ML20027D708

Provides Documentation of Discussions Re Humphrey Concerns on Adequacy of Design Margins of Mark I & Mark II Containment Sys.Documentation Should Close Issue

ML20027D708
Person / Time
Site:	LaSalle
Issue date:	10/29/1982
From:	Schroeder C COMMONWEALTH EDISON CO.
To:	Schwencer A Office of Nuclear Reactor Regulation
References
5346N, NUDOCS 8211080268
Download: ML20027D708 (23)

Text

-_ _

j' N Commonwealth Edison s

) one First Nttional Plus, Chic 2go, Ilknois

\\ O 7 Address Riply to: Post Offics Box 767 (j Chicago, litinois 60690 October 29, 1982 Mr. A. Schwencer, Chie f Licensing Branch #2 Division o f Licensing U. S. Nuclear Regulatory Commission Washington, DC 20555

Subject:

LaSalle County Station Units 1 and 2 Concerns Regarding the Adequacy of Design Margins o f the Mark I and II Containment Systems NRC Docket Nos. 50-373 and 50-374 Re ferences (a):

R. L.

Tedesco letter to L. O. De1 George dated July 2, 1982.

(b):

C. W. Schroeder letter to A. Schwencer dated August 30, 1982.

Dear Mr. Schwencer:

Re ference (a) listed 22 concerns which Mr. John Humphrey had identified regarding Mark III containments.

It requested a response to those concerns which were identified as being potentially applica-ble to LaSalle County Station.

Reference (b) provided Commonwealth Edison Company's response to those concerns.

On October 4 and 19, 1982, Commonwealth Edison representatives discussed 'NRC questions on Re ference (b) by telecon with Messrs. A. Bournia and Farouk Eltawila.

The pu rpos t: of this letter is to provide you with the enclosed written dc:umentation o f the discussions.

It is our under-standing that the N.lC concerns were satisfied and that providing this documentation would close this issue for LaSalle County Station.

Enclosed for your use are one (1) signed original and thirty-nine (39) copies o f this letter and the enclosure.

If there are any further questions in this matter, please contact this o f fice.

Very truly yours,

& w)2AlWL C. W. Schroeder Nuclear Licensing Adcinistrator 1m Enclosure cc:

NRC Resident Inspector - LSCS 0

\\

5346N 8211080268 821029 PDR ADOCK 05000373 P

PDR

LASALLE RESPONSES TO NRC QUESTIONS ON THE HUMPHREY CONCERNS Question 3.1 - What load combinations were considered with RHR relief valve discharge?. What is the reference to the condensation map in DFFR7 Response - The RHR steam condensing mode is a secondary method of reactor cocidown not normally used in plant operations.

By virtue of the time required to align RHR into the steam condensing mode (greater than 20 min.), it is not expected that the reactor would be maximum pressure (conditions for highest loads) nor in a transient when SRV's are needed to control pressure while RHR is in the steam condensing mode.

The postulated combination of events, ie. RHR in the steam condensing mode, reactor at) 500 psig, pressure controller fails high, plus LOCA was judged too improbable to include RHR relief valve loads in tne DFFR load combinaticns.

Further, the SRV loads bound RHR loads by about 50%, so LOCA plus SRV combinations bound LOCA plus RHR.

Question 3.3 - Provide justification for fcCm-1/3 Response - The justification for the formula for frequency is given in the following reference (copies attached):

1.

Devin, Jr., Charles, " Survey of Thermal, Radiation, and Viscous Damping of Pulsating Air Bubbles in Water", The Journal of the Acoustical Society of America, Volume 31, Number 12, December, 1959, pages 1654-1667.

2.

Foody, B.E., Huber, P.W.,

"On the Radial Oscillations of Multiple Gas Bubbles in an Incompressible Liquid", Journal of Applied Mechanics, Volume 48, December, 1981, pages 727-731.

I 1

3ygcPO Reference 1 f

=

2*Ho P

Equation 2-1 3

7Rd Vo =

4 3

Vo =

m RTo Po R To h @

Ro =

3_.

m 4Y Po j

4y Po

) 1/3 f3y9c Po 1/2 1

1 f

=

27 3 m R To jo f

1

~

. 1 f 4w N

f 3y9c t 2 f

=

2 tf m

3 RTo

}

p j

1

[4 W

)b f 3y9c 1D c

=

2W (3 RTo

(

p j

b C

Pn f

=-

mA/3 f ~ PoS/6 f ew m-1/3 Question 3.6 - When the RHR heat exchanger is in the steam condensing mooe, what happen s when a LOCA occurs?

Response - When a LOCA occurs, the valving in the RHR system automatically realigns itself to the LPCI injection mode.

The action is taken on either low RPV water level or high drywell pressure signal.

Dithin the first minute of the accident, all of the valves would be realigned.

Question 4.9 - Are there instances when the drywell spray and pool cooling modes are used concurrently?

Response - It is functiona'1j possible to run one independent RHR loop in pool cooling and another in containment spray if conditions warranted.

An RHR loop, however, cannot run in the two modes simultaneously.

System pressure in the pool cooling mode will not allow water to flow up and out the drywell spray ring header.

Question 5.2 - Is there any interlock between the drywell spray and recombiner?

Response - There are no interlocks between the drywell spray and control recombiner.

j Question 6.4 - In certain conditions, the drywell may reach 3400F.

v! hat happens to the air monitoring system?

0 Response - The analyzers are kept at 300 F.

At design conditions of 45 psig and 3400F, the steam is superheated.

At 45 psig the saturation i

temperature is equal to 2920F.

This temperature is below the temperature at wnich the analyzers are maintained.

It is anticipated that no condensation will take place of the superheated steam, only cooling.

t

{

l

3-Quest! ion 9.2 - Last sentence is unclear.

Response -.Last sentence.should read:

"The RHR wetwell-spray mode can citigate the effects of prolonged leakage into the wetwell having no effect on suppression pool cooling."

4 1

Wa1-w s---

y

+

m

-~ - - - -

yw-

,vww,h-vy-r-ar?

r--

+

+%.

m,-

p m-ey w

• cwy e-e7 rr y v 90 -re-*-*y

+

~*w-

-+ - * -

VQ /t a

\\

e t

?

I J.

f

.i I

D.E.Foody.

A1 T

81 im.3 n")aJag"1 oC "#M10no 01 A

n P. W. Huber

.~.

Mulu.. le Gas Eubbles in an i

p

__.r,

,incorrryre=m.ele L.

.d a_ s ts.

rui censne e.ussa. cans n

s The radial oscillations of mult'iple gas bubbles in arn incompressible liquid tr:,unded b.s plane solid nr free surfaces are analyzed. Dericatior: of the normal made inquencies and mode shapes il two. three and four bubble confa'urations in an unbounded liquid silu n.

trate the method of anal.ssis. Expressions for the osed!ation frequencies of cubb es nsor lrre and solid surfaces are derived.

r.

1N t

i 1

l Introduction The dynamics of a single spherical bubble oscillatine radially in a bo.mded liquid. The liquid flow is sssumed to be incompre<4ble aad liquid hase b*o mensisely investirated. Ila>leigh [1] derised the irrotational Gravitational and surface tension effects are ig.ored. ft.r basic equation gr vernin;t tha motion of a gas cavity in an unbounded from the bubbles the pressure P.14 unitnrm.

fluid. Minwrt [2]. using a simple energy method, determined the Itadial displacements of the bubble surfaces are taken to be small natural fre-.pncy of an oscillating gas bubble. Since that time re.

cornpared to bubble radii r search on vapor bubbles and casitation h.u motivated studies of vapair bubble co!! apse, heat tran3fer effects, and a wide range of other topics.

• # I*

lisich [3. 4j reviews snme of this work.

If the velocityin the fluid varies with a period r and is characterized No previous ana!ysis boweser. has dealt generally with the hy.

by an amplitude (~oon the bubble surface. then 3r is of order (5 r. The drodynstric relations goserning mteractions among multiple oscil-St rouhat number. r/l'or is then much greater th.in 1 and cons ectise lating bubbles and their surroundings. It is weit known that a solid accelerations in the liquid c an be ignored. Equation s 16 also impii.s surface attracts an oscillatinz bubble while a free surface repels it ]5).

that the time sca!e for o<ci!!ation is much smaller than that for con.

Cole [6l ar d Friedman 17 hase derived expres. ions for the oscillation vective distortions of the hulhle zeometry. We shall as,ume in.it the period of a sin:le. espimivelv iormed bubble situate <t between a free bu'-ble geometry is imariant bpharical or ell!;Mdh dur:n: the l

and solid surface. Since surfaces can he mterpreted as the ;> lanes of oscillations analynd. The gas is treated as ideal and adiabatic..th l

symmetry in a multiple bubble system the-c expressions al>.. deter.

a ratio of specific heats 3. Equation (1) can then be written mine oscillat;un periods for a restricted set of multiple bubble eun.

i figurations.

JP/37P.

1 (2)

This paper presents a met hnd for estimating the radial osallation where 3P i the maximum change in hubtle pressure from the equi-natural frequencies of any number and contituration of gas ')ubble' librium.

In an unbounded liquid or in a hquid bounded by pl.ane solid or f ree j

surfaces. The analysis as limited to imcar, small amplitude owilla.

Analysis tions.

The kinetic energy T of a liquid bounded internal!y by surfaces.4 P" **

Model Assumptions l

We consider the radial o<cillaticns of N gas bubbles in an un-7,, 4 (, 0 8dA 13)

JA on where c is the velocity potentialand p is theliquid den ity.The n"r-mal sector n is difine<1 as pointing enrn the hqui.i ei e..outaar I tr. m Contnhuted by the Appi'r.1 %han.c. D.sitien for puhticati..n in the the g.is hubble<s. In our analy>is, the integral is taken m er the b..:Ac Jotasas.nr An t es p \\tti,H aa. :..

surfaces.

D.,cv a.n un th p n-r n. w bc ad.he

.! t.. the M.o rul(knviment.

AShtE. Un ted Eng.c..ru.at s' ntre. u?. Ea. 47t h.vreet. Ne. bra..V Y.

The rate of charge o' the ith bubble's volume is gisen by

• Innt;.arvt..tl e.

coe..s u:.us r...m..nen. ner on.1 put.l u

.n..I the p user itulfin the.los pius i+ Te #1 a o str. to wn

\\t. nu,.,p. e..m.s ta A AIK q, =

- dA.

(4)

Appl ed hierhan. is u

.n. Dr. ember. IN. t.nal reuuon. Star h. l'an.

A ort Journal of Applied Mcchanics DECEMBER 1981. VOL. 48 / 727

- ~ ZEP

er> the integr:1 is t: ken over the surfao of tl.e ith bubth. W2 E" " P.y (15) derme tro potentiah on eod 02,whomum is e,such tint ei s uniform pg i

ove r sath bubble surfice.

Lagt:nge's equitions, with af, substituted for q. require:

- dA, = q, for i = 1,2,.. N (5a) d'OT' uT b V*

odi JA. On for s = I,2,..N.

(16) de ab(nV.),

o(bV,>

ctiV,)

If M does not vary with time.se c,btain

- dA, = 0 for i = 1,2,.. N.

(56) p g + :gy. o, (g7) b62 SA.On The constant term T;in the kir. tic energy disappears in the differ.

Physically, $2 s the amnimt by which the actual potential o departs entiation.The eigenvalues of equation (17),

i from the aserage potential e.. at any point on a hubble's surface. Es.

cept in ti.e case of a single bubble m an onconfine d liquid, oc/on will A + 4 = 0*

W not be uniform over exh bubble and thus acdon may vary over each are the natural frequeneirs,e of the N hubble system a bubble.

vectors define the n.ude shapes ifIhe einem estors are known a prir,ri, Using the relation w can be deduced from the Itagigh Ritz equation tS) fes dA = l es dA (6) yg r

\\ Mv ee3v me obtain where 87 is an eigenvector. This formulation is particularly useful 4 - dA (7) when the mode thapes can be desjuced directly from considerations be, oi dn86, dA + [A T = 1p [A

2. dra of symmetry.

The Stasi Slatrix. The mses rnatrix. equation (12)," relates e t.nd The kinetic energy can therefore be defined as the sum of two terms-r4 For a single spherical bubble c f radius to T: which depends on ei only, and T uhich depends on 42 Euler's equation for higiay unsteady motion.

(2'))

& = 4rro

-VP = pV(as/at),

(B) and equations (10) and (12) yield requires that co/0t be uni'orm on the bubh!c surface since the pres.

sure P in the liquid is uniform on that surface. Any initial nonuni.

Af =

p (21) formity in 6 mill thus persist through a number of oscillations. As a result,02 and therefore T win renam roughly constant oser the time Sub>tituting equations (21) and (151 into equation (19) yields the 2

scales (several bubble osci!!ation periods) of interest in this anal.

well.k nown Rayleigh bubble natural frequency (2) ysis.

(22),\\

g 3p,3 We note here that, as a consequence of equation (7), the kinetic energy of any bubble system wi h gisen volume rates of expansion is 68* 7

t a minimum when e is uniform over each bubble surface.

For a single ellipsoidal bub 12e with semines a, b, c, Smythe [?]

Since et s unifotm on each bubble i

gives T = -lpe e + T (9) r 4 8)(e + 6)l-8hde.

(23) 2 Ba2 + eHb2 where q is a column vector of the e, and di s a column vector speci.

i a

fying the uniform potential oi on each surface. Each q, is related to

'"I "

di y Laplace's equation and hence saries linearly with each term of b

Br Joe-[(a + es(62 + 6)(e + 6)}-thd8 (24) di.The relation can therefore be expressed as Af = a f

2 g = Cf (10) if the ellipsoid is pretate, equadon (24) sith a = b can be inte rsted where c is a square matrix dependine only on geometry. c is indogou.s to give to the capacitance matrix in an electrostatic problem with charged "fe 2

' in

i Af = p leg (c/mi-1-.-1 c>a (2*c) conductors.

4 ra gsc/ap - I k

The kinetic energy may now be written T = 1g ug + T (11) r d

e < a.

M Af tan Mr-pC-8 (12)

. I-Q

\\j)e 2

u

• b'M dra ep

-I.

The potential energy of the ith bubble V,* is Fig.1 compares bubble frequem ies of sintre spherica! and ellipsoidal bubh!cs et eyual volume cuuimed u4ng equation i19, s.tn equaum v.+ 4 v, V,* = - [* *

(P, - P.)dV (133 (21) or (2M. respectiseic Tr.+ depencence of bubb!e frequency on bubble geometry is toimd t,Imerv weaic.

where P, is the pressure in the ith bubble. Vo, is its equilibrium vol-It i* rnore ditficult to deiermme ahe eiements of M arsaatically far ume and 6V, is the chante m solume from Vn.. Assumma reseruble conteurations imolvine tw. ne more bubides. 6msthe bl uses a adiabatic expan-mn of each ;:n hubble the total puent tal energv U*

ruethod ofima;:eupor,m h M..) to solse the anaho.us ebtrwtat.c of the multiple bubble sptem can be tound f rom equatwn (th and proh!em: the putent A s swro d two char;:cd sondatmg sphero radius roeparated at nier tenter by r:2.1 bc wiution can be es-expressed as e

y..g3yrK3V ta4)

• [ 2-W A

g where 3Y is a vector of &V, and is a diagonal matrix who e elements 3g, 3g-s urea are Transactions of the AC.'.!ii 728 / VOL. 48, DECEMBER 1981

1 IEI 5

5 5

'l'B I

I

' 3 8 3 '

L l

g g

9

\\

b/

8)

~

90C*

e CU~

f")

lCCR

~

)..

l,,

hl.

e*

, moot 3.evemas out or e=as

=*

1)

I IIS-3 /

' toos..

~

e

,a q

seat s.ematas as eiist

~

~

f f

a gg f

n e

y e. r.

,3 to 30 40 30 60 gg,

's, J

2

.4 6 e so _ lo ao 6 0 8 0 aCO tutei.t StPanarce egee, ele P*

Fig.1 Clitpsoldal butble frequency (w

) corrpated to the frequency of c spherical bebble of the sarne volume; othpsold semiases are c, a, a g

kl p

't,.) 4 8 M = Mai = 4 xr (266) ns s w AT-B%

}_

{ jr, r,

r

]

where i

A= sinh $ k esch [(2n - 1))l

,)

==t ri,

r,a 1

Fig. 3 Three-buttle configura.lon B = sinh #.E.: csch [(2n)#]

0) cash p M.

2re Mg=#"d 61)

D Solving the eigenvalue problem of equation (18) we find that the natural frequencies are Here ry la the center.to-center separation between the ith and jth bubble ifi e j, and the equilibrium bubble radius ifi = j.

5) 3#,*T (A - B)

(2Ge)

Use of the approximate mass matrix leads to an overestimate of T.

g.

i P

and thus underestimates w 4quation 1190.The accuracy of the ap.

O 3

proximation is espected to increase with increasing bubble sep.tration.

JP.y (A + B)

(26d)

#8 "

The examples i'n the following section demonstrate that equat.on idll provides a remarkably accurate e.timate of w even for bubb!ca in c'use I)

The two eigenvectors can be given as the columns of a square matrix p,,,,;gy, A2:

'l l'

Multiple Hubb!c Configurattuns (26e)

1. 8 "

,g

.g Consider again two spherical bubb'es of equal radii. Usin;iequation 3

(31) fur M and taking A = 1/w, equation (18) reduces to the eigen-2 The first mode describes two bubbles osciliating in phase;in the problem pd second mode the bubbles are 1W uut of pha3e. Buth the phasing and the relatise amplitudes of the bubble oscillation in these two normal

- Y _ _g,,,_

p l.

modes could, of course, hase been anticipated from the problem's Vo 4rro 4 rrg h) symmetry.

Apg p

Exact analytic solutions of M for more comptex configurations are 12 W

Mo not available. However. a very accurate approtimate expremon for M may be derived for spherical bubbles by assummg a unif orm source This yields natural frequencies l

$)

distribution dy' un each bubble's surface in any multiple bubble g

3p,y i

configuration this assumption implies a nonuniform potential fic!d W=-

(33a)

I v on each bubble. It fulluma from equation (7) that p 1+

Ta < - 2p f e'av al 1

dA.

(27) and Ja un A

The definition of the divergence, implies q.1 3P.y l

(m}

'tY

~

p dq' = vtsdV.

(28) p b

r422 a

he Equations (27) and (28) and Green's theorem then yield with the mode shapes defined by the columns of A:

SI 7: < -) pf e'dq'.

(29) rg g'

8-A=

(33e)

Integrating

,1

-l<

Fi 7, < QQTyh (g) t 2 compares these approximate w. lotions (equations Cl.tal and h)

(3db)) with the esatt volutions sequ.ition 1261). The approumste l

mhere the approtimate mass matrit is solutions underestimate w fur Imth mode shapes,as espected. flow.

L II Journal of Applied Mechanics DECEMBER 1981, VOL. 48 / 729

~

__m.m s-

-m y.

ai

R 46 s

t r

~

~

'~

~

r2 t

ro r

%6 Il2 O

7 Fig. 4 Four-bubble configur'ation h

ever, for bubble center-to-center separations as low as three bubble radii, the error is less than 1.2 percent.

Consider three aligned bubbles with center-to-center separations rir (Fig. 3). The approximate M matrix is

+

1 1 1

ro ris 2rt 1 1 1 M=p (34)

~

4w iss to rs n

1 1 1 27t2 inx to The eigenvalues are found to be 1

3P.y 1

(35a)

%=-

~

p 1 + 1.6861 i

r12

. 1 3P.y I'

(35b) son = -

1 - 0.5 Fig.5 subble betweci tree and send sataces r

rz a

6 1

3P.y I

(35e),

The four bubb!e case shown in Fig. 4. provides a final test of the 403 -

}g approximations inherent in equation tau.The same proctsiure as in r

ro g,g 6

r:3 the previms examples yields a matrix of eigemectors and the mode shapes are given by the columns of As 1

1 1

1 Il' 1

1 l 1

1

-1

-1 A=

1.1661 0 -1.6861 (35d) -

g

_g g

_g 3

kI

'I I

E-1

-1.

-1 1

J Again, the symmetry cf the problem requires that each bubble have In the first mnde. all the bubbles mciLtc in phaw. In the secnnd.on!v the same amphtude and that the normal m. des invehe p..a* dar-the two outside bubbles hase finite amphtudes of oscillation. Th' e

third mode ha> the cer ter bubble IN1' eut of phaw with the ether two.

bothof thete requirements.

u As espected from the ss mmetrs uithc contieuration.the end bubbles I he eigemalues are found to be p

have the same amphtude in each male. Comparnon with t he resuits f

of a nu.nericalcalculat een t:a *hich tbe mass matrix mas accuratcay 3

3p,7 l

P (36b)

W*-

computed by a method of images an.l.wid shows that the estimated

[I + 2.7 M

• I 2/r 2:

p natural f requencies ate in error by less than 1.7 percent for r:

o 3 (11).

rt L

Transactions of the ASME l

730 / VOL. 48, DECEMBER 1981

• ^-

• i 1

318 7 l

The r:tb q,gidq, c ill equal i ! d;pending on the t ypes of"r Act.

a ],

(3Ge) in2"surfaes inm!ved. Thi et:ments of the stiffness rr.af nx n wJi be

1. 8 " "*

• 7" 3

f*,,

e p 1 - 0.707U1 given by equation fIM ts befurr.

L

'ta Fig. 5 shows an cumple of a sinrie bubble situated mih letaeen '

a free and solid zurface separated by a diatance A. For this cue.

p 1

3P.y I

s eds= -

(3Gd) s I-1.29a h A'"h,1+E,(-13'h (38) i raw and equati n (19) then >ields De identity of w; and w results from the geometric identity of s

snode shapes 2 and 4 (the second and fourth columns of A l.These 1

3P.y i I 3P.y I

i

• {

results'are within 2.7 per< ent of an accurate numerical calculation for **{

b p ! 1 + E.: (-1)*

p p 1 - Oi33 r /r = 3 (11).

i n

,kha hj es o 6

Free and Solid Surfaces Cole (C] and Friedrnan [7] use a similar technique to describe the k

De normal mc-les for 2N bubbles in an infinite pool will implicitly motion of an esplosively formed bubble in a pool of finae cepth.

determine the wrmal modes for N bubbles adjacent to a plane polid 6

L or free surface that replaces any plane of n mmetry in the 2N bubble Concluding Ilcraarks configuration. A first enarnple is presented in equations 433a) and The small amplitude omllations of a single spherical hubble in an (336) which can be interpreted as the frequencies of oscillation of a unbounded liquid are analogous to those of a one sprint.one.rnas single bubble adjacent to a solid or free surf ace, respectively. Two systern with K =.'.7/W and af = p/4xic. Nonspherical bubb:e bubbles aligned parallel to a free surface, twice as far apart as their shape has only a weak effect on predicted frequency. Muit pie distance from the surface.will hac rwo normal m des correspending sphencal bubbles in an unbounded liquid hase norn ai mwe to Modes 2 and 3 of the four bubble configuration. equations 136c p frequencies and mcde shapes tnst can be found by forr-u! air; K and and (364). If the surface is replaced by a solid surface, the two normal 31 matrices as outhned in this paper and analping ihe matr:t 3f *K.

modes are given by Modes I and 4.These results demonstrate ex-Use of approximate mass matrix elements greatly s:npiiries the i

pected trends: prosimity to a free surface raises normal mode construction of 31 with little loss of accuracy in the irequen:. g re.

frequencies and brings them clmer together; a solid surface lowers dictions.The effects of plane pool b6undanes on tne elements ct Af frequencies and spreads them apart.

can be determined by a method of images.

The analysis of bubbles oscillatine near free and solid surfaces is s

greatly simplified by a rnethod at images approach. When more than References one free or solid surface is preaent,the number ofimage bubs!es re.

1 Ravtenrh. O. St IVu fasophnal Ma carine. Vol. 3 4.1917. p. 94, 2 bl.nnsert. St.. PNI.aopancal Marni9e Vol.16.19rt. p. m.

quired is usua!!y infinite, but the amplitudes and pha<inir of the image 3 H sieh. D. Y., AS.\\tE Journal c/ he En/inecrms. Vol of..h 4. Dec.

bubbles can be deduced in advance by inspection. Imares are always 19C5. pp. 991-1f45.

of the same arnplitude as the " object" bubble. in phase for a sohd 4 Weh.D. Y ASNtEJournat o/Buic Engince uig.bl94.No.J.5ept.

surface image and IeO' out of phase tot a free surface. When the mode 1972, pp. 6WA).

shape can be determined directly, an analysis of the natural 5 B;rkhoff and Zarenteito.Jers. WaAes.and Cocities. Acade=ie Prm.

I frequencies ba=ed on equatir n iIW is considerably simpler than an kR Undermater Esplosioits. Princeton t'nisew Preu, attempt to identify the appropriate egenvalues m equauon t id) when 1950.

the dimensions of M and K are infinite.

7 Friedman'. B.,Comm.on 1%rc end Applied Afarks Vol. 3..W 2. June For a system with N independent bubbles and an arbitrary 19%.

mb. H Hydrodynamics.6th ed.,Crnbridge Unnersity Presi. Len.

boundary of free and solid plane surfaces there are N normal mndes of oscillation.The problem can always he reduced to analping an N 9 Srr the. W. R., staric and Ihnamic Eintricity. Stccts.Eit. New X N mass mattit w hose 3f., element relates the potenttal on the ath

%.rk. tw. shima. A. ASNt E.lournat o/ Aesec Ensincerins.W1. M.19 1. ;p.

bubble to the volume tius from the jth bubble and allits images. If 4W44 alw add e> e* tha prnb:em.

U hI'***'"P" I hk N"'"" N)d' d)**"uce. Stani, b thej + ANth bubble is a retlection of thejth.

York.19G.

p

'qgr 11 Foody. B. E. *0n the Dynamics of Babb!es in Cime Psumity.'s?.1 1

(37)

Thesis. Department of Stechan: cal Enineerw. >!amcNet.s ir.r.c:e ti My = E.o :,4 mr.p.yi a

q, t Technulcg/.Cambndge, Stass Stay IN.

1 Journal of Applied Mechanics DECEMBER 1981, VOL. 48 / 731 m 3 y--

N lWN i

O r

un: ;nvn.x,u. og un: w>t mc.u. t recna v or.numica vot.mnai, xunrt.n is in:cun u.

t,,,

Survey ~of Therma!, Rm!iation, and Viscous Damping of Pulsatir.g Air Utibb!cS in Water *

,q i g ;,,,..

i CHAF.I M IhAIN, ju.

Wa

• m.

kl.! T.r>fer.11eM frnia. IVas!< ion.tso:, D. 4.'.

,;gt,ubbir.,ti

~ -

(

(Reccised June 17, l'/.W) g do,;i!ct.

l A thenrctical e:ivunion is persented on the islamental procon Iw which pulsating ras bnb!.fes in

'9,wn that 3

liquids diw!pite their tncrry.'! he survey is lirnited to tbe ca<c w here the a'm1.titudc of abe whur.c pe!,ations I'UI'b[

W'

'~

are anunied to be rufriciently > mall that the putsations rnay be elocribal by liacar equati< n*.* A p.rt'e.n e.f if 4

the encrgy a f the l'ubble systen is Imt by the ra.liatie.n of spherie:st 5.eumi ssaws, a psrt is in't by heat shave 4 3 ' i*

conduction alue to the poi > tropic cuanpressions an 1 capu.>i..ns of the encloud cas, snd a p.rti..n is 1+t

.;ie dtgtw by siscous sfiuipation attra.utol to vise:.us forces actin-at the gas.tiquid interface. A survcy is rna.te e.f

--suation t f r" the procedures for nw.i<uring the remnant ebrap'n;; c mtant as elmribel ei the nwthnJ4 of succesire oscillations, tidth of the reinr.ance rcepurae, standint;-nave ration, an.! resonance absorp. ion. Inperirecntal f* "'.:.#

-i e

results scrify that the darapir.g at resonance is due to therraal and radiativn, and possibly visco;:s dampin,;.

ar a ma. fy

.atly o pang 3e intrt na GLOSSARY OF SDIDOLS R

radial distance

i5nc55 I' II A

attent:ation Re mean bubb!c radius der radiati.

g p,p,j,,,y R) instantaneous bubble radius Squency 11 M/2 Rayleigh's dissipation function R

nonre30nant bubble rad:us change m radius imm the mean babb!c radia.

b di33ipation coefGcient r

C gcncratized dtiving function S

net stress dyadic

. dare & in :

c velocity of sound s,

specific heat at constant prenure Nic prem-1 Il thermal diffusivity 8;

spectfic heat at con-tant volume 7.,.y i, the r.

l 4

thicknc>s of bubble screen 7e eqmlibrmm absolute tcmperature

.t the bNiA l

Et incident sound energy 7'

absolute ternpe ature

.w,g tre I

t,ime j

E,. reflected sound energy.

U microal enyrgy

.gge ye,;=

t F

chatacteristic frequency of the gas bubb!c

,!!ab:itic vi.

Y'

'f3r Ennaert's resrsnant frequency V.

equilibrium tubb;e volume 95tte, jn r fe resonant freautncy Vt instanta::cous Lt:bble volume do used e n

M G

universal galconstant IV work done on the bubble

~

.;.L.crial ?.a

.l a factor which takes into account the (ffect of X rate of pure strain dyadic sq m cy 9 g

g A

height above the bubb!c producers y

amplitude of the change m, temperature from -

de ma,e ' -

l surface tension equdibrit'm tem;>crature kr a ratia e 7,

idem factor E'

tube radiu5

. a ob' ate e a factor which describes the departere ni.t -

I.p!y.:e w;;h :

i K thermal conductivity a

I k

restoring stirTness bubb!c t.tiffness from the adiabatic stitine 4 S t larg. l l

I, I.agrangian function

[(Re/R')'- 12 nry,mdi..

.1 mt rnass of the gas in the bubb:e i

ratto of specific heats in ad. %

l m,

generalized m:u 3

1/Q dampm; constant

ny be.M m'

rnass' of the gas contained in the volume v' 3

reonant damping constant utue,a (;o; angic buwcen the incident sound ray and .

,,,cag;oa 1.,

ne avert.ge number of bubbles per unit volume e

n rmal to the bubble screen a5 um! n p.

static pressure P.

instantaneous prcrute in the tindisturbed liquid v

polytropic aponent

< sci!:sth e.

JE e

change in temj>crature from tise utud.! rh

• complex amplitude of >inu-oidal pres-ure p'

..,cos,m y

,l P ' in3tantantous per*. ure on the bubble >uriaee temperature -

%.,nau Ll P,

preuure inside the bubble A

natural Ingarithmic dr.erement Eme tr*-

l p

sinusoidal pree ure on the liquid -urface A

wapndt kn Ge r

p' sinusoidal prerure on the bubble >uriate p

cocthcient of v, cos.ity shi;h can-e a

l p.

ueou. stir pressure on the bubble >uriat e P

den $ tty 4 3 fue b.bbh i.."-/

p4 ' incident soutal prewure surface ten 3mn e

change in vo ume from the equ!'ibn,am bul -

a kd,b:e< p t

, p, reflected >ound pressure v

j Q

numl tr of tydes requiral for the amplitude of volume motion to rolute to c-of i:s origin d vahte v'

inGnitesinial clenient i,f volume in the gas Web,

,,j,}i,'ti t a

3 ra, /.

f q

amount of heat ener;'y tran<ferred e

(w/2/>d (jw//))I l r"%

• Itased on a thesia subm!stol to the l'.wntty of the t%!umfiin

' M. M r.s*

n VI ou.ty potent;ag 4

• t t t,.,..

I Cotterr of The Grnrge hhinri..n t%ivrr>ity in partial uti4.te-g, p,;f,,

tion of the requircou nt, hit the dq;tu: ut Sla>ttr vi &isntc.

urcul.tr frequency a 31, sm.

.l w

a

• i 1651

{

f.'v7Fme&~r:&..wM:7 - m m ' m m + - s.

c.- ~ w ' c -.. tic "m. &

--u cau-m e,0 of*.,

. D 5f.11,'IN G O P 1*ULSATING A I It 1;U!!f1L1:S iN WATliR 1655

!. INTItODUCTIO i entrained bubb!cs through a pi e past a c 4.striction; l

agg[; earliest reference to hubl,!cs as sound sousees " "d " " I"'Id'"1

• 8"d "

• C-L:,8 made by lir. igg.' w ho attributed to entrained I. @k contyning a ps po-co liWier >ound darn pernnents nauhuled by F6tensen' sfu,wol that

" Q,,t,t,les the murmuring of a brook and the "phnik" character,istics than do thu e whkh are gas free. Ju t

" gun

,ttis falling into water..Tlinnaert: has since a that the sound genuated by gas bubb'es in " I'"'.".I'.I/ di-g> erred bubb!cs, which are o small a3 1[a,14 awoeiated with simp'e volume pul,ations of to he mvisible, can have an appro iable aco t,ubble without change of shape. The bubb!c M, en a large rmndjer of thee smaH bubbles are

. pes as a simple darnped osciliating sy.4 tem witti I resent, the h,qu,d '.vdl be nearly op.ugue aenu. tical'y.

,,sdee of freedom. Therciere, the ditierential blmaH impunn,cs ni l,ugu,nh, such an s.i.in of motion for the bubble >v3 tem has the same have negi, g, e, duence m compamon unh th n

in

.3 e the second.ordtr linear d'ifferential equation damping inercase due to bubbfes. Therefore, bubb!cs

, onus fastened to a >pring. As the bubb!e periodi-have a considerable importance in the transmbsion of

, espanda and contracts, the surrounding liquid b "nderwater sound. In order to under.<tand the attenua.

..r.crt mass which is set into vibration, whi!c the l.'on ( sound by gas bubble m hquals, the fun.lamental s.cs is due to the gas in the btibble. This zero-pr cenes by which pubatmg bubb1cs di stpate their

.a radiator has a sharp!y defined resonance at the C""EF Sust be known. 'l his discussion wd, l m, vestn; ate

.:.sacy the portion of the ener;'y radiated in ti:e fccm of sphen. cal sound waves, the part which is tran:formcd f.ir= (37Pc/p )l/2:rR.,

(1) into beat energy during,the polytropic comprenions ec R. is the rnean radius of the bubble, P. is the and npansions of the enclosed gas, and the part of the energy lost in viscous dis ipation. It may be that these

.. pressure at nhich the bubble has the mean radius thrte' processcs comp!ctely account for the total I

ais the ratio c f the specific heats of the gas enclo.cd damping of gas bud!cs in liquids.

.c bubble, and p: is the density of the liquid. This

.,unt frequency dcrived by.Tlinnacrt a:aumes an j

H. THEORY Attic equation ofitate for the gas ia the bubble.

.sc vo!ume pulsation frequency of nonsphen. cal gas (c). Equation of Motion l

-tirs in liquids has been coniidered by Strasberg*

Periodic cnforced changes in the pressure on a

/

3 u ed oblate sphcroids to approximate the nnr.-

bubb!c rcsult in vo!ume pulsations of the bubble. If dical shal es. This determination indicato that the-i cancy is only 3 lightly dependent upon the ratto of the ranplitude of the volume mtsation is arna!!, the

==

a motion of the bubble system is discribed by a second.

unjor to the minor axis of the spheroid. In fact, e a ratio of two, the volume pulsation frequency of order linear' differential equation. I'or this 3ystem et, fate spheroid ditiers by only 29~e from that of a pos3cssing one degree of freedora, the condition of the

{, ".,

nre with the same volume. Observations have shown from the equilibrium volume l'.. The instantaneous bubble system is defmed by the change in volun e v

.: brge bubb!cs are generally nonspherical while volume l's of the bubble is the afgebraic rum of the umall bubbles tend to be spherical.

hddition to simple volume pukations, there a140 mean volume li and e. In a.<imilar manner, the u be o<ciHations in the 3hape of the bubble. 'lhe instantaneous radius Ri of the bubble is the ah;tbraie sum of the mean radius R. and the radial incrcrnent

-:241 frequency for the higher mo&s of shape

r. The bubble is assumed to be in an incompre.-sib!c

,,a m dation has bten calculated by 1.amb'; Stra: berga liquid. At the surface of the liquid, a sinusoidal pressure

'*yl this analy3is to demonstrate that >hape p is applied:

.' attoris do not seem to result in eignificant sound

.,,;. a Nare cicept perhaps very clo<c to the bubb!c.

p= p exp(jet),

(2)

'c sound prenure resulting from excitation of where P is a constant. Ia. quids are sli;;ht!y compro ib!e,

' me pubation3 by several mahani-ms has terentiv but, as long as the bubble me s sm.ul compared :n m discussed in the littr uure.* The mechanism!,

th". wavelength of the pnwum wam the hquni is d* cause bubbles to puhate and radiate cund, are

'"";'utered incompre ibleg,,,1 ne mstantaneous pre.- ure Ne forn6 tion, ceale cence, or division; tne motion Pe ru the undi turbed h, quid 8 the um of the -muzo;dal i frte stream of liquid containing entrained gas P"'ure ? exp(pt) and the >tatic pre-ure f.,:

'desFnt an obstac!e, or the !!ow ofliquid containing p,- p exp(jet)+ p,,,

(3) f 'kagg, T4c tl'y,M.fLm.,f (G. Ih tl an6t Sons Ltd., I.unilun, u n.

llowever, at the bubble surface, the in3tantaneous llinnaert, I'liit..We, 26.235 (1o11).

presum N p the Nantanue [eum O in b

• ' I4ti), &c.* N 1.I.arnh. Ih.fr. /pm ths tD. ate l'ublir iions Inc., New undisturbed hym,d minus th

}

trahrt. J..trousi. Soc..tm. 23. 53t. (1953).

m i

Dirasherg, J..\\w%t. Soc..\\m. 2S, 20 (1956).

a C. $3remen,.\\nn. l'In il 26, 121 (1936).

{

8 &c referen(c 4, &c. hu.

I P#

.I

... w c. s.,m.v.7-- ~; c.

=rm. 7 T www w -- -~~ - - p = -

i Il 10M C I' A li t,1 S D EVIN, J R.

y liquid in motion af,out the bubble. l'er the moment, therefore, a velocity potential exists. The vit,,,;..

f.bentherEt h

~

until the inertial rutction t,f the liquid is detern.inul, potential of a !! quid lurtiste at a di.tance E, due in M*i' P"95 the instant.,ncou< pre %ure P ' at the bubMe w/ac is, simp!c source,in a liquid at nest at infinity,is'

'r#8.

IN

• 8LC

.in'i>oidal od.a's dermed as the sum of the >inu-oldal presure p' and the U"(0/4*N flh pr the ampli:u.h static prenure f.:

"{

P '= p'+ P.- P' cap (j.d)+ P,

(4). and the vdocity of this liquid particle is ahcro P' is the cominx am! tude of the driving N " ~ W " N /4 ' N*)-

(11

3"CY d Ihe o-c pressure p'. The bubble, which is in this uniform but The kinetic energy of all ti.e liquid volume elemu.s

" ".'f Ms O citernatim; pre-sure fictd, camiot be in equitibrium of density pt is with this oscillatin;; prewure unten the hubble iacif 15 pulsating. l*niform pics-ure in the gas bubb!c K.E =(pd2) F"' ?)*(4rr)dR.

(l (11

.herefe i< the re-imp!ies that the intrtia of the ga< is ne;;!i.;ible. The liquid surrounding the bubble provides the incrtia for n:namm; yncr:y

's the bubble system. The equatir.n of motion for the The integration is extended to infinity becau.e !,

• 1. H # " Y.',

fE bubble is assumed to be surrounded by a verv L[.

'[.M bubble system is derived in terms of generalized co-liquid volume. Upon intgration, the fciegoing'expr.

crdinates by using Lagrange's equation. When there cre no disstpation or forem, ; pressures present, sion yields the kinetic energy as stich I e ener;y Lagrange's (quition is

. K.E.= (pdSrR.)(0)'.

(g's breed oscilhthna (d/dt)(OL/04)-OL/Ju=0, (5)

Mnedas Accord,mgly, the Langra ngian L,is where the Lagrangian function L is defined as the kinetic cnergy minus the potential enercy of the 1"(P:/8rNo)(@-(7 e/2F M, a lt, P

5 W !* *[

system. \\yhen dini ntion is present, the dissipath,n and the equation of motion for the bubb!e sv t.s-i pressure is assumed to be proportional to the bubb!c when a sinusoidal pressure P exp(f t) is app!!cd'at e

'.l'[c.'

T v:htme velocity e. lbsipation of this type rnay be surface of the liquid sad dissipation is present,is

[C,-(e y c.f,:

der,ived m, terms of a function B, known as Ray!cign,s (p:/4tRS ba+(3tPc/4rR/)u=-P exp(jx!). :::

/edr.ci as he r-dissipation functign, and denned ast c5 times the n. :

~

D= [6(0)']/2, (6) The forcing function P expijut) is precedcd 1..

a g

mmus sign as a decrease m pressure results m t

[

. where b is the dissipation coefiicient. The equation of increase in the bubb!e volume. The term fp:/4:3..

i 7**.c..,gI:'..;

,n V.

motion for the bubb!e system wnen there is disupatma the geneuti ed mass m2 of the bubble system. Ib

'~

and a generalized driving function C, where neithcr.stifincts of the bubble system is defmed as the &.-

!~

arise from a pottntial, is in pressure on the bubSe surface associated with r

.1. The:ma! 6 n between t. r (d/dt)(OL/00)-OL/Ju+0B/Of = C.

(7) change in bubble volume; thcrcfore, the term (yP.- I-p;u,.d.

s is the adiabatic stmness ka, TIIe potential energy of the bubb'c system is obtained 2 Sund rd by assuming the has in the bub $e undergoes an Ad"-(8Ps'/d f )" (7Pd fe).

(18' L ncous dr i

cdiabatic process during the volume pulsations of the Consequent!v the linear second. order difiertm!.

r!yd W bubble:

equation of', motion for the bubble system is d--

L total den f p:csses:

p,'y,v = p, r,

(s) written as m&+l0-1-k n= -P expfjvt).

il"

,I)01,.i (9)

OP,= -(yfef When the bubble.is slightly nonspherical, cach t.r.-

i

(

^>

" Ps'-Pe = - (yPd Fe)v.

(10). in Eq. (19) is nearly indepIndent of s!: ape when W l

Therefore, the potential energy is mean radius R. is tal.cn as the radius of a sphire In the de:.n

} wency, the L the same volume.2 Transient volume pu!<atio,.

given by the so!ution of Eq. (19) when the richt -

- Ihe Pren 2 e 3' P'

P.E.= -

(P:'- Po)du = (yPc/2 V )u'.

(11) of the equation is :,et equal to zero. Furthtrmort.

• )ac angha 9 the dissipation !< negligible, Eq. (10) boomes agabat:c cye, As the bubble periodically expands and contracts, the' md+kau=0

(-M g3. r.

surroundmg hymd i3 det intu vibratm. The maximum 4mmP*

' kinetic entrgy i,f the liquid p.utides occurs at the and the resqnant frequency of the bubje >y, tem I-1 case there.i-I

,g g monient the bubb'e ha:, arain ricowred.its equdil2rium fgjg.jp g7

,g. pgg g,3 i vohime i. t he 11uw of the hquid,is irrotational; f

. L k>.. -

which is 3finnaert's expresion as given in Pq. (lh d-81I.Gotilsten.Umkal.tfulmnks (ildima.htey Pul,tMing

'See reference 4, Sec. 56.

D')

• M F r] Coyny, Inc., Camterat,:e,199), t,. 21.

%d e

" ?"'N'

_m

.. sQ,.

- <- n.- a as a u s. w au.n - m a.4.;4 O.L.su M.u&udw.a.J.W kdC O Ab* N D A M P I N G O r p tJ L S A T I N G A llt li tJ IJ il l. E S IN W A T I:lt M57

(

t.

y, ten the ri;;bt side of Eq. (19) is zero, i.e., the.cxpansion. The work done by ti,e driving prewure in

/

es

~

4

• ping prewure has been rem <>ved, the 3ound pul3e ' compressing the g.u space is jet equal to the work j.;

4s prn the bubble consists of a damped exponential poidal cecillation. The number of ryeles required done by the expaneling gas in moving the turroundin M

liquid. Ilowever, for the ra,e of a real bubble, the gas b

sp i.r the amplitude of moti<m to reduce to e-' of its in contact with t'.e liquid (loxty f., dows the i othermal k.

jginal value 13 the Q of the bubble >ystem. When the wipation is small, the difference between the fre-equation of state since the liquid has a large speci6c y{p heat and thermal condectivity. In the center of a real p.

pency of the o<cillation and the resonant frequency bubble away from a sub3tance having a high specine s

1the bubble systm without dinipation is riegligible; heat, the gas nearly fo!!ows an adiabatic equation of 3e g of the bubb!c system is exprcssed as state. Therefore, the thermal procc3s is po!ytropic for p

r.h r.r Q-2rf mdb, (22) " .a buWy, and a 1,hase dij7erence exists betwcen the increase m pressure per umt ongm:d presure and g{-

is.

derefe is the resonant frequency. The fraction of the the decrease in volume per unit original volume. This IJ sensining energy In3t in cach cycle of the bubb!c phase difference causes a hystercais eficct. The work f

3,cillation is just 2r/G. In order to maintain a constant done on the gas volume by the driving prewure durin:;

G h

enplitude of o,cillation with time for forced oscil!ations compression is more than the work done by the gas

,X 1a bubble, the rate at which energy is supplied to

> pace in moving the surrounding liquid durin; expan.

's 3e bubble system must be just equal to the rate at sion. This diffstence in the work done represents a net 3

shich the energy of this ' system is dissipated. For flow of heat into the liquid. This net flow of heat into g

g larced oscillations of a bubble, the Q may be alternately the liquid is characterind by the thermal damping f:

15ned as

, constant.

. P The subicct of thermal damping has been investi;ated E

Q=fd(ft-fd, (24 indepen<iently by Piriem," Willi 4," and Saneyosi" and h

shere f and fa are the two frequencies respectively all have obtained somewhat similar results. The f.

dove and below resonance at which the average results of both Willis and Sancyosi are availabic. but g

eund power of the bubb!c has dropped to one. half its unfortunately their derivations are not ca<ily accmible.

nsonance value." The total damping constant 6 is now Consequently, the derivation as outlined by hriem yf p & fir.cd as the reciprocal Q of the bubb!c system or

['-

a wdl be more or less followed.

P.

2( P times the natural logarithmic dccrement A:

In deriving the exprenion for the thermal damping g

constant, the gas bubb!c is as.sumed to be in an in-4-1/Q= A/r.

(21) n y

compressible liquid, and is excited to volume pulsations r

Re total damping may be explained by losses ori;;i.'

by a sinusoidal pressure P'cxp(jwl) applied at the y

rating from thrce processes:

surface of the bubb!c. The !!ciuid has a large_speci6e g

1. 'l.hermal damping Jd. due to the thermal conduc. heat and thermai conJunivitv. and behaves as a heat p

i tion betuen the gas m the bubb!c and the surroundmg re-ervbir. Con 3ecuently, in the liquid adjacent to the f-

)

~

quid.

gas-liquid interface, it will be assumed that there are f

2. Sound radiatmn damp.mg 2,.4 no changes in temperature. In addition, the temper-(

(

3.jiscous da:npmg ca. due to vhcous forces at the ature in the cen!er cf the bubble is nnite. The esci!!a.

1 i

its-hquid interlace.

tions in the pre <sure, volume, and temperature of the i

The total damping constant 3 is the sum of these three gas in the bubb!c will be assumed small. Cun.<cquently, I

the equations relating these three thermodynamic f

processes:

coordinates are linear. In addition, the density an!

1/Q= 4 = 8 d*3,.a-i-3,a.

(25) the spccine heat > of the gas are regarded as constant.

[p-In the gas, the pres ure is not a functior. of po.sition (b) Thermal Damping but only of time. Thcrefore, the gas is in a uniform

{

L

. In the der.tvat. ion of E.q. (1) for the resonant fre-but alternating preuure 6chl; the incitia of the gas is Tnncy the adiabatic equation c. state was a<sumed. mdi iW. The k a tMr pmu is whim.

lhe pr, essure and volume ch.in.;es are in pha<c with Convection is unimportant as the time factor for k

Sac another so that JP'df. equals td!'i/I'.. l'or the c> taw 3hnynt of tlu.s pmess is con.uderably lar>;ct

Miabatic case, there i.s no tran fer of heat. In the than the tmie con.,umed durm, g a half-cyde companion y

, tther limiting of the bubble.

i the gas space,ca e of a purely isothermal prom.s in the pre ><ure and volume chan.:es are In order to calculate the thermal contributiens to apin in phase; JP'dP. otua!s -d!'d!'.. For this We, there is just as much heat flowing outward inom the total d.unping of the bubble system, the change in I

( '*t bubbic*during comprenion as flows inward during[, 'h'."* ;. )

'6 led.N in a li W h'*

i t

k ColuenW l'nnrrdir 41 Le of.Nirutdie Resc. tub amiI W h.p.

J

,.l gay & Sons, Ire., New kk,1950), p. 2 4.*I-Kinster an.1 A. t*rcy. Fund.nuratds of Auunits (John rnent, licpt.1705, s r. No. 6.1 >r 'n 'ns. July 15.1913.

t 68 Z. Lngosi,1.lutrotes h. J. 5,.19 (1981).

.o

- w..

w c-

~~

~~

9r 1653.

C il A lt l. I;S 1) E V I N, j R.

bubble volume must first be determinal. When the Equation (.13) may be I;fferentiatul with rupu t i...

bsa

• I drivin;t pre >ure at the bubble surfate comprewes the time to yictd bubble, work i< done on the gas space. This wmk dbne I'IU"'/U')d*T/M[U(#)/U3 U "

8 cn the t.as opne increa-es the intern d energy of the

.tates of th Il gas, ami al.o rc~ uhs in a transfer of heat energy thneu;th

-v'(JP//01) 0:

t:.e gas. The addul hcat i< transferrol by om;luctirm (pjf,,)(a,,,fgf),(,,gjp)[g(gg,)fa,3_gpjjg, from the g.n bubble, to ti.e surroundmg hqual. t he m

compawion proecss must' obey the conservation of (P//v')(0v'/al) a (pisy,/R)[0(Rh)/at]

.,' {

cnergy prmople as stated m the hr.st law of thermo-

_(pg,,fR)[0(Rg)fg;]

te,.

dynarm,c4:

p P. cxp(j.,t), tw

. A U= ag+ All",

(26) ci -

where sci n the speo.ne heat a.t constent pre ur.-

where oI* is the increase in internal energy of the gas Therefore, the lin.ar difierential (quation den td.m:

The espa-9 s the heat added to the gas space, and air the tcmperature ne!d within the g.ts bubb!c is Therefore.-

i space,3 is the work done on the gas > pace. When each term in a(R0 )/0I-Di[62(Rei)/0R']

g i

Eq. (26) is divided by an induitesimal time at and at is. allowed to approach zero as a ilmit, the rate of

+j(wR/psfri)P' exp(f 4), (.t; id"]

increase in internal cnergy is given as where the thermal diffusivity Da is Kt/p;s,i. A. sol.:ti....

X}f dU/dt= dg/dt+dir/dt.

(27) of this differential equation mu<t sati2fy the b:uadar; i

condition;. At the cutter of the bubb!c, the chance h i

At a po. t m the gas space, the rate at which wor,n,s m

i temperature d must be snite, and the gradient of :'.

i

lone,per umt voturne by the driving pressute on an change in temperature must be zero. The chan.

e Ede' z

l mfiru,tenmal element of volume u, of the gas is temperature mu.t be zero at the gr.s-licuid intuia..

j " #'snT, i

dir/dl= -(P//c')(av'/01).

(2S) white the gradient of the change in teriil erature nm i

i be finite. The solution of Eq. (37) ma be obtainW hs l

p.s.

)

For th.is small e!(ment of vo!ume at a pomt, the rate of several methods. One rncthod is to assume that X l2 I

incicase m internal energy per unit volume is change in tcmperature d is i

p JU/dta pini(d6/dt),

(29) g, y, g. ;)

(33, p,,g g I'It FdJn-i*

where su is the speci6c heat of the gas at constant where v.is a funct. ion of R only.1 Th:rcr. ore, 1..q. (.s.

, P :-

I vo ume, and di s the chan;e m. gas temp (rature trom..

becomes z

i

\\{

the equilitm.um abso!ute temperature To..t.he rate ot -

r. _

l tramfer of heat energy per unit volume fcr an infini-jw(Ry) = Di[a!(Ry)/aR:]4 jwRP'/pis,i.

t.19

, P, s

tesimal votume at a point as a result of conduction is I

proportional to the divergence of the temperature The solution of Eq. (39) is i

X l

gradient; the proportionality constant is the thermal Ry=%[R/R -sinh (hR)/s%R )'3, I ;"

l conductivity Ai of the gas:

l

#' i5 R'P'I#'srt and '. is (je/Di)l. Thcrti -

ag/Ot= Kirvi.

(30) the temperature 6cid with,n the whole gas.pau- -

l v

l l

Since the temperature is a frnction of time and radial nc,w k nown. The change in bubble volume y can nua I.

~F"-

distance, Eq. (30) becomes determincd for a chan;c in the drivin:; pre-ure at

• bubble surfre and a change in tbc temperature -

ag/dl= (Ki/R)[02(R6)/dRQ,,

'(31)

The total change in volume of the ga ne i the -e X

l-where the spherical coordinate system originates at of the changes in volume of uit conce.tnc 401 6 -

the center of the bubble. When the above exprecions radius is R and thhknen JR. A shell ci thicknc-d Mui the ci l

are substituted into Eq. (27), the ditierential equation t It.is evilent that I..+ O M.is not en euct snfutun n,,.,

• r,

• i) r-L bCC

• C8 physical pro!.'em, alti.us:,:ii it sath6es :. t the conoition..d

• it unit or!E matheinatical c.tuati..ns. t he phym.at rsa-.ning inoa a9 G **

4, n,"..,..E' (94+t/R),.d(E9)/UI) the center or tbverature u4ide the bust.ic m t insrs.t e "* *

.fareaon the encan trin 8

u 8

e ht.hhte. i t..u u r. acint.to.: t:, t 4 i." i. 4:

= (Ki/R)[d'(R0 )/hR:]-(P//v')(Oc'/Jt). (32) bubble Suriate. it.e ' mp. rature tramo:t is t !v lK' i sp,-

.rvts r.[;rt-i e

i A. subilitriion can be made for the second term on the with a time aurare sa!ae ni icra. u hereas,u tis,rc as w l**

4 7g g shn:.ht b negatise n.r a not out.s.ir.i d.w in :. cat. I hi n.-.iG s

l tight htdc of the foregoing u[uation hy om,dgu,n;; the camn,d..ut in hr. In tin. attai. w uend e.t k r iron-m. '.a *

, Crt rtrnai a

ideal ga% eqtialion:

obtain 2 ihurris.lial uguatinn with lier.tr o.ohsii nt. a" l

• 3 * *.c hulge g.

in awuniing that 6 an.1 e arc wmwud.d. 't ne usul '. r an n * ** "

at 3 i

'" i in e."

i P/v'= m'C(T.+ 6),

(33) la ' he ".'ed" v"I"' *d 6 i"d" di" d-" d": ".o r n".n"i

.."f i t.c -J ',

I ttraies o harnwnn. it.mntr. it.c nh npa nt nr i

where m'I the ma of the gas rontained in the volume

,g l'[,',n

. 'l' ',,,

i"

.l n g.itt:

t -

i i.:

1. - c

.I n ord ci-the um,ver.-a gas con tant.

appn.,im.u;. i.

v element v, and (,

is

'$ 7 "

"~'W-,

,e%CY

,.-].

gp 7

'F-g

. ~

..p

f, L.

..,_.a....,...m..c......~.........%.

H.,... :.xy;.r.sw.a r,a.w.i. a ;. uan,.aw:a.: yaktw m.%.w a,s m,.,.,,

a.

s, _.:.;._..%..a: w. L*: y..

.. c:.

t i

j D AMPING OF P til. S.\\ T I N G A l lt 111111111. C 5 IN W A r E lt 1659 e

i w a vulume

^

b t t..

differential equaih,n of motion for the bubble sy> tem re=4rRVR.

(41)

.' occord$nce with the ideat g'as law, 'two different is considered. Since the inertia o

""b

' ' " " '

• W'""

, f,2tes og gse gas are exprewed as tial equatioa of molton for the total ga3 Space m tne

{

P/v-Pee.T/T.,

(42) gg

dere Tis the absolute teraperatute of the compnsed g4f,,, _ gp,xp(j;,),,j), _ p.,,p gjw,), (49) ps. When Eq. (13) is differentiated; the result a,s 4

m P'exp(jet) is the sinuwidat excitation prewure i

den (c/Fe)dT-(v /Pe)df/

at the surface of the bubble. When the expression for P's 8

(43) the change in hubb!c volume v, as giver, by Eq. (47),

j e

fribin:

du=4x exp(jxt)[1Fy/7.-R2P'/P.]. R.

(41).

is difierentiated and substituted into Eq. (49), the t

1 uurr i

followmg expremon s.s obtained:

he exptmion for Ry is obtained from Eq. (40).

Screfore, the total chan;;c in the bubble voiume is 3

y#

m_

it, k+jwh qPe (3h

' 'I"M X 1+ -(7-1) [f R.,coth($tRe)]-1 3

e 1

"I" iP i'

f/R

.(50)

"J X

,P' "R' sinh (p R) ~

P'R'

--R, h(p R.).

P. )

Since the parameter pi has the symbo! j under the JR (45) n jir 7s.R.

n sm y,p'cu squate root, which is undesirable, and there is the i

need to separate Eq. (50) into real and iraaginary e-

' I'"

4 ris To components, a substitution for tl: is introduced:

n must

$/= (1+j)%*= 2jd/=jw/Di JX 1-

--Y/RU'['f Re coth(4:Rc)]-1 (46)

Y'"(IYl)0'"IIYSf"! O' '

I jj Fe L

b V

Accordingly, the parameter c5R, is R,(e/2D )I or

, ust. ion (46) is further su.nplified by not.mg that T.

(3.c Ro(:pis f/K )l. When the frequency and the radius of n

i

.f/pi(sn-s.i). Therefore, e is the bubb:e are kept constant, the q'antity oiR3 varies I

i 8

(3p u

J as the square root or the density of the gas, or alter-

[

}~y,piu natively as the square root of the avera;;e pressure c

(3t 4 yPe

. inside the bubb!c since the specitic heat at constant '

{

t r 3

- pressure and the. thermal conduttivity are ir: dependent X 1+ [ -O{[piR coth(fi.)]-1\\

(47) " of density. Another condition exists for diR v. he b.

R excitation frequency is constant and the radius of the f

2 (4m p/K/ \\

/;

t.

bubbic satishes Eq. 1) for a resonant buab!c. Ior I,c I

this case, etc parame(ter 6.R. varies as the ave

["".i

<e w (

'v Ve-1, 1,3 - P.-

/ ^

pressure inside the bubb!c. By introdue:ng the I'#

the r,

r.

yP, substitution for' pi and noting the identities for I

b sinh (diRo+jciRe) and cosh (piRo+jci

), Eq. (50) l R

6 becomes X 1+ (y-1) [fi.,coth(5:

{. -

3 sum R

R )]-1 (43)

A-jwk J.

go.c f/R/

F.

=-[1 3(7-1)' sinh (23 R.)--sin (20 R )

f.

gg 4tn the change in volume per unit ori:,inal volume k'+(wb)' yP., + 26:Re l cosh (2ciR,)-sin (2 sir.)

[

2 j

U.c l-ri)/V, is plotted against the cham [e in presure

. sinh (26 R.)+ sin (26 R,,)

1 i

jl' l** cnit original presure' (P/-P.)/P. on a lyressure-7[geo33(;j,g,)_co3(yj,g,)4,pff l(

g.

.crae graph for the real components of Eq t.

,,.g

. (#:).

{i

• e area ciwkned by the compresion and espan ion t t.<

i' Q,,; f ts represent 3 the net to s of encry,y by heat conduc-(51)

[

1;.4. The change in hubide volume is Even though Eq. (31) is separated into real aml l

now hnown.

l'tre remains now the task of relating the thange in imaginary terms, the form is still not suitable for f ta 3 '*

lNure at the surface of the hnhble to the vibrational

• hubble vo!um I

7

/

in Eq. (22), the damping constant is given in terms of l

6 *, isities of the gas I.

h' ting attributes.

hubble, i.e., the stiTnos and the diwipation codlicient b, roonant circular frequency

(;,*,

w., and the generaliecd m tss m.. Ilowever the general-l.*.I ized mass i4

. iorder to dactmine the. stiffness and damping, the simply the.tiriness k divided by the square of the res.mant circular freynency. Therefore, 4

(4.

~e g, _ -

--w x

-1

[

b a u.

i CII A kl.!:S D EVIN, J R.

1660 g

the dimcu,iemk.s damping constant, which at etwnance clearly illu.<trates this tran4 tion reghm b,;tw. n ***

es.

t h,the reripioral p,is ab/h. At roonance, the snaximum thermal and adiabatic states.

value of the thrrm:d damping ton' tant is nbout one.

The t,tiffnem of ga* bubbles is also imp,,rt.n.:

tenth; therefore, (ebe/kF is scry small and can be the re onant frequency for the bubble sy tim i,.g...

g.

neglected with m. pert to unity:

proportional to the rquare root of the ic torina..p

. Ily compering the real tctms on both sides of E. '.-

0

I (1/h)[1-j,(uba/k)]

the stifiness is expressed as i

- Ov(ld, )[t->.(wba/k),r.. (52) y, y3,,

s,

3

[1+ (ubo./h)']

p

} e,

1+ -

-m Accordingly, Eq. (51) becomes A y, 3

[

(53) x 34 (y-1) sinh (24:R.)-sin (26 R,.)

l' wbo. ~

1 -).

k-k 24:R. co:.h(24:R )-cos(24:R) sinh (2ci e)-sin (26 Ro) -

For large values of'24:Ro, i.e., f'or lar;;c bubh!..,.

F.

3(y-1)

R 20i. \\ cosh (2c R )-cos(24,R.).

the stiffncss approaching the adiabatic s:itin.... -

~~yP.

R dimensionless stitTness yPa/kl e is given to wi:h;a..

sinh (2+i e)+ sin (26 Ro) 1 percent by-R ci o/

yP.

3(y-1) 1+3(y-1) pproaches u.hy

\\ cosh (2ci e)-cos(26 R.)

R R

.uing stifincu c:

er 1'

1-j 20i., sinh (2i,i.3)- sin (2&i c)

+.

X R

R kaV.

2ctRa 20iRo rastant cre nr.'.

( (y R i[f((

1) ' cush'26 Re)-cos(26tRo)L Pfricm, due to an oversight, used only the fir. t paw e

of uba/k in btaining Eq. (3S), and, therefore,.! -.

3.hr-ald.% ;.

Therefore, the dimensionless thermal damping constant an crroncous result.,l'he stmness, as given 1.;. -

The corrret en g,

foregoing equation, it used to determine the !.rr-

.troduud H > -d sinh (24:Ro)+ sin (24 Ra) 1 damping constant at resonance. Conecquently.

8 Minnert dd.ed-theimal damping con < tant at re.tonance obtah ! -

I

,.-,3

, Nj..,,C-[cM cash (20i.)-cos(2?i.)

R R

R di.

Piriem is incorrect For small va!ues of 26 R.it.

u5 (3I) small bubbles and the stitinem-

,I

~

k.1+[2ci c/3(7-1)] 2ciR.

stiffness approaches 7-5 for small values c,f 2ciR R

The facier a d:-

R oit

.ti. ness fr. '

Large values of 20i. corre pond to large bubb!cs or the adiabatic ca e. The dissipation ari<ing from heat Er;. (1) bccomes c:nduction vanishes. The thermal ciamping constant far very small values of 26 Re, i.e., equal to or ! css 4

!"6 than 2, is given to within one percent by j

oos q

!a the di-cun e5o,/6[(y,1)/y][(26 R,,)'/30]7 (36) idd the l' W'

r 1 tantar.w.

4 Yery small values of 26 R. corrc< pond to small bubb!cs T

IInever, w'.ta t

~

er the iwthermal ca<e, and rgain diWpation arising I

,,.., J.i u....<--

Utssure increa--

from heat rondu tion vanishes. llowever, there is a

?

,c,

• cquent:y. tb transition reyinn betwetn small and large values of I,

'4 tie is rrta:..

2(i oR where the thermal di, ipatian is a mannum.

A

'c bubb*e

-t r.

In this transition region, the relation between the

!! c.on,n.9. :. t r 7

pressure :uul the total gas vohnne can be exprewd as e

4 P/VO equal, a son-tant, where the esponcut n varies Qr%i*l$

f from unity to y; the state is polytropic, l'igure 1, a Fic. l. Thermal [bm;Jng e..mtant es d mensionte

• 1%" R. RA"A plot of tlie thermal dampiitg constant who./k rs lfiRo, pmmeier 2piR 30, m (tW.

1y

\\

N,- p s p ~ --- ~ -~ q - n... o..

. ~. ?,-vr

w.. S.=...

.g

.,..n.

. newnw.v * % ne. -- ---

L a

~

u

.k. m

.....m.

w.. n n m, n G na m xpng y

i.....v.

.. w. -....

.v

....,. : o...n

..u..U. *.c M.s.m.M: G.&.w.;.m.s.a wr= w sM, *a ZV.;

• A *'~%. : Lf,6 k,, v~..

.:.w.:q,;,,,,.w u.ut.a,.

r...

s y

D A fs! P I N G ' O F PULSATING A ! !t IS U li ll L E S IN WATElt 1661

^

\\

c' among some of the inve3tir,ators who have d!>cumd

_ l the efftet of surface tension on the bubble stifint.w.

O a

?

The prohkr1 is to relate the p.more on the surface of s

e f.,.

- {*

the bubble, which is a.-.ociated with the thant;c h.

r c.r i..

f.

bubble volume, to the prenure inside the hubide. The N 8:

/

stiffness l' is defined as k'= -(OP//0 V ),

(ISb) j i

i..o.....

The pressure P4 inside the bubble is

% or P.= P/+ (2.r/R ),

(62).

i

(

wherc. P/ is the pressure on the bubb!c surface, a is the

)

surfacc tension, and R:-is the instantaneous bubb!e radius. Tbc polytropic equation of state for the gas.

c6, i,

%N 8 8,'.

inside the bubble is m;,,,.j...

rio. 2. Dimensio.ikmtirnen rs dimensionkss P,=.[P.+(2a/R.)ZVe/ V ].

(61) g stametcr 2AA..

Accordingly, p,> roaches unity for large values of 24 R. The re.

P/= P.-2a/R =[P +(2e/Re,)](VdV )'-2e/Ri (61) p nririg stiffness of the gas and the thermal damp,mg l

L mstant are now known. The only rcmaining task is and, as n is y/a, thcn sintroduce the correct expression for the stir? ness into k'= (yP./V.)[1+(2a/P.Ro)-(2 /3::PJ.)] (6a. )

a h8't l' *'

te equation for the frcquency, and then determine the oy gfy, yp /yo, (655)

'rd" "d hnnat damping constant og., at resonance.

Is p

g The correct espression for the stmness wdl now be 5"

where

troducul into the equation for the res.onant frequency.

"d' E "*

tanaer t derived the equation for the resonant ircquency g= 1+(2a/P.R )-(2s/3 P R ).

1 b,' '"" ' '

ipulsating gas bubbks in liquids by considering an Therefore, the correct expression for the tcsonant

.fabatic equation of state. However, the state is frequencyfe is

"* ' h ' ' "

.ifytropic; the stifiness constant 1 for large bubbles, fe= (3yPcg/p a)l/2:R3=/3r(g/o)l.

(66) w is r

dich is of practical importance in discussing under..

isttr sound transmission, can be obtained from When the bubbles are very large, the stiffness is. the adiabatic sthTness and al<o surface tension etTects are

'a. (5S):

(CO) negligible; consequently, the ratio g.'a is unity and the

)

k=7 e/Vp, P

V resonant frequency is 'given esactly by.Tiinnacrt's dere

.p!ott,de.

3(y-1) 3(7-1)

,"l lamping is of prime importance at resonance, the thermal damping constant will now be dc:crmin.ed f",.h" 2c R.

24:Re /.

3 as a functmn of the rc<onant frequcacy. At rewinance, le factor a describes the departure of the bubble the parameter ci. is R

,finess from the adiabatic stifiness. Consequently, 3R =R.(n/2D )i= (3tPg/4:p D /c)I

~

t:

'4 (1) becomes

= (Fg//s)l, (67) f= (37Pe/pft)l/2rRa.*f;i/a.

(61) l where I,. (37Pc,Alep:D ). I.or a given preuure, the i

is i

a the discu.9 ion 50 far, the ir.stantaneous pressure characterinic frequency F is a constant for the ga*.

Lide the bubble has been considered the same as the When F.q. (67) is squared and the exprewton for a 4tantaneous prc+ure on the surface ef the bubble. Substituted, a qua lratic equation for ci re>uks in R

hever, when the bubb'c i.4 > mal!. the surface tention terms of tl'e resonant sequency f,, ch..racteri-tic frequency F, and g.11 sub-titutimt this value of oi.,

R

%ure increa<cs the pre"ure inside the bubbie; 3

%tquently, the in<tantaneous 1 rewure inside the into Eq. (55), the thenuat damping constant J.

at

.'4ble is greater than the instantaneous presure on re onance is found to be

(

• bubble surface. Smith," liriggs,, john >on,and 16 Fg l 6hq

\\..

hson,"_.Spitect," and I;obinson and !!uthanan" are

)

-3 H7 ~ I) [8 T-N (08)

• If. Smith, I'l.it. 31ag.19.1117 (10.15).

Idt" g "2 hl 4 Jotumn, and Sla ain, J. Asuust. Soc. Am.19, CGI g

7 4,nt,

-4

• k. R.,1,;

ui and R. Iluchanan, l' roc. Phys. Soc. (f.ondon) gI,m (19%).

9(y-1)',./e m --

--r gpg _

^~

-~

(7 16T CII Alti.ES Dhy t N, j lt.

Thcrefort, th.

s proportirmal to the 3.itiitre runt of the remnant fr.

3,g,i(b,i..,

i

,,. 7/

quency at low fiequenuo.

i J,

/

sh'fC N: I* *

{,~

/,

7 (c) Radiation Damping

he rati.l; u i ghe radial:.ra In a comprnsible liquid, a bubble excittil i.,.

xndent v.i fr.

1 i

},

g l

volume pubations esptads a partion of its ar;.y L.

r I

radiating spherical sound wavu. The bubb!c i,....

siderol as a simple sount! murce; the bubble r.uliu. /:

The v.u c..s is considend small compued to the wavelem:th A..

3 c o, u

g !satm(p

e..de s,....d. i, 1,o.u.. u.",

@c radiatcd sound. Smith" has calcalatol the r.uh u...

damping for gas bubbles in liquids. In ordtr to dtr.,

aquid wu.-

Fio. 3. Therroal demp:gr constant for it?mant air tutt;fritru. - the uIires.-ion for the radiation damping con tant, t!.

'ater, Sph u r -

6 1ri in watts. ~1h. upper curve is it.c cri mus rault obtainct! by I y

p velocity potential for a simple s.mu oldal :.ourse ia ycr a p ^... ;.

c mirenib!c liquid is stated:

if[ ult 7, J

As Pfriem used an incorrect exprc wiun for u, he dcived I

the fo:!owing cnoncous ruult for the~re:>onant thermal -

og (j,.vi/4:R) esp [jw(t--R/c:)],

i i.'

,ipation of c-I damping ccnstant:

,"h"#

t where c is the velocity of sound in the liquid. R i. ti.

radial distat.cc, and i is the complex amplitude of e.

l 1

3M=-

change in bubbic vohime v. The charg in bubb

[*".:'

i 6"

37 - 1 1.g volume u is simply In the :'t.

3(y -1) Je g.: exp(jet),

IIk raitted ir-b 2

where the ccmplex emplitude ci is dcfined bv 1:q. # 4

• !*[""'

X 1-(69) The acoustic presure is p (6%'at). Consequent v. -

! * 'F *

• I 3y-1 Ff tl e bubble surface, the acounic pressure is

+ :r.omeatu:::

3(r-1) 3(T-I)-

-1 g,g.

I' p.--(p;c*ci/4 R.)[1-j(eRe/c )-(x'R//c/23 pr unit v A L-

+l(W'h'd'?3!)3 'XP(lde

'U

• • I When the following values are used where oMy the fint four terms are hept in the expan !.

l=n.tvo,.u=<

i ribu en -

Pe = 1 X 10' d/cm '

2 of exp(jeRc/rr). The acou < tic prenure on the bul!/

! a nony's o*2 a = 75 d/cm, surface is just the difierence between the drivL.

43nce of Kr 5.6X10-s c:.1/cm.sec-deg c, pressure on the surface of the liquid and the chau l geo;;;,.,

each tm'it m.-

~ = 1.40' in pressure on the bubble surface associated with *'.

a e

change in bubble volume. The change in prevure.

h s,70.24 cal /g, the bubble surface is k'e. Therefore,

'{d/d! Md p = 1.00 g/cm', and

,g, 9p;, exp(jwf), _ p,xp(jyf) (73

= - Q,4

j pi= 1.29X1Cr3 g/cm*,

d.ere d J' or aI the redonant thermal damping constant 30, calculated

• • • • ". P hR((** RET" tk hqu.

P 5l using ):q. (6S) and the one calculated u-in.: pfriem's c/2!/

^"".6"*[

• 7" it equatian can be compared. Fi; ure 3 clearly shows that O

the result obtained by piriem is 50 to 657' too high.

*3""**'

uRe e'R/

f

.The rewmant the mal damping con tant determir.ul in

+ p:o 6 t fv = - Pee '. i J.

he mo: ion -

I this. survey agrees exactiv with Willis' curve as given M'

'8

1. 23!

'* be c.r In the report by SpitEr. For air liubbtu in water t.

]:I littger than 1*3 p raditts am! with re onant ircquencies Since the term w.R/c is approximately l'73. E '

ti i.. !

less than 240 kc/ wee, the thermal damping constant " terms of higher order than wR/c will be nc$$til resonant frequency a, few than about

.N.

,. i,.

m

.t -

. When the 15 y.iven to within one percent by 1:q. (6f, )ke,3ce, i.e.,

and Eq. (76) becomes l

I 4

lI bubb!.s larpr than 500 p in radius. the resonant (p:/ trR )ii (p.d/4:r:)6+Fu= - P u;)(j d.

III 3"I'd Ih' As long as the hubb!c rad.ms is small compared 8.> t.

?,,.e";', '.9

",I.

jI thermal constant is given to within one percent by i

g h 7, the s.imple relations!u.p:

wavelength of the radiated round,..it is >cen th..'. d -

i * --

.-(,

3.q [9(Y-1)'/4 F]l(f,,):

(70) generali,rd mass term for the case of a comi.re-Jy G. A!,f,{,'

i g

b" d*

w as de contsponding tenn in M, t (,

{ psa;,rc.,<3 where the h."".l was omsideral,meomprewb!c..

i-ki.= 4.41X10 'j,,l >cel, (71) c p quu distfosing that the thermal damping, omstant is radiation diwipation.coedkient b,.4 is p.d/152 b.-.sm..,....r..-.~.--..

--T-=.*.-*=--"e**mm*****w*********8""**e.********'*"**""**********.""'~'#**

w,

> ~ * -

n

.c w....~.,.-. 4

....s..<..

..a.;. 4.,... ;....q,, ;,..

}'Jys,ia d a. W M/C.c.s.6?ar..is;.a 4 nl.h.a_p b ac

Z..W 'h.a;.LP.

Ms/c,.e4,M.J.,4.. ;,,

D A P.! P I N G O F PULS ATING AIR ll U D Ill. F. S IN V A T!;ll W3

'\\

l erefore, the resonance radiation damping constant is. any element' of vofmne internal to the liquid. The

(

th Navier-Stokes ccguation is not applic.:b'e for discuoin,t B Itt.

= (b,,J/e m:) = (w.R,/c:)u (2rR./u/c )(g/$)l, (78) the eficct of viscinity for the case of a pu!>atmg i st:er-mr is the generalized maw.1'or large bubbles, spherical bubble.

t 3e ratio g/a is unity, and since 'brRa/u/c:is a con tant, Ilowever. even thou;;h the net vinous forces in the l3c iadiation d.unping constapt at resonance is inde.

) in:..

liquid, vanish, thue are viscous fmtes acting at the ty h;.

'yadent of frequency.

~

surface of the bub!de where they exert an exce S g

pressure. 3fa!!ork gives us a physical picture of the

[

(d) Viscous Damp ~ng effect of viscosity on a put.atins; bubbla by cunsiderin::

tor,.

' u s R.

a small clement of a >pherical shell of liquid at the The v.iscous damp.mg constant at resonance for a x e, bubb!c surface. This ocment has delimte radial and

,ulsating sphen. cal bubbic.m a viscous, mcomprenible

, g!"*

)luid will now be discuned. 3f a!!ock8' in 1910, and, jatcral dimensons at the. instant the bubb!c rah.us s cine f

t.-

at its mean po3ition. Whcn the bubb!c expands, the ater, Spitzer,2and Por;. kv5, m.vutigated th.is prob!cm.

('Uy*

For a puh:ating bubble, the etTect of viscosity is perhaps small h.qmd cicment is th.storted; the radial thickness

. talicult to visualize. Lamba tates, "The only condition decreases while the lateral dimension merca<cs. Like.

^

s l2r.dcr which a liquid can be in motion without dis-O' n tb bulMy contracu, the h,qmd element i<

?

agam distorted; th,s time the radial thickness increase.<

1' i

(8.,4 3 pstion of energy by v.iscosity is that thuc must bc and the lateral d.: mens;on deciencs. c..mce the h.qmd:.s mthere any extensions or contract. ions of h.ncar

)';"*

dements;in other wcrds, the mot on mu.st consist of a inc n nress:oic, the distoruon.as net caused by a chan;c i

t4,>!e ram!ation and a rotation of the mass as a whole, as in in the volume of the hquid clument but by viscons y,,,,e. f rigid body.,,

stresses. Consequently, more energy is required to o

i In the presence of viscosity, momentum is trans. c minos the bubble than is regamed in the subecquent

• *P* * "'

"* 1 Ihe raWal : notion and sphencal (it!

siitted from one region of the liquid to another moein-it a different velocity. An clement of liquid moving synunetry, tk pn.nc: pal d:, rect:cns of stress and rate I

I 54"I" "*

dvadic Sn is-lt be radial. Acc ramgly, the net stress f

Sp.Jty in a pariicular d:rection tends to trammit its ly,o

.memcatum to other e!cments of the h.qmd.,The f

{

} Navier. Stoics (quation of motion describes the force Sn= -(2p/3)v.(dR/d!)I 2p.L, (S0)

- (~

l;ict unit volume acting on an infinitesimal c!craent of whcm I. the.dera factor and A.. the rate of pure

{

is i

is 4!ume,' at a point in a viscous liquid. The force per h"("73 i mit volume is due to the instantaneous pressure str m dy dic. Smcc the rate of pure stram is the

!htribution of the surroundin; liquid as in the case of gradient of the radial velocity and the hquid is m.

uhh!<

. i nonviscous liquid, and is also due to the rate of c rnpressib!c, the radial stress at the su: face of the bubb!c is

("V"4 hange of momentum caused by the presence of

'3'[* Meosity. When there are no external forces acting on Sn=2 V(JR/dt)=--(pe/rR/).

(81) i 2ch unit mass of the liquid, the equation of motion is re ca When the effect of viscosity is included, the equation

>{0/at+(JR/dt).r](dR/dt) of motion for the bubble system is l~

(i.h

=~ rte + (p/3) ! v[r. (J R/d!)]) +g r2(dR/dt), (79)

. (gf3g,);j4( f,.pj)gy._p expgjyf), (gy)

.f iltere dR/l!t is the radial vclocity vector, fa is the where (u/rRJ) is the viscous dissipation coefiicient h

ean pressure, and 2 is the coeflicient of viscosity. As b.;..

Therefore, 'the viscous damping constant at le liquid is considtred to be incomprecible, the re:.onance is hergence of the velocity vanishes >o that the second crm on the ri;;ht side of the equation is zero. The 6.;.e (ba./m.nr2) = (Srpfy,/3 P.g)

~

7

'unaining vi cous term pr2(JR/dt) is also zero since

= (Stufu/37P.)(o/g)l.

(83)

(ifd ie motion of the !iquid is irrotational and the ve!acity y

The v.iscous dampin:; con-tant at rc<onance is direct:y 4ft be eyirt33ed as the gradient of a scalar velocit,

g 6ttntial. Therefore, there are nu net viscous forces

• PNI"8' nal to the resonant ircquencyf l

fg,

" tint; inside the liquid for the case of a pulsatin-L s

E

.&crical bul.hte in an incompressil,:c, viscous liquid' (c) Total Da:nping Constant at Resonance

.{

sue to the presence of viscosity, inomentum is trans.

The total damping constant 3 at resonance is i~

9

(;7'

'atted through the liquid, but each infinitesimal I

' uncut of liquid volume reteivu just as mui;h as it 37 3u.-F3 s-F 3.i.,

(SI) trl ib

%s.Thertfore, there i.s no net viscous force atting on where the thermal, radiation, and viscous dampin:t fl M.

" A. mit. 6 t, Pi..c. unv..w. (Iominn> As l. 3'st (t9tm.

constants are given by IGgs. -(6S), (7S), and (8.0, f

T

11. Purie Ly. Pr..c. l'irst N.iiinut Congem of Appl..\\fe.

J I' f,L. l hiis; p. 813 tjunc 19311.

8' L P.mc. hirclwt!..= fa T4roretical l'4 r,l. e ( D V.u L,t ramt I

.I

%c Arenu 4, &c..iN.

Comgany, tu., Prinas.4n,.%v Jir ty, v6h, p. m.

i Irs:

i

~

i i

_.m

.~s..%-

,9.%a -h.+

+

-~*-4.-

N

W M64 C II A R L l's D E V I N. j it.

l gg photoc!cctric method, ami, for mall bubble.<. the ra.l..!

o n

- l' ***}pjyl velocity can be measured usint,a hinil of vetority. rih!.a

'oroco i

o ncoo 7[-- -

microphone..-

oo co 1.1%:ocIcd ic 31cilw.I

O.1o

/

}*',,,,,,a eoroo

[ 00'"

A single gas bubble pulsating to a mnic c.ritati,,n ;.

.j joo9.o illuminattd optic:.ity and the scattered light m.a ur..!

21

{ '

fi L.

by a 1.h9totlectric "". The chan:c in trou >u t.,n..,

y

/}

the bubble image modulates the quantity of t...

y o,ob g",',,

/.

amplified and recorded on a suitable rcce,r&r. I::.

/

rcccived at the photoce!!. In the photo:cl! sin u..

a

/

. there is geacrated an alternating current whhh

~~[

j L

, varying.the ronic excitation frequency and notin 9 0.02 ;---

changea in the bubb!c cron section, the band wi.!9

'"*'...,o and the resonant frequency can be determincd.

.o

,co

.co,osa ruw.c,e....

.,o...u." W Fm. 5. The :L Fac. 4. Tleuretical thermaf, ra fistinn. viscoue. and total d:rnping tonstants for rewnant ait tmbleles in water.

2. RiMon 3ficropl:cne Met lrol A single bubble is caught on a small was -ph.r.

respectively. l'igurc 4 is a plot of the thermal, radiation, fastened to a platinum thread which is p! aced latu..:

visemn:, and total damping constants as a function of the pol:s of an electromagnct. The bubMe puf-at;..

the iemnant frequency for air bubb!cs in water. In are produced by a constant frequency magnetn :r!. r,

this figure, damping coastants are given for bubblcs projector. As the l'ubbic pulsatc5. the platinum thr.a.

l ranging in radius from 3 y to 3 mm.

is carried along with the oscillation 5 and this nt.:..

l of the thread produces an alternating cmf c.h!.h l shue 2 is t_e =

s 111. EXPERii.1MTAL METI!ODS proportional to the radial velocitv. Since the gem raL.

,There are essentia!!y four methods by which the mass of the bubble system is co$siderably great. r t:u -

i eend pren ru cf resonant dampmg constant can be dcunm,ned exper" the vibrating part ei the thread mass and wax m.r j 35pectively. Thr.

b' I< * ", " '

ment.dly. T!ost of these methods are mdirect ones the thread and was are assumed to exert ntn!idc.

I U~P M.C#5C '

mvolema the calculation of the dampmrt from certain imiuence on the resonant frequency. The bubbb-i

• #."d l

, measumt acoustical properties of the bubbies, allowed to groa slowly and its diameter mea 4uto! u "

l *F CY ","' I:

a microscope; the voltage produced bv the t.bi...

t j rodtces the raa p

miciophone traverses a masinmm as th'e diameirr -

(a) Succc:;s.tvc Ose.dht. ions the bubble incrwes. Therefore, the resonant freynci..

(d' R e

The snethod of successive oscillations is a direct and damping con 3 tant can be determined.

process for determining the damping constant. The This rnee:d c signal fiom a hydrophne. which is placed close-to a (c) Standing-Wave Ra'tios" $

onstant den 5 pulsating bubble, is ampli: icd and applied to the input mnd by a scr:r:

. terminals of a cathode. ray oscilloscope; the bubble m;te bubble.is allowed to rue freely.m a L.. ;m~

. he, the atte..a.

A s.

pulse :,ppears on the screen as a damped sine. wave. If idled tube and puh. ate under the induer.cc oc. a p'

mimum at tb the aieplitude of succes ive occillations is plotted as a E*E'#" "* '.wnd wave. The diameter of the I'dr dec or - J r-

~

function of the cyde number of oscillation, the Io; arith.

• * ","*

• d* * " *I F l :TpWte sido c mic derrement, and, therciore, the reciprocal Q can be !'". c ns" tant crer the cron rection of the td-5 faced tow::! t detcunined from the tope. ile knowin:t the time scafe-I 5"'I*CC "'# P"C"d by an ab orpa; ave frn:n :

turbances by re": lect:on of th:s sound w de paint of i.M cero.s the screen, the re-onan't frcquenc'v of the bubb!c on deu.

w system can be found.

by usm; a putre techu.v[ue. The sound entrn /

Attin data a U I

which is radiated by a transdacer at the toact iwi 4 s; d i nn c (b) Width of the Resonance Response"-"

the tube, a p.ittly retletted A, by the bubb:c. -

I..f tb bub"~ '

! gh] *.

n tr Previously, the Q for forced o eil!ations of a bubb!c recorded by a probe hydroph ene arr.mged lectuu was dctinut. Since the uund powcr is prr.portional'to, transducer and the bubb!c. The resonant damp ;

l., M p da r-the square of either the radial velocity or the radial con.stant can be mea-urtd from the relative redM ",

1., a pc;z3t 3 ;3 4

di placement. the re3nnant damping constant can be coedicient (E,'E.)i of a bubble patsating at its re-ons-g g.3 3, )y found by ph4tntg the square of ci:hcr of the-c parame-frcquency t t.pg,,,9 ter. as a functiop of the fre.tueng. The amplitude of 3.= (R./2)(4 E,/E.)l = (Rap,/Zp.)(2)l, N-Lmts or pd-r e.tediat,mn of large bubbles can be found u ing a-r'wd a'm &

121. Mrpr aml K Tamm. Atust. Z. f. 145 (1939L M M. Esner, Atu t. IMh. No.1 25 (lo.tli.

n. n a., A

. m, ii tiei 3. 07 (IC3b.

p[,q,"','#

C8

'8 M. Es n r an.1 W. II.impe, Acu-O If. f.wcr. Mu t. Ikih..h, m,I.12 (IWlb_. A_m s. m om.

,i.

n..A.i., xmu.. _

s g

' " "' Y

~-- -, 7.,-, ;,.,,..,.N.v.~.m-~ n r,-e 7sv.

n m& m W:. ~ f

.--m u

.u t 4.:.....

~- - s.

,, M.si r;:Aam

• m 2 d'.*1a.s.;;;/JJ3:-4..a u a.24. %.$.'.q'd,;*(;,;d.,j, ;,... ;-;.g, y,g;-f<,2g g,' g i

.g.

- li A hl P ! N G O l' P Ul.S A T I N G A I It 11 U 11 B !. I: S IN W A T1:It 1665

(..

x l-oro s..,

9 h

a

", u

,f.

e I

d'o,g o 4

0 t.

y F

s a

D.,,.

E e

o

/

F v

,t O.04 q

[Ib i:

/

l g

hh',

0.02 8

4 2 ~

I esonant.I"restricy f

[h 4

IO 20 40 1o0 200 400 I

o in knocycles ger second 8

yac. 5. The 11.enietical curve and esperimental valm for the total elampiv cc.nstant r,f reenant air buht>!cs in water.

j The points are from faired curves through the operimes.tu's data.

I Symtal Esperiinenter Mett.od l

X Meyer and Tamm Width of resonr.ce ruponse f,n' }

O Carstensus and Foldy Reronsnce ahnrption i

A IJauer Sxec.s ive oscil:atic,as

,n.

T) 1.auer Width of resonar.ce tcspense M

e Exner Stan ling. wave ratios A

unct and llampe Standin;; was e estirs

.ad 31 liscsle Standi:,g.v as e r-ths r.

shcre Z is the radius of the tube. and pg and p, ar? the, using the transminion hydrophone. This method is cmd pressures of the incident and reflected waves, repeated for scccral pm,iector frequenei.;s. Carstensen

('.upcctively. The incident sound cncqy E,is corrected and Fo:dy" give the resonant damping constant in 5e hittion lones occurring in the tube. Therefore, the terms of the attenuation.-i as

%- braping constant can be me:u ured from the standin;;-

L

save ratio, hnd the resonant frequency is that fre-ore 9e/(R')'3,a]

(,,

ency of the phne progressive sound wave whi h

'I"4 3I"f sec(e),

(S6) c og. poducts the maximum pulsation for the bubbic.

g

,I cc

~

where n is the average number of bubbles per unit l.

-(d) Resonance Absorph.on"J' o

vo!sme, d is the thickness of the bubble scrc<.n, R' is

?

'this method of determining the resonant damping the off-resonant bubbic radius, and J is ((Ro/R')2-1]2

enstant depends upon measuring the attenuation of In deriving this cr;uation, Carstensen and Foldy j

g.

' ound by a s:rcen of bubbles. For bubh!cs of a single assumed, for a buhbie screen containin;; bubblo of Ig" ize the attenuation throutth a bubble screen is a

".' !'m,imum at the rc<onant irhuency of the bubb!cs. A essentially uniform size, tbat thc off-resonance dampin;;

constant is 4,(R.v'R')8

';rojector and tran minion hy'dmphone arc located on

p..

t['osite sides of the bubb!c scrcen. F.ach instrument IV. COMrai:ISON OF TIIEO!1Y WITII i

• Iaced toward the othcr, and the line joinin;; them, at IM TAL RESL:LTS

,h c point of intersection with the babble screen, forms Thc theoretical damping constant and the experi-(

4 angle < with ti.e norm! to the screen. In onler to mental valucs for the damping constants are plotted j

g} j,%SIIe the rate of riEtain data as to the distribtition of bubblu according as a fun yI r.e of behhles, whii.h is a functivt g

4 the bubble radiu<, is determined. If only a very indicatin,; how well the es[Nrimental re.<ults agree with 9

the theory of damping.

%rt butst of hubb!n is aitowed to ocape from the 3 feyer and Tamm" have used ti.1,tdth of the p].

hbble pmductrs and the resultant screen is obwrved resonance response method to obtain.he da.nping

,"" hCI ht h and time i !ater, the.*creen contains only I-s

.%e bubb!n who-e rate of rise is h/t. When the constant; the>e :esults are estremely high. In using I

r.

8 hbbka are allw..ul to rise freelv but in the:c dstinite the ribbon mioophoae, conJderable damping rnay

~

have been due to tiie oscillation of the p!alinum thread R

,:ttsts or pule, attenuation ncasurements :s time in the magnetic ticht. In addition. the espaimenters j

41Ned after the !nitiation of the pulse.sercen are :.ade themselves state the bubbles appeared dull and blurred

}

7. Carsten<cn and f l'ohly, J. Acoust. Soc. Am.19, 451 near the reumant point; consortuentir, the diameters t

of the bubbles may nnt have been measuted acettratvly l

!cycr and E. Studtr> L,'Acustica 3,43 8 (t953).

With the mitruseope, w hish would a!Tect the dett rmina-l t

{

"".e-.,,

-e,-

3,s,,c-- --

1' V

lo; C ll A lt !,1? S D EVIN, J R.

1666 l A.$h C**'I i*

g.

increase in the sti'iness. Straderg" tentativeiv un.

5 i

"'P"I"'" ' A ' "' 'r ion cf the resonant frequency,'Ith lamping con-tant

. h ' sor targe bubble t was determir>cd usin:t the photoctettric gested this behavi;r raay p I

P'0 ** "* "I l resonance i<

3 3'urface iscittations of the bubble since, for very.mi method by 3 feyer and Tamm, and later Lauer." They 'bubb!ct, the frequeidy of surfate o cillations may i used a thin w re anndus to hohl the bubbles and of the >ame order as the frequency of ordinarv vnium,

"" k",y*tJ -

P -

md evenn prevent them from rising to the surface while measure-pulsations. The ewitation of suiface usci!!.[ tion, I,s M '? ""d I "*"

ments weie bein;;.made. Indeed, the high damping i

sonic excitation uay require >ome non ymm. t r.

tc the particubr

. constants found by Slger and Tpmm may be due to the wire annuti adding to the dar6 pin;t of the bubble supplied by the doit particles. When Ifac-ke p.r

.nu.H ih-nt p.uu y system. At low frequencie.4, the damping constants formed his esperiment, he took cxtreme care to n of Dauer aral I.;n tneasured by Lauer are about twenty.6ve pavat clean esperimental conditions, and found no trare..i betuten the ra.h.

' " anomalous" bubb!cs in the 1t. 400 kc/*ec range.

0 hi'her than the theoretical prediction.

Carsten3cn and roldy used the reson.,nce absorp:!..-

[ sch suay re:nq the volut.e pu. c Bauer, formo!v of the David Tavlor 3!odel lla<in, method to d::termine the resonant ducping con.t.,m.

tsed the succes3ive orcillation method for determin~ng The damping coratant re<ults are very hi.;h. In tiu tha damping constant at reconance. In this experiment, method, a large number of bubbles are present, at.:

j the damping constant for a free bubb!c was measured; the exact distribution in size, space, and numhtr i. !

therefore, there is no additional dampin;* due to a ditheult to determine. There may ala be inttratti e.

t Lubble holder. The bubble was formed at nozz!e; the n!ume pulsations > tarted ju<t as the bubble closed between individual bubb!cs;it is didicult to state h...

cnd srparated from the nor.z!e. The unpublished the<e interactions aficct the damping constant. Fin.d!y the assurption of Carsten<en and Fuldy v> to er damping comtant measurements of Bauer are about behavior of the ofi resonance damping coastant m.q twenty percent higher than the theory predicts.

The.rnethod of standing. wave ratios was u3cd by be invalid.

Excluding the re<ults obtained by 3 feyer and Tanm Exner," Exner and llampe," and Hae.ke" to determine and Carsteinen and Foldy, the experimental dampN.

the resonant damping con < tant; the resuhs of Exner, cnd Exner and llam;)e agree very we!! with the theo-constants agree very well with the theoretical curt.

reticaP curve. Accurding to the theory, the vi+cous The damping constants at Irv frequencies obtained I.3 l

,. ' damping becomes irrportant around 200 kr.lsee; Dauer and 1.auer, using difierent experi:nental

'1!ac3ke luts measured the damping constant in this agree quite we!! with each other, but their result.< ar.

k" indican.

i

. frequency range. At resonant frequcncies of 200 4 00 ' hightr than the thcory predicts. Rennt wor I

that for large amplitude radial pulations, there nu)

' Ac/see, the damping constants determined by Haeske be some coupling between the radial puhation and tb cte four to cight pcrcent towar than the theoretical sharie oscillation..This may result in the removal.4 curec. When the theoretical damping con < tant cur e some of the energy anociatcd with the pulsation. &

does not include the viscous damping constant, but high frequencies, IIaeske's work does r.ot corainc enly the thermal and radiation damping constants, whether or not viscous damping is important.

the experimental results of IIaeske are four to eight

~

ptrccnt higher than the theoretical curve. Ifoweser, V.

SUMMARY

AND CONCLUSION the recasurements by 1 f ac.-ke appear to be only accurate "to within about ten percent. Therefore, n. deinite-Bubbles escited to volume pulsations have a Idy-conclauon can not be formed as to whether viscous tropic equation of state for the enclosed gas uhis damping contribute, or does not contribute to the total results in a pha>c diderente between the chance ia dampmg.Now, the value tor the coemcient of visconty, pressure per unit original pressure and the chaner -

l which enters the calcuhttions for the vi*cous dampmg. volume per unit original volume. Therefore, the wofi was obtained experimentally for steady tiow. At high done in compressing the bubble is raore than the wo i frequencies, the value for the coenicient of vscosity done by the bubh!e in expanding; this di6erence in :h may be considerably smaller than for the steady tiow work done repre<ents a net 11ow of heat ener.gr in9 case. Above 40 kc6ee, Esner and llampe vety often the liquid.

~

, found " anomalous" bubb!cs with much lower damping Pulsating bubbles egend a portion of their sr'er::F c:nstants than the regular bubbles. The measurert in the form of >pherical sound waves. The radiati-resonant frequency did not agree with the frequency d.unping is just this to45 of entrav.

calculated from the mea ured diameter of the bubble The eficet of vi-cosity on pS4ating hubt.h> in

• when Eq. (Ud was used. The "anomahm," bubl le*

incomprenible, viwous liquid is under-tood throer

, have higher frequencies than th:s equation predicts. th es q.mions al tk boundary condie"*.-

rather than the Navier Stokes equathm i,f motion. N There was nottsl. that in almost all ca es the on t he,ir ti bubide surface, there. are viwous forec*

• d d

" anomalous" Imbb!cs had du t part n.hs surfaces. TI.is increa c in rnmant frequency conh! not

" 3r. strekrc..w.itu t. W (ten be explained by a decrea e in the gemralized ma-s as g,["$,

gg g'" I I

u-the du31 parthkx would add to this maw. aml there g

.does not >cem to be a log ca esp anat on or a possible coneg, camta;.tce, nn,Soa i l l

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D A M 1* l N G 0 1' l' U l.S A T I N C.

A 1 It 13 U 1411!.1:S IN W A T l;11 1607 i

l ACI:!'0WI.EI)G?.!ENTS f.,h exert on exec >s 1,imure. Thi:; results irs the "Jgntiori of energy.

The writer gratefully aiknowiniges his indebte<!ne s pperimental results verify that the damp.m; at b> Dr. T. II. I:rown,'. Phy-ics Profi s or F.meriru in Maance is due to thermal amt radi ttiin, and gwissib!y g,e.-Idence, for the pleasure of working directly under

,.cous daminng. l'he di-erepancies between thetary hi< supervision and for ouggestions as wtll as encourage-l' pl esp rinu:.t-d results fouw! m measurtments by ment throughmo th cour>e of work. Ile abo wi..hes I

• )cr and Tanun and Catstensen and l'ohly are due to thank I,ruftwar I.. ~y!ack for in.s nyny sug;;e< tion.+

the >a r t.icular cotuh..tmns of the. expernuent..l'he at the be;;.mmng of the course of wor,x.

1 J.ull discrepancy between the t worv an t it resu ts The author w. hes to thank Dr..',I. Strasber;; an<l a

'g Itauer and I.auer may be due to some coupling s

weren the radial motion ami the shape o-:cillations the other members of the Acoustics Dividon. Daviil

.dh may semove so.ve of the ener y aaociatcd with Taylor.Tiodel liasin, for their many beneficial g!

g y volume pub.ation.

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