavior. The observed oscillations of the spectraltransmission curves are therefore interpreted as resultingfrom the oscillation of the scattering coefficient as thedroplets of a monodisperse cloud increase in size. Calculation of droplet number density and size at any timeduring the growth of the droplets is made using thetransmission law. Details on the method, of interpretation of the optical transmission curves used to obtainfinal values of droplet diameter and concentration aregiven by Cole et al. (1970). The acoustic pressure at the fixed[ end of the chamberis measured by a 1-inch diameter piezoelectric microphone (Bruel and Kjaer type 4117) mounted externallyto the cloud chamber. The microphone was expresslydesigned for use in humid atmospheres and was modifiedto assure good low-frequency response after the suddendecrease of pressure resulting from the motion of thepiston (Cole, 1970). The microphone housing is connected to a port in thefixed end of the cloud chamber by flexible plastic tubingwhich isolates the microphone from mechanical vibrations generated during the operation of the chamber.4. Acoustic tests A fog created in the cloud charaber completely disappears within 10-15 sec after cessation of piston motionas a result of heat transfer from the chamber walls.Therefore, a method for determining acoustic attenuation resulting from the presence of the fog must permit:measurements before any significant alteration occurs:in the state of the fog due to heating from the chamberwalls. Total acoustic attenuation in the,' chamber (i.e.,attenuation due to both the fog and the wall losses)immediately after cessation of piston motion is meassured by observing the rate of decay of the fundamentalmode of acoustic oscillation which is excited during theexpansion. During the motion of the piston, pressuredisturbances propagate to the fixed wall of the chamberand reflect back producing an acoustic field whichcontains many frequencies. After cessation of pistonmotion, the acoustic field in the chamber begins to decay(Fig. 3), and the frequency content is limited to resonantfrequencies of the chamber. In almost all runs, the onlysignificant frequency present in the chamber during thedecay was the longitudinal fundamental frequency of---80 Hz. Occasionally, the second harmonic was superimposed, and in these cases no quantitative datareduction was performed.MARCn1971 JOttN E. COLE, III, AND RICHARD A. DOBBINS 205 In the theoretical analysis of Cole and Dobbins(1970), the acoustic pressure is assumed to decay asp(x,t)=p(O,t)e-~x/'~, where a is the spatial coefficientof acoustic energy attenuation for a plane progressivewave, p(O,t)'the amplitude of acoustic pressure atx=0, and p(x,t) the amplitude of acoustic pressure at x.In the experiment the microphone measures the decayof acoustic pressure of a standing wave at the fixed waltof the cloud chamber as a function of time. This temporal decay is expressed as p(O,t) =p(O,O)e-at/2, where fiis the temporal coefficient of acoustic energy attenuation, p(0,0) the amplitude of acoustic pressure at theend of the chamber at t=0, and p(O,t) the amplitude ofacoustic pressure at the end of the chamber at t. Therelationship between the spatial and temporal attenuation coeficients is aa =fi, where a is the speed of sound(Lindsay, 1960). The measurement of the decay of thestanding wave in the cloud chamber is thereforeexpressed as p(O,t)---p(O,O)e-"~/~. The reduction of the acoustic data on the oscillographchart proceeds by fitting a smooth curve through thepoints of maxlmmn amplitude along one side of theoscillation, e.g., the negative side. A point of maximttmamplitude on the opposite side of the oscillation nearthe beginning of the decay of the signal is noted as thezeroth cycle, and time is called zero at this point. Thedistance from the point of maximum amplitude to thesmooth curve at t:0 is measured. This distance istwice the anaplitude of the acoustic pressure at the endof the chamber at t = 0, or 2p (0,0). Similar measurementsare made for at least seven cycles following the zerothcycle in order to obtain 2p(0,t). At any particular cycle,the time is given by NX/a, where X is the acousticwavelength and ~Y=0, 1, 2,-.. is the number of thecycle (e.g., ~V=0 corresponds to the zeroth cycle).Therefore, the measurements of the acoustic decay areexpressed asp(0, NX . -~)=p(O,O)e-'''~x/~, ~V=0,1,2, ....The solution of this equation is2 / /0NThe parameter of interest in the acoustic theory isaoa/w, where a0 is the speed of sound through humid air.The dispersion resulting from the presence of thedroplets is sm~ll, so that a-~ a0; therefore,a0aThe final measured value of aoa/co is taken to be theaverage value of the calculation over the number ofcycles measured.PISTON X=.365/zm~2o0 ~oo time (m sec) o J PRESSURE /.546H, mDISPLACEMENTs. ~' V /[ X 'I I I I I I I I I I I I I I I400 .300 time (m sec)3. Tracing of an[oscillograph chart from a typical test. When the total dimensionless attenuation is large(e.g., aoa/co=6-8XlO-~), the amplitude of the signalbecomes too small to be accurately measured on the17.8-cm wide oscillograph chart after 12-16 cycles.Normally, use of the first two to three cycles immediately following the end of the expansion leads to spuriousresults, and a constant value for the attenuation coefficient is only obtained after this amount of time. Tests were conducted with dry air in order to obtainmeasurements of attenuation solely due to wall losses.Initially these attempts did not produce repeatableresults from one test to the next. The lack of reproducibility was presumed to result from the variationfrom run to run of the final concentricity of the clearancebetween the piston and chamber walls. Therefore, wallattenuation in the chamber is measured after each testwithout disturbing the piston. It was found that resonant modes of the chamber can be excited by placing anelectro-mechanical vibrator on the support frame nearthe chamber and tuning its frequency of excitation. Theacoustic mode easiest to generate in this manner is thelongitudinal fundamental mode. In order to demonstratethat this method does excite acoustic modes in thechamber rather than transmit mechanical vibration tothe microphone, the vibrator is excited with a resonantfrequency of the chamber. The microphone thenprovides a signal in response to the acoustic field at theend of the chamber; however, when the flexible connection between the microphone and the chamber is pinchedshut, there is no signal observed from the microphone.A spectral analysis of the continuous sound field in the206 JOURNAL OF THE ATMOSPHERIC SCIENCES VO~.UME28 .08 3 .04 .02 0 I ~ I I I I I I I I .2 .4 .6 .8 I ~ 2 4 6 8 I0 20 Fro. 4. Dimensionless attenuation vs ~. The theoretical curvesapply to the data points as indicated where T~=281K, T2=276Kand Ta-~271K.chamber excited by the fundamental frequency, usingthis method, shows no significant generation of otherfrequencies. The second and third harmonics are theonly other frequencies observed, and these are down atleast 45 dB. The test to determine attenuation due towall losses is performed after every run by first tuningthe vibrator to the fundamental resonant frequency ofthe expanded chamber. A gating device (General Radiomodel 1396-A Tone Burst Generator) allows a nearlysteady sound field to build up in the chamber and thenstops the excitation. The resulting transient decay isrecorded by the oscillograph. The same procedure ofdata reduction is used for this method; however, theresults are averaged over at least 20 cycles. Comparison of the results of the two methods (i.e.,excitation by piston cessation and excitation by electromechanical vibrator) for determining acoustic loss inthe chamber was made for tests using dry air. Theattenuation was first determined by measuring the rateof decay of the fundamental mode excited by the rapidexpansion of the dry air. After thermal equilibrium atroom temperature occurred, the attenuation in thechamber at its final configuration was also determinedby use of the vibrator. The results of both methods agreewithin the limits of experimental accuracy (<q-10%).Values of wall attenuation obtained are 20-100% largerthan the predictions of the Helmholtz-Kirchoff theory,as would be expected to result from the presence ofseveral ports and small external volumes connected tothe chamber. These tests to determine wall losses areperformed at temperatures differing by ---22C; however,if the temperature dependence of the wall attenuationcan be scaled in the same manner as that given by theHelmholtz-Kirchoff result, this effect is small.5. Thermodynamic state of the fog The initial thermodynamic state of the humid air inthe cloud chamber is known from measurements of theamount of water added, total pressure, temperature andvolume. For the temperatures and pressures encounteredin the cloud chamber, the assumption of perfect gasbehavior for both the air and the water vapor as well astheir mixture is known to be realistic (see Dufour andDefay, 1963, pp. 55-59). The relaxation time for masstransfer from the droplets (r~/C,~) is much larger thanthe relaxation time for either the thermal or the viscous.transport processes. Since the large:st value of rt/C,~ is20 msec for the fogs created in the chamber, completethermodynamic equilibrium between the droplets andtlhe gas exists following the expansion 'within 'a timeequal to two periods of the acoustic oscillation. This istlhe minimum amount of time required for the acousticpressure signal to begin to display a uniform decay;therefore, mass, viscous and therma![ transport processesbetween the droplets and the gas that are establishedduring the expansion are not important when theacoustic measurements are made. The final thermodynamic state of the fog is determined by first equating the total pressure at the finalstate to the sum of the partial pressures of the air andthe water vapor; then, the partial pressure of the air isevaluated using the perfect gas relationship. Thus, wehave P/= P,~-t-Pw' = (maR,T//Vy)-J-P~i.The mass of the air in the chamber (m,) is constantthroughout the expansion and therefore can be evaluated at the initial state. Measurements of final totalpressure (Py) and volume (Vy) are obtained from theoscillograph chart. The existence of complete thermodynamic equilibrium between the droplets and thesurrounding gas specifies the relationship between vaporpressure (P~y) and temperature (T~). Calculations byDufour and Defay (1963, pp. 70-90) show that, forliquid droplets of diameter >l/~m, surface-tensioneffects are too small to significantly influence the valuesof the thermodynamic properties for water applicablefor an infinite radius of curvature. Final temperature isthen determined by trial and error using tabulatedvalues of P~ and T.6. Results and conclusions For most of the fogs created in the cloud chamber,~=0(1) and C,~ 10-2; therefore, the simplified resultsof the acoustic theory of Cole and Dobbins (1970) forCm~<<l are applicable. In this lirnit viscous effects areunimportant, and the attenuation of acoustic energyresults from the heat and mass transfer processes. Theresult for the dimensionless attenuation aao/cO that isapplicable for Cm~<<l is (~L - ~-) D (L - 1) - L-I~ / 0/co / ~L (L- 1 F~L':~+/- (! -~)q~/ *~+/'-'=---'--=--'/~, (1) L AL~+BN~ _1 Jwhere a is the spatial attenuation coefficient of acousticMAUC}~1971 JOHN E. COLE, II1, AND RICHARD A. DOBBINS 207energy (equal to twice the attenuation coeffident foracoustic pressure), ~=p~o/po (where the subscript 0indicates the equilibrium or mean values), L = L/(R~To),.~=c~,~(%--l)/ct%, and l~--cv/ct. Since the Lewisnumber is of O(1) and P<~l, the second term in thedenominator of Eq. (1) is nearly equal to unity andtherefore independent of temperature. The coefficientof v in the numerator of (1) is dependent on temperaturethrough the latent heat and the saturated vapor density. The calculation of ~ from the experimental measurements is made by subtracting the wall attenuation fromthe attenuation measured in the presence of the fog.The results of the measurements of ~ performed over arange of e from 0.4-16 appear in Fig. 4, where thetheoretical curves, calculated using Eq. (1), are presented to compare with three particular data points thatrepresent the extreme and mean fog temperature for allof the data. Values of droplet size, droplet numberdensity, acoustic frequency, and temperature of the fogfor the three indicated points appear in Table 1. Thedifference between the maximum and minimum fogtemperature encountered in the data is 10K. Dropletdiameter varies from 1.8-1.0/~m and droplet numberdensity correspondingly varies from 2X10~-i. IX104cm-~ over the measured range of ~. A detailed table ofall measured and calculated values for several fogs isgiven by Cole (1970). The dependence of Eq. (1) on temperature can beaccounted for by plotting OF(T) vs ~, where ~g (L- 1)~+~iV~ F(~')--- (2) (~iL-~)[, (L- 1)and is the inverse of the coefficient of e in the numeratorof (1). In Fig. 5 a comparison is made of the theoreticaland experimental values of oF (T) vs e. The experimentalresults are plotted using a mean value of Lewis number(N~.~--0.839) for the temperatnre and pressure rangeencountered in the data. The amount of scatter in thedata, however, is not measurably reduced in thispresentation. In both Figs. 4 and 5 the value of thermal conductivity for the nfixture of air and water vapor is takento be that of pure air. Values of the binary diffusioncoeffi-ient for water vapor and ah' at ambient conditionsare obtained from the observed values reported inFuller et al. (1966). The binary diffusion coefficient isassumed to vary with the 1.75 power of temperatureand inversely with pressure.T,x~,~. t. Values of properties for fogs indicated on Fig. 4. D C, f TaFog (#m) (cm-a) (I2Iz) (-K) 1 4.36 1.7lx10s 80.5 281 2 1.84 1.52X 10- 79.0 276 3 4.94 7.74X104 81.0 271 0,5 0.4 ~0.3b.Io O.Zo.I + I I I I t I I I I I I .10 .Z .4 .6 .8 I 2 4 6 8 I0 20 .~Fro. 5. Dimensionless attenuation normalized with respecttemperature vs rr where T~=281K and T~--271K. ]-For some of the data indicated on Figs. 4 and 5,~10 and C~10-s; therefore, Eq. (1), which isobtained from the exact theoretical result in the limit ofC,~?~<<I, is not strictly applicable for these data points.The exact theoretical equation appears as Eq. (24) inCole and Dobbins (1970), and it agrees within 3% withthe results that are calculated using (1) over the rangeof f from 0.2 to 10. For values of ~ > 10, the results ofEq. (1) begin to diverge from those of the exact theoretical equation.~ Some of the critical relationships between the theoryand the experiment are as follows: 1) The droplets created in the chamber are of uniformsize, as indicated by the observations of Tyndallspectra and the internally consistent results obtainedfrom the data reduction of the optical transmission testsassuming monodispersity. Therefore, the acoustictheory presented for single-size droplets is applicable. 2) Measurements of total acoustic attenuation aremade within 0.20 sec after completion of the expansion.Within this time, pressure in the chamber and opticaltransmissions are nearly constant (Fig. 3), indicatingthat the state of the fog is very nearly uniform duringthe time when the acoustic measurements are made. Theexistence of a warm thermal boundary layer resultingfrom wall heating is an effect not considered by thetheory. Tests using dry air to determine wall loss, however, show agreement between the results of the twomethods of determining wall attenuation, previouslydescribed, within the limits of experimental uncertainty.Thus, the acoustic attenuation in the chamber owing towall losses is not measurably influenced by the wallthermal gradient which results from the rapid expansion. 3) The magnitude of the wall attenuation comparedto the total attenuation varies from 0.5, for the highervalues of ~, to 0.1 in the region near to ~= 1. Therefore,especially in the vicinity of the Napier frequency, valuesof ~ are not very sensitive to small errors in the determination of wall losses. The attenuation of sound byhumid air is also insignificant at these low frequencies(see Harris, 1967).208 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME28 4) The presence of gravitational settling of the fogdroplets does not influence the interpretation of theacoustic experiment as long as the particles fall at avelocity corresponding to the Stokes' flow regime. Inthis case the convective terms in the equations of motionremain negligible and superposition of solutions to theequations of motion is possible. 5) Since the diameter of the droplets is at least eighttimes larger than the mean free path of the surroundinggas molecules, near-continuum transport effects wouldbe important only if the thermal accommodation coefficient or the evaporation-condensation coefficient weresmall. Calculations made to investigate the effects ofnear-continuum heat and mass transport on acousticattenuation are described by Cole (1970). A value forthe thermal accommodation coefficient is taken to benear unity (qOT=0.70), as seems to be indicated by theliterature. From these calculations it does not appearthat the experimental data are compatible with a valueof evaporation-condensation coefficient -m as small as0.01. However, the scatter in the data would preventthe recognition of a smaller departure of ~m fromunity. There is a moderate amount of scatter in the experimental data, which is plotted in Figs. 4 and 5. Scatter ofthis order of magnitude (less than 15%) is understandable in view of the difficulties involved with performingthese experiments. The data also deviate systematicallyfrom the theoretical predictions for attenuation, andon the average, the attenuation measurements are 35%below the theoretical curves. The cause for this deviationis not apparent. In Figs. 4 and 5 the maximum measured value ofdimensionless attenuation occurs in the region neary = 1. The specific location of the maximum dimensionless attenuation cannot be determined from an examination of only the measurements because of the scatter inthe data; however, a decrease in ~ is observed for valuesof y much different from unity. The measurements ofthe maximum dimensionless acoustic attenuationresulting from the presence of the fog are approximatelyi00 times greater than corresponding measurements ofacoustic attenuation in a nonvolatile aerosol (seeTernkin and Dobbins, 1966). On this basis, the deviationof the measurements from the theory indicated in Figs.4 and 5 is not as significant. The relatively large amountof acoustic attenuation at low frequencies, which hasbeen measured in these experiments, reflects the importance of the mass transfer process. The appearance of themaximum dimensionless attenuation in the region nearoort/C,~: 1 is in agreement with the predictions of thetheory.Administration through their graduate traineeshipprogram. Special thanks also go to Mr. Thomas A. Raso for his technical assistance with the experiment. APPENDIX List of Symbolsa frequency-dependent speed of sounda0 speed of sound in the gaseous mixture of air and water vapor (a0 equals the speed of sound through the fog in the limit of infinite frequency)c~ specific heat at constant pressure per unit mass of gaseous mixture of air and water vaporc~ specific heat at constant pressure per unit mass of water vaporct specific heat of liquid waterC~ mass of liquid per unit mass of gaseous mixtureC~ number of droplets per un!it volume (number density or concentration of the droplets)D droplet diameterf acoustic frequency (f equals the number of acoustic oscillations per unit time) latent heat of vaporization mass of air Lewis number, ratio of diffusivity of heat to the diffusivity of massp acoustic pressureP total thermodynamic pressureP~ partial pressure of the airP~ partial pressure of the water vaporR~ gas constant per unit mass of airR, gas constant per unit mass o:f water vaport timeT temperatureV volume of the chamberx distancea spatial attenuation coefficient for acoustic energy or intensity (a equals two times the spatial attenuation coefficient for acoustic pressure) dimensionless attenuation parameter (~=aao/w) thermal diffusivity of the gaseous mixture of air and water vapor temporal attenuation coeflScient for acoustic energy or intensity specific heat ratio for the gaseous mixture of air and water vapor specific heat ratio for water 'vapor wavelength density of liquid water equilibrium density of gaseous mixture of air and water vapor equilibrium density of water vaporLma~Le(~T~-~~,~xpzpopro Acknowledgments. The authors wish to acknowledgethe financial support from the Office of Naval Researchthrough Contract N00014-67-A-0001. Additional support was given by the National Aeronautics and Space REFERENCESCole, J. E., 1970: Measurements of sound attenuation by a warm air fog. Doctoral dissertation, Brown University, Providence, R.I.M~u~cu197t JOHN E. COLE, II1, AND RICHARD A. DOBBINS 209 , and R. A. Dobbins, 1970, Propagation of sound through atmospheric fog. J. Atmos. Sci., 27, 426-434. , and H. Semerjlan, 1970: Time-resolved measurement of droplet size and conccutratlon in cloud chambers. J. Appl. Meteor., 9~ 684-689.Derham, R., 1708: Experiments on the motion of sound. Phil. Trans. Roy. So-. -ondon, S, 380-395.Dufour, L., and R. Delay, 1963: Thermodynamics of Clouds. New York, Academic Press, 55-59 and 70-79.Fuller, E. N., P. D. Schettler and J'. C. Giddlngs, 1966: A new method for prediction of binary gas-phase diffusion coeffi cients. Ind. l~ng. Chem., SS, 19-27.Harris, C. 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