NUSC Technical Document 849524 February 1989

AD-A206 537

Low Frequency Noise andBubble Plume OscillationsPresented at the

114th Meeting of theAcoustical Society of America,16-20 November 1987, Miami, Florida

William M. CareySurface Anti-Submarine Warfare Directorate

J. W. FitzgeraldKildare Corporation

Naval Underwater Systems CenterNewport, Rhode Island / New London, Connecticut

Approved for public release; distribution Is unlimited.

Preface

This report contains the oral presentation "Low Frequency Noise and BubbleOscillations" given at the 114th Meeting of the Acoustical Society of America, 16-20November 1987. in Miami. Florida.

Reviewed and Approved:

W. A. Von WinkleAssoicate Technical Directorfor Research and Technology

RZCURITY CEMSIFICATION OF; T141S PACEj A17 ~po-40-REPORT DOCUMENTATION PAGE

!s. REPORT SECURITY CLASSIFICATION lb. RESTRICTIVE MARKINGSU NCLASS IFIED ____________________________

2a. SECURITY CLASSIFICATION AUTHORITY 3. DISTftIBUTION /AV/AILABILITY OF REPORT

2b. ECLASWIATIO/DONGRAINGSCHEULEAPPROVED FOR PUBLIC RELEASE2b. DCLASIFICAIO~jOWNGAOINGSOIEUI.EflSTR IBUTION UNL IMITED

4. PERFORMING ORGANIZATION REPORT NUMBER(S) 5. MONITORING ORGANIZATION REPORT NUMBER(S)

NUSC TD 84956a. NAME OF PERFORMING ORGANIZATION 6b. OFFICE SYMBOL 7a. NAME OF MONITORING ORGANIZATION

Nc AE SS LONON State,andZIP Code). 7b. ADDRESS (Otey, Statip. mnd ZIP Code)

NEW LONDON, CT 06320

Sa. NAME OF FUNDING I SPONSORING 8 b. OFFICE SYMBOL 9. PROCUREMENT INSTRUMENT IDENTIFICATION NUMBERORGANIZATION O (f ap,ikaWOe

a8. ADDRESS (City, State, and ZIP Code) 10. SOURCE OF FUNDING NUMBERSPROGRAM IPROJECT ITASK IWORK UNIT

ELEMNT N. NO 1 N. ~ A~SSIN N.

11 TITLE (include Secunty Ciassficition) LMN O O O CESO

LOW FREQUENCY NOISE AND BUBBLE PLUME OSCILLATIONS

12. PERSONAL AUTHOR(S)

16 SUPPLEMENTARY NOTATION

17 COSATI CODES 18. SUBJECT TERMS (Continue on reverse if necessary ndidentify by block numcier)FIELD GROU-P _SUB-GROUP

19 ABSTRACT (Continue on reversit it necoisay and identify by block numnber)

The evidence for a wind dependence in low-frequency ambient noise (200 Hz) is sparse dueto the corrupting influences of the radiated sound from ships. Measurements made withhydrophones below critical depth, in sparse shipped basins, or at high sea states showindications of two distinct regimes associated with the occurrence of breaking waves anda variation of root-mean-square pressure with the square of the local wind speed. In aprevious paper [W. Carey, J. Accust. Soc. A\m. Supi . 1 78, Sl (lgSS)ij, the generatiocn ofsound by wave turbulence interaction at low sea states (WS <l0m/s) and collective bubbleOscillations driven by turbulence at high sea states was proposed (WS >l0m/s). This papershows that recently observed bubble clouds that penetrate to tens of meters below thesurface of the sea result in regions of low sonic velocity described by Wood's treatmentof air-bubble water Mixtures, but with a density close to that of water. It is shown thatsuch a region can be treated as a flexible body with mixture speed and density. c4-+e--,radiation from such a body would be nmonopole and dipole in nature; but due to the proximity,

DO FORM 1473. 84 MAR 83 APR cition may be used until exhausted. SECURITY CLASSIFICATION OF THIS PAGEAll other editions are oboleIte.

)Ecum1Tv cL.ASSIFICATO. OF TNis Raac

-- of the sea surface, only the resulting dipole source would be of importance. Theseregions could also have a resonant characteristic, and when driven by the turbulenceresult in sufficient radiated $ound. These results are similar to and consistent withthe analysis by A. Prosperetti - "Natural Mechanisms of Surface Generated Noise in theOcean", SEA SURFACE SOUND, B. Kerman (ed), Kulwer Acad. Pub., Boston, 151-171, 1988.

A- ,,... .5

£gcumgy"w CL,,ASIICATION OFr TWIS IIGg

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TABLE OF CONTENTS

Page

Background .............. .......................... 4

Summary ............... ............................ 7

The Low Frequency Approximation: Wood's Result ..... ........ 9

Experimental Verification of wood's Result ........... .I.11

Bubble Clouds as a Source of Sound? ................. ... 13

The Surface Image Interference Effect ............... .... 15

The Radiation From a Fluctuating Volume V ............. .... 17

The Oscillation of a Compliant Sphere ................ .... 19

Bubble Cloud Resonant Frequency .................... ... 23

Summary ......... ............................ ... 25

Bibliography ........ ......................... .... 26

I .

i/iReverse Blank

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LOW-FREQUENCY OCEAN SURFACENOISE SOURCES

W. CAREY (J. ACOUSTIC SOC. AM. 78(SI),1985)

CONCLUSIGAIS:

0 f< 2 Hz: WAVE-WAVE INTERACTION (KIBBLEWHITE).

0 20 < f< 200 Hz: DATA (VERTICAL/OMNI) SUGGEST WIND NOISEIS IMPORTANT.

* f< 200 Hz VERTICAL DIR. DET. BY DSE AND SSC.

* 20<f<200 Hz: THE SOURCE OF NOISE.

- BF < 5,SS < 3,WS < 8 - 10 m/s:WAVE TURBULENCE INTERACTION (YEN &PERRONE, GONCHAROV)

- BF > 5,SS > 3,WS > 8 - 10 m/s:BUBBLE NOISE-THAT COLLECTIVE BUBBLE OSCILLATIONSDRIVEN BY THE PSEUDO-SONIC AND TURBULENT PRESSUREFIELDS. DIPOLE DUE TO PRESSURE RELEASE SURFACE.

- TURBULENCE SPECTRA AND VOID FRACTION DATAREQUIRED.

The evidence for a wind dependence in low-frequency ambient noise (<200Hz) is sparse due to the corrupting influences of the radiated sound fromships. Measurements made with hydrophones below critical depth, in sparseshipped basins or at high sea states, show indications of two distantregimes associated with the occurrence of breaking waves and a variation ofroot-mean-square pressure with the square of the local wind speed. In aprevious paper [W. Carey, J. Acoust. Soc. Am. Supl. 1 78, Sl (1985)], thegeneration of sound by wave turbulence interaction at low sea states(WS<10m/s) and collective bubble oscillations driven by turbulence at highsea states was proposed (WS>10m/s). This paper shows that recently observedbubble clouds that penetrate to tens of meters below the surface of the searesult in regions of low sonic velocity described by Wood's treatment ofair-bubble water mixtures, but with a density close to that of water. It isshown that such a region can be treated as a flexible body with mixturespeed and density. The radiation from such a body would be monopole anddipole in nature; but due to the proximity of the sea surface, only theresulting dipole source would be of importance. These regions could alsohave a resonant characteristic and, when driven by the turbulence, result insufficient radiated sound. These results are similar to and consistent withthe analysis by A. Prosneretti [Nato ASI, "Natural Mechanisms of SurfaceGenerated Noise in the Ocean," (1987)).

N . • •| |

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(a)

(b)

2

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BACKGROUND:

This vu-graph shows the results of Su et al (1984) andillustrates the production of bubble clouds when wave breakingoccurs in tank water. The picture is striking and is not whatone would observe when wavebreaking occurs in salt water. Thedifference being the coalescence of micro-bubbles in fresh waterto produce larger bubbles. The ionic properties of salt water(Pounder (1986)) prevent coalescence; consequently plumes ofmicro-bubbles have been observed to penetrate to depths on theorder of tens of meters.

In a previous paper (Carey (1985)) a review of measured verticaland omnidirectional ambient noise was presented. Experimentalevidence obtained with hydrophones in the Northern Hemispherebelow critical depth in the deep ocean (Whittenborn (1976)) andin the Southern Hemisphere (Burgess (1983)), Browning (1982))showed the existence of wind produced noise in the 200Hz region.The data showed agreement with the observations of Kerman that atfrequencies greater than 500 Hz the noise produced at the seasurface has a different speed dependent characteristic above andbelow a critical wind speed of 12 m/sec (Uc 4gT/ ), thusimplying two different physical mechanisms for the production ofsound.

Examination of the possible ways the wind may produce sound(Carey (1985), (1987)) showed that wave-wave interactions wouldbe important at ultra-low frequencies <2Hz, individual bubbles,spray and splashes at the higher frequencies >2kHz, andwave-turbulence interaction in the low to mid-frequencies200Hz. However, wave turbulence interaction could not explainthe levels observed at higher wind speeds or explain the rapidtransition in levels observed in experimental data at thecritical wind speed.

Thus a possible explanation of the observed effect was that forwind speeds greater than 8-10m/s, noise produced near the seasurface was due to the collective oscillations of micro-bubbleclouds driven by the pseudo-sonic and turbulent pressure fields.If the bubble plume or portion there of could be considered as acompact source of sound, then in the absence of the pressurerelease surface it could be represented as a sum of a monopole,dipole, quardrupole, plus higher order terms. The addition ofthe pressure release surface would result in the monopole term

3

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BREAKING WAVES

SPILLING PLUNGING

RESIDUAL BUBBLESN(Z) =Noexp( - uZ)

BUBBLE PLUME

4

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becoming a dipole-like term that would govern the radiation characteristic.These concepts are discussed farther in a recent paper (Carey (1986)) andare similar to the analytical work of Prosperetti (1985, 1986).

A5

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SUMMARY

" THIS PAPER EXAMINES THE POSSIBILITY THAT BUBBLEPLUMES FROM BREAKING WAVES CAN PRODUCE MF/LF(f-- 200 Hz) SOUND.

* THE PROPOSED PHYSICAL MECHANISM IS BASED ON:- THE PROPERTIES OF BUBBLY MIXTURES- THE RESONANT OSCILLATION OF A BUBBLY VOLUME,

CLOUD.

* WE SHOW THAT THE RANGE OF RESONANT FREQUENCIESSPANS THE REGION OF INTEREST AND THAT IF VOLUMEFRACTIONS IN THE CLOUDS (PLUMES, AS OPPOSED TO THERESIDUAL BUBBLE LAYER) ARE IN THE 0.1 TO 1% RANGE(VF=.001 TO .01), THEN LEVELS ON THE ORDER OF 60 dB RE1 jtPa CAN BE REALIZED.

* WE RECOMMEND THAT VOID FRACTION MEASUREMENTS BEPERFORMED.

6

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SUMMARY:

This paper examines the possibility that bubble plumes frombreaking waves can produce sufficient sound in the low-frequency(LF) to mid-frequency (MF) region (f < 2000Hz). As shown in thisvu-graph, we are distinguishing between the residual bubble layerwhich decreases exponentially with depth and the plume of

micro-bubbles which can be convected to depths on the order of

ten meters below the breaking wave.

The proposed physical mechanism is based on the acousticproperties of a bubbly-liquid mixture and the resonantoscillation of a bubbly volume (cloud or plume). We show that

the range of resonant frequencies spans the region of interest(20 Hz to 2 kHz) and that ii the volume fsaction of the bubblesin the cloud is between 10 (.1%) to 10- (1%) then sound levelson the order of 60dB re i/ Pa can be realized.

We also conclude that void fraction measurements need to be

performed beneath the breaking waves rather than bubble size andpopulation measurements. Finally we observe that "a goodradiator of sound is also a good scatterer of sound."

7

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LOW FREQUENCYAPPROXIMATION:WOOD'S RESULTS

1600 41500 m/sec

1400' C-2= xP.-{I - x ) P, I xK,,-(1- x) K i

1200 PARAMETERS:

P1 1 O3kg/rr 3

* P9 =1.2 kg/m 3

t,- 1000- K,=4.54 x 10 - 0 m 2/N

E K9 =7x 10- 6m 2 /N

S800 C. P 002 x .94800 Px(1 - x)

-I> 6001

400- 340 m/sec

200.

10- 7

10-6

10-5 10

- 4 10

- 3 10

- 2 1 ' .

FRACTION BY VOLUME OF AIR IN WATER (x)

8

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The Low Frequency Approximation: Wood's Result

A. B. Wood (1932) showed that the sonic speed could becalculated for an air-bubble/water mixture by use of the mixturedensity (Am,) and the mean compressibility (km). The mixture canbe treated as a continuous medium when the bubble diameter (d)and spacing between the bubbles (D) are much less than thewavelength of sound. In the case of low frequencies, for themixture with a volume fraction (X) of gas we can calculate themean density and compressibility as follows:

~ -'I) dVQp = (- Y~5p~/)(VP(/.>~~*Z17

This imples that a state of equilibrium prevails and the mixturemass is conserved, and the pressure, P, is uniform throughout themixture (a low frequency assumption). Since the sonic speed is

we have c t Effrl9c EiC-/) kx #-;/]

The expression for the sonic speed poses the question ofwhether the gas compressibility is described by an isothermal oradiabatic process, especially since the single phase sonic speedis known to be adiabatic. However, in the case of an air-bubble/water mixture, the controlling physical factor is the transfer ofthe heat generated in bubble compression to the surroundingliquid. If the transfer is rapid, then the bubble oscillation isisothermal,( VAp=-V/*,Kg= Yp), as compared to the adiabaticcondition 4v/ap =-Vfp,%=P. Thus, in use of the above equationsone must use either for the adiabatic or isothermal case,C.e 4e/ 40Isothermal conditions are most likely to prevail for air-bubble/water mixture due to the large thermal capacity of water. _,Examination of the above expressions shows that as X-Pa , C - Iand as % c,;- Ci - , as one would expect. The strikingcharacteristic revealed by these equations (shown in thisvugraph) is the snarp reduction in the sonic velocity at smallvolume fractions; i.e., X=0.002-. Cm=225 m/sec. These equationsmay be approximated for the air/water mixture:

9

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1 1500 340 36

11001.0 KC 100,0

1 C 0.5 KC 34-C0 EXTRAPOLATED TO

0 ZERO FREQUENCY z //

9go 32-

80 130- -

704 - 200)0z>; 28-

0 60 -

30050 -J 26-

I-

i24- 0400/6030

2 0/20

2010

0 . . ... 18- -0 20 40 60 80 100 0 500 1000 1500 2000

PERCENT AIR BY VOLUME FREQUENCYVELOCITY OF SOUND VELOCITY OF SOUND

IN WATER CONTAINING AIR BUBBLES IN WATER CONTAINING AIR BUBBLES

_ (Ix 2 1S- X)2

C2h C2 C21 ((1 - .2/,2) + 2 ./0 )

2500-

2000-

1500% 2o

1000 "* NEAR

RESONANCE Cmh f C

5002 P

-- Cm t = XPlx(1 - x)

o 30 60 90 120 150 180 210 240 270 300 330 360 390

f jkc'slPLOT OF PHASE VELOCITY vs FREQUENCY SHOWING

ALL OF THE RELIABLE DATA OBTAINED

10

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Experimental Verification of Wood's Result:

Karplus (1958) used an Acoustic tube to determine the standingwave pattern as a function of air volume fraction. Karplus'sresults are shown in parts a and b of this vu-graph. Closeagreement was found between calculated and measured sonic speedsat frequencies between 500 and 1000 Hz using Wood's expression.Similar results have also been observed at low frequencies byCampbell and Pitcher (1955).

Sonic speed changes with volume fraction are also observed athigher frequencies above and below the mean resonant frequency ofthe micro-bubble distribution of the mixture. Part C of thisvu-graph shows the phase velocity measurements of Fox at aconstant volume fraction as a function of frequency. Oneobserves the progression from low values at low frequencies, tosupersonic values near the resonant frequencies of the individualmicro-bubbles to sonic values at frequencies much higher than theresonant frequencies of the individual micro-bubbles (the sonicvelocimeter region).

It is important to note that most calculations performed at thehigher frequencies have used km = k1 +kg rather than the Woodapproach, km - (l-x)k I + xkg. Near the resonant frequencies ofthe micro-bubbles which form the mixture and at low volumefractions, this difference is unimportant. However, the oppositeis true as one approaches the low frequencies of interest to thispaper. One can show that the correct frequency dependentbehavior for a mixture of mirco-bubbles radius ( 1 ) is

In this expression hf andl; are the high frequency and lowfrequency values of the sonic speed. Cmif is determined fromWood's expression; C is the sonic veloci y of the liquid;6f isthe dampening of bubble pulsations; and W, is the radian resonantfrequency of a micro-bubble with radius 0 , and ;X is the volumefraction.

The measurements of Fox et al shown in "C" are compared to hisexpression (the solid line) assuming km = kI + kg. Near theresonance frequency region agreement between measurements andcalculation is observed. However, a lack of agreement isobserved at the low frequencies and low phase speeds. The exactlimiting expression using the Wood approximation is also shown.The lower and higher frequency limits as well as the trends arecorrectly predictedly the expression given here and Wood'sapproximation. Care must be used in extrapolation of a varietyof results which have been published to this low frequencyregion.

11

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BUBBLE CLOUDSAS A SOURCE OF SOUND?

THE DENSITY FIELD CAN BE WRITTEN AS

47rC2 (p-Po) = 4"rP = I[aqiat] dv/R

'COMPACT'1 a

- - jqdVR at

BUBBLY MIXTURE: COMPRESSION BECAUSE BUBBLESCHANGE VOLUME. LIQUID COMPRESSIBILITY NEGLIGIBLE

q = -p DIn(1 - x) Dx= - -- ; x<<l

Dt Dt

BY DEFINITION AP = C 2 A(1 -x)p=-pC 2 Ax

-1 AP TURBULENT PRESSURE FIELDq C 2 At PSEUDOSOUND - pu 2

L-LENGTH SCALEFOR dv = L3 L/u -TIMESCALE

1 2qdv _ 1 q L47rC02(p-PO)- At = R L/U

L - -1R ) pu=/L/U

-L U 4 C 2 -L 2P oC *-= m4. (C oCm)

47rR Co Cm 4rR

"A CLOUD OF BUBBLY FLOW RADIATES VERY MUCHMORE EFFICIENTLY THAN THE TURBULENCE ALONE .....BY (C0 /Cm) 4 . (CO/Cm) 4

2 % -1975

12

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Bubble clouds as a source of sound?

We can examine the radiation of sound from a region of turbulenceby following the treatment of sound generation from a turbulenceliquid-bubble mixtures in FFOWCS- Williams review article"Hydrodynamic Noise".

If the mixture density is given by/Oe/-X) o -' where X isthe volume fraction, then the equations for conservation of massand momentum are

mass: - 5 - - - *2Id9- x/i

momentum: a(p i'/~ '- / / *) (4 4',,y) - .

Take the time derivative of the mass equation and subtract it fromthe momentum equation to yield:

14.

where

;zj

These equation describe the density fluctuations and as shown byFFOWCS-Williams, when the Mach number is small

Consequently the solution for the density field may be written as

where we have assumed j is a compact source. We may estimate thesound production by replacing the infinitesimal quantities withapproximate values.

and within 0 V we note ? = u2. With these approximations wefind

13

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I/' z '

The turbulence will radiate sound with an intensity as shown in theabove equation when C --> C . Hence, the ratio of the soundradiated from the bubBly mixtures is seen to be

-'y /i=.

Thus, a bubbly mixture will radiate sound 1,975 more intense thanthe turbulence alone in the absence of the pressure releasesurface.

THE SURFACE IMAGE INTERFERENCE EFFECT* THE VELOCITY POTENTIAL AT R,

, - = exp(ik. E - iwt). 1)

P(Z = 0) = 0 4 TRS

o THE RADIATED PRESSURE AT R0 DUE TO THE POINT SOURCE, S,

Z Zs PR Rs ipv0 exp(ik. R- iwt) 2)

RP0__ exp(ik. )

ZR IRIsinO = ZR/Ro

R01 .RI IRi THE PRESSURE FIELD AT R, PT = Ps + Pi.PT10=exp~i!. e) + Eexp(ik. Rei 3)

=TP - RI + Il

12 A2 +2 2A - 4p sin2(kZsZR/Ro)lIP"/P0l R2

1 I + 2 + 2A - 41(167r2) (Z,/X)2 (ZR/R)21 16w2 si4)

R0 0

14

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THE SURFACE IMAGE INTERFERENCE EFFECT

When a monochromatic point source is placed a distance l beneath apressure release surface, an interference occurs which is referredto as the LLOYD's mirror effect. Shown on this vugraph areequations (3,4) for the radiation characteristic of a point source,'Z$/A :/ , beneath a pressure release surface. WhenL = -1, the

expression for this dipole radiation

shows that I -- > 0 as Is --> 0, where Iis the point source ormonopole intensity.

In our particular problem, each incremental volume of bubbly liquidwill have an image in the pressure release surface which results ina dipole-like radiation characteristic. HowevIr, regions near thesurface radiate at a reduced factor of (s/) .

When a dipole source of sound is placed beneath the surface, ratherthan a monopole source, a quadrupole results. Futhermore, if thebubbly-region is considered as a complex compact source representedby a superposition of monopole, dipole, quadrupole, and higherorder sources, then the proximity of the pressure release surfaceresults in a source represented by a dpole and higher order poles.Consequently for the ambient noise problem, the monopole and itsimage (the dipole) acts as a source of sound.

15

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THE RADIATION FROM A FLUCTUATINGVOLUME V WITH A SONIC SPEED Cm MAY BE

DESCRIBED BY

P(R,t) = Po V (t)/47rR WHERE V = 4/3 7r(r 0 + S) 3 1)

THE DRIVING FORCE CAN BE TURBULENCE 1/2 p 2 AND A SIMPLEHARMONIC OSCILLATION ASSUMED.

(p, .6137rr3 )'s + hs+Ks=Pf0 4rr2 , V (t) = -4rw 2r " 2)

CONSEQUENTLY

IP(R, t)j = 1X) 0(620 (Cm) Rt~'') ~ / 0 3

WHERE lf(w, wO)l IS THE RESONANT TRANSFER FUNCTION.

A FURTHER ASSUMPTION OF A SIMPLE IMAGE INTERFERENCE EFFECT YIELDS

=3p 2r3 M2 (C 0/Cm)2 41f 2 2 Z s s 1JP(R, t)12 0 (1 - x) _r) I 2 1 [ n ] 4)

THUS WE HAVE A POSSIBLE SOURCE OF SOUND WHICH HAS A DIPOLE NATURE.WHAT ABOUT w0 ?

16

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THE RADIATION FROM A FLUCTUATING VOLUME V.

If we have a region of space with a volume V that has differentsonic properties than its surroundings and if the volume oscillatesunder tne influence of an external force then the radiated pressurefrom such a region is given by PXt) =V/9 1l77- (equation 1).The driving force could be the turbulence itself or an incidentwave, both of which would have a wave length A > e . Twoapproaches can be used to derive an expression for the radiatedpressure field. The first is heuristic and is shown on thisvugraph. The driving force is taken to be the turbulence 10= 'Vand a simple harmonic oscillation is assumed. This simple harmonicoscillator equation (2) yields an expression for the radiatedpressure with the ?(f/c/,)1 describing the efficiency of the regionsas a radiator and J (w,wo) I describing the resonant harmonicoscillator behavior (equation 3). This monopole when placed underthe pressure release surface becomes a dipole with an intensityradiation efficiency on the order of (equation4). Thus the degree of turbulence, the compressibility of themixture, and the proximity of the source to the pressure releasesurface determine the radiation characteristic.

171

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THE COMPLIANT SPHERE

PS

I /i '. , '. \ \

Pi -- Ii .

P.!. I

p, c

X> >ro >>rb

THE INCIDENT PLANEWAVEPRESSURE oPi = Po exp(iwt - ikr) = Po exp(iowt) E _ im(2m + 1)P(O)Jm(k, r) 1)

PARTICLE VELOCITY

vj w = 3r 2)BOUNDARY CONDITIONS: (Pi + Psjro = P(ro)

(Vin + Vsn )r 0 = Vn (ro) 3)

WHERE0o

PS=ex0wt amP(0)Gm(kr)exp(-iem(kr)) 4)

o

=exp(iw~t) E amlP(6)Jm(kr)m=o 5)

REFERENCE RSCHEVKIN, (350 TO 390)

18

• • = m m = |

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THE OSCILLATION OF A COMPLIANT SPHERE

The generation and scattering of sound from a compliant sphereimmersed in a fluid can be found in classical texts on the theoryof acoustics such as Rschevkin (1963) and Morse (1948). Thetreatment concerning the radiation from compact source regions(Lighthill, 1978) when the characteristic demension, 1 </2T showsthat a compact region can be considered as a superposition ofpoles, i.e. monopole, dipole, etc. In the case of significantvolume fluctuation, the resultant field is described by anequivalent monopole source whose strength is proportional to thevolume pulsation. This reasoning was applied in our previousestimation of the source strength to describe the radiation from abubble plume.

Here we assume that the bubbly region is compact with a generalizedradius, and composed of micro-bubbles with resonant frequenciesfar above the frequency of excitation. Bouyancy forces and arestoring force such as surface tension are not required. Theproperties of the bubbly region are described by a mixture speed cand density/F . The compressibility of the mixture,6 6FO . Themicro-bubbles supply the compressibility and the liquid theenertia. The source of excitation is assumed to be an incidentplane wave expanded in terms of spherical harmonics (eq. 1).However, any forcing field such as the pseudo-sonic field inturbulent flow, which has a wave length greater than the compactregion, can be expanded in such a series and yield similar results.The physical reason is that the properties of the bubbly regiondetermine its ability to radiate provided there is a source ofexcitation. This means that a radiator of sound is also a goodscatter of sound.

As shown in equations 2 and 3, continuity of pressure and velocityare required at our generalized radius, r a Futhermore, aradiation condition is required at large 8 istances and the fieldmust remain finite within the bubbly region.

The procedure is to assume the scattered wave or radiated wave is asum of outward propagating spherical waves described by equation 4.The pressure field inside the volume is expanded in terms ofspherical Bessel functions of the first kind. When the region iscompact and X >t, we can solve for the unknown coefficientsQ., and by substitution into the boundary condition equation 3.Equating each m term, we find the results shown in equations 6 and7 to ((lft1 3) . The CLo term is seen to represent the resonantvolume oscillation of the bubbly region.

Resonance occurs when, as shown in equation 8, the real part of thedenominator is equal to zero. The result, equation 9, isinteresting as the resonanct angular frequency is proportional tothe compressibility of the bubbly region, generalized stiffness

( ) and the generalized inertia I -y3 rr, 3 p) . This equation

19

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THE COMPLIANT SPHERE(CON'T)

FOR THE CASE OF kro < < 1, WE SOLVE THEBOUNDARY CONDITION EQUATIONS FOR EACHmth TERM SEPARATELY TO YIELD O((kro) 3)

a0 = iPo(kro) 3/3 (1 - x/X) 6)(k 0) i(kro) [x (k[,x 3 x 3+

a, = P°(kr°) 3 ( -p) 7)(2i +p)

RESONANCE OCCURS WHEN

_3X 3 2 8)(kro) 2 =(27rfo ro/c) -= PC2

x pC2

fo3 A2 92irr0 PC227rro / pC2

o- C (1_ x)/2 0.276 yP 10)27.\ro / ro

20

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is what one would find for a single isolated gas bubble. In thecase of a gas bubble I/ ooi3 ) the resonant frequencyis 1 =I (5C-ML -m/ r. In the case considered here C - 200 m/sec,and we find 4c - 110 Hz-m/2r . We note that the resonant cloud ofmicro-bubble has a resonant frequency (-f€/Ab - 110/680 =0.162) ofapproximately .2 times a gas bubble of the same diameter.

21

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10

SPHERE------------------------- CYLINDER

"'x 0 - ~ .ool1 x- 0.001- ",," - CYLINDER --SPHERE

SPECA x = 0.01 .\

"',,AIR BUBBLE SPHERE"-.\

Vx

10 10O0 1000

FREOUENCY (Hz)

THE SPHERICAL AND CYLINDRICAL BUBBLE CLOUD RESONANT

I-. N

FREQUENCY VERSUS RADIUS AND VOID FRACTION

22

-If N

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BUBBLE CLOUD RESONANT FREQUENCY

Shown on this vugraph are curves of resonant frequency versusradius of both spherical and cWlindrical volumes with bubble voidfractions between 10 and 10 . These void fractions areconsistent with measurements of bubble plumes but must bedifferentiated from the bubble densities and void fractions withinthe near surface residual layer.

The curves for the spherical bubble volume are taken from theequations 9 and 10 previously derived in this paper. Also shownare curves for a long cylindrical volume. The estimates for thecylindrical volume are scaled by a comparison of the resonantfrequency radius at specific void fractions. This ratio of(Ros/Roc) the spherical radius to cylinder radius is shown to beapproximately 3. The results show that the frequency radiusdependence of a cylinder with a X - .001 corresponds to aspherical cloud of X = .01. These curves show that structuresbetween 10 and 50 cm can have resonant frequencies in our range ofinterest - 100Hz.

These calculations show that portions of the deep diving bubbleplumes have a resonant behavior within the frequency 100 to 200 Hzwith reasonable radial sizes.

23

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SUMMARY" WHEN BUBBLY MIXTURE VOID FRACTIONS ARE

10-3< x 10-2 (N = 109/M3(-50Am)

THEN318 -tCm 101 m/sec

WITH.0063 pC2/pwCw< .062

UNDER THESE CONDITIONS, COMPOSITE OSCILLATIONSOCCUR

p2 = 68.104(APa )2 [f2 sin 20

WHICH CORRESPOND TO SOURCE LEVELS > 58 dB AND WITH[f]2 = 68.(.09), ON THE ORDER OF 66 dB re tPa.

" RESIDUAL BUBBLE LAYERS HAVE DENSITIES AND VOIDFRACTIONS LESS THAN THE ABOVE. HOWEVERMEASUREMENTS OF BUBBLE CONCENTRATIONS IN PLUMES ATTHE HIGHER SEA STATES ARE VERY UNCERTAIN AND COULDBE ORDERS OF MAGNITUDE GREATER THAN LAYER DENSITIES.

* VOID FRACTION MEASUREMENTS ARE REQUIRED.

24

TD 8495

SUMMARY

This paper has built on previous results indicating that winddriven noise was important at frequencies near 200 Hz. Prior towave breaking, this low frequency noise was found consistent withwave-turbulence interaction. After a critical wind speed of 8-10m/sec (Kerman) bubble plumes composed of micro-bubbles have beenobserved at depths of tens of meters. The collective oscillationsof the micro-bubbles in these plumes driven by the pseudo-sonic andturbulent pressure fields could produce noise which due to thepresence of the sea surface would have a dipole character (Carey1985,1987).

This paper has presented details of the analysis reportedpreYiously. 3We have shown that when the void fraction are between10 and 10 that sonic velocity changes are large and can resultin sizeable levels of radiated noise as shown in the summaryvugraph. The key parameter identified for measurement is thebubble plane void fraction.

25

TD 8495

BIBLIOGRAPHY

Browning, D. G., et al., "Vertical Directionality of Low Frequency AmbientNoise in the South Fiji Basin," Journal of the Acoustical Society ofAmerica, vol. 70, no. S1, S66a (also NUSC TD 6611, Jan. 1982), 1981.

Burgess, A. S., and D. J. Kewley, "Wind-Generated Surface Noise SourceLevels in Deep Water East of Australia," Journal of the Acoustical Society

of America, vol. 73, no. 1, 1983, pp. 201-210.

Carey, W. M., and M. P. Bradley, "Low Frequency Ocean Surface NoiseSources," Paper A.3, p. 51, Journal of the Acoustical Society of America,vol. 78, no. 51, 1985.

Carey, W. M., and R. A. Wagstaff, "Low Frequency Noise Fields and SignalCharacteristics," Ocean Seismo-Acoustics: Low Frequency UnderwaterAcoustics, T. Akk and J. M. Bedson eds., Plenum Press, 1985, pp. 753-766.

Carey, W. M., and R. A. Wagstaff, "Low Frequency Noise Fields," Journal ofthe Acoustical Society of America, vol. 80, no. 5, 1986, pp. 1523-1526.

Carey, W. M., and D. Browning, "Low Frequency Ocean Ambient Noise,Measurement and Theory," Sea Surface Sound, B. R. Kerman ed., KluwerAcademic Publishers, Boston, Mass., 1987, pp. 361-376.

Fox, F. E., et al., "Ambient-Noise Directivity Measurements," Journal of theAcoustical Society of America, vol. 27, no. 3, pp. 534-539, 1955.

Karplus, H. B., "The velocity of Sound in a Liquid Containing Gas Bubbles,"Armour Research Foundation, University of Illinois, Proj. cOO-248, TID-4500,1958.

Kerman, B. R., "Underwater Sound Generation by Breaking Waves," Journal ofthe Acoustical Society of America, vol. 85, no. 1, pp. 149-165, 1984.

Lighthill, J., Waves in Fluids, Cambridge University Press, New York, 1978.

Monahan, E. C., Oceanic Whitecaps, D. Reidel Publishing Co., Boston, 1956.

Morse, P. M., Vibration and Sound, McGraw-Hill Book Company, Inc., New York,1948.

Pounder, C. V., "Sodium Chloride and Water Temperature Effects on Bubbles,"Oceanic Whitecaps, E. C. Monahan and G. MacNiocaill eds., D. ReidelPublishing Co., Holland (also Kluwer Acd. Pub., Hingham, Mass.), 1986, p.278.

26

TD 8495

Prosperetti, A., "Bubble-Related Ambient Noise in the Ocean," Journal of theAcoustical society of America, vol. 78, no. 51, 1985.

Rschevkin, S. N., The Theory of Sound, Pergamon Press, New York, 1963.

Su, M. Y., A. W. Green, and M. T. Bergin, "Experimental Studies of SurfaceWave Breaking and Air Entrainment," Gas Transference at Water Surface, W.Brutsaert and G. Jirka eds., Reidel Press, 1984, pp. 211-219.

Whittenborn, A. F., Ambient Noise and Associated Propagation Factors as aFunction of Depth Wind Speed in the Deep Ocean, Tracor Report T76RV5060,DTIC(AD 006902), Alexandria, VA, 1976.

Wood, A. B., A Textbook of Sound, G. Bell and Sons Ltd., London, 1932/1955,pp. 360-364.

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