S AD-AG88 894 HARRY DIAMOND LABS AOELPHI MO F/6 2 /6I THE EXTENDED RAYLEIGH THEORY OF THE OSCILLATION OF LIQUID DROPL--ETC(U)I MAY 80 C A MORRISON, R P LEAVITT, D E WOATMAN

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20. Abstract (Cont'd)

from that found by previous workers. This difference isattributed to the assumptions by previous workers that thedrops, under the influence of an electric field, distortedinto ellipsoids of revolution about the field direction. Thedynamical equations are derived, and the solution for smalloscillation is given.

II

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I,

CONTENTS

Page

1. INTRODUCTION ......................................... 5

2. THEORY .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1 Derivation of Rayleigh's Result ........................................... 62.2 Electric Field......................................................... 8

3. DISCUSSION........................................................... 123.1 Static Electric Field.................................................... 123.2 Dynamical Equation for Varying Electric Fields ............................. 15

LITERATURE CITED.................................................... 19

SELECTED BIBLIOGRAPHY ............................................ 21

DISTRIBUTION......................................................... 23

TABLE

1. Critical values of y (E'a/y) are lowest values of y for which

Ak(yk) = 0Ofor k = 4. 6, 8, and 10 .......................................... 13

FIGURES

1. Amplitude of second mode, x, as function of y (E'a/-y) for dielectric constantc = 78.2 (A) and E (B).............................................. 14

2. Amplitude of fourth mode, X4, as function of y (Ela/-y) for dielectric constantE =78.2 (A')and E= - (13).............................................. 14

3. Quantities ao/a, a,/a, and 1 - A 2B obtained from normal modeanalysis versus field strength y............................................... 16

ACCOSSIOU 7or

* DC TABca

_DiStrjVU _~

Dist. spitaldO

ID * ~ .. ~ -3

1. INTRODUCTION should be half that required for resonance of thedrop. O'Konski and Harris6 extended this work to

The first work on the dynamical theory of the include in their analysis the effect of electricaloscillation of liquid drops appears to have been conductivity of both the surrounding medium anddone by Lord Rayleigh.' In his investigation, the drop. No consideration was given to theRayleigh considered small displacements from possible effect of conductivity on the damping. Inequilibrium under the action of forces due to their analysis, they found the rather surprisingsurface tension (capillary forces) alone. Later, result that, under certain appropriate choices ofLamb' included damping ofthe small oscillation of conductivities, the equilibrium shape of the dropletdrops by the inclusion of internal viscous forces remained a sphere. As in their previous investiga-and showed that the rate of damping was depen- tion, O'Konski and Harris assumed that the drop-dent on the size of the liquid drops, becoming let under the action of an electric field becomes anextremely large for very small drops. The possi- ellipsoid of revolution about the field direction.bility of instability of the lower modes of oscilla-tion was first investigated by Rayleigh,3 who Later, Garton and Kra,ucki' investigated the

showed that the resonance frequency of the nth stability of bubbles in a static electric field andmode of a charged drop was questioned the correctness of previous theoreticalwork on electrostriction. They showed actual

W2 n(n - 1) +1 photographs of bubbles in various stages of dis-a 4- a 3, integration that vividly displayed the physical

nature of bubbles breaking.

where p is the density of the liquid drop, a is the In a series of two papers, Taylor8 discussed theequilibrium radius, y is the surface tension, and stability of conducting drops in an electric field. HeQ is the total charge on the drop. Instability showed that it is necessary to introduce motion ofoccurs when w,2 _ 0; or when the charge the fluid inside a drop to attain the sphericalQI > [41ra'Y(n + 2)112/, then the nth mode is (undistorted) solution of O'Konski and Harris.'unstable; the lower modes become unstable for the Some experimental work was reported in Taylor'ssmaller amount of charge. The condition for insta- papers that agreed quite well with the theory.bility is sometimes written in terms of the electric Sozou9 later extended Taylor's theory to includefield at the surface of the drop, E = Q/a2, so that time dependent electric fields. He gave a number ofthe condition expressed above is frequently given results in the form of equations with a few cases ofas JEl > [47f(n + 2)y/a112 and has been used as numerical results, but further work seems neces-a starting point by later investigators. sary for comparison with experiment and with

Considerably later than the above work, the previous results.

investigation of the dynamics of small droplets was Rosenkilde'0 investigated the stability of drop-begun by Thacher4 and O'Konski and Thacher.' In lets in an electric field using methods of tensortheir investigations, they considered the distortion calculus and Chandrasekhar's" virial method. Heof droplets by an electric field, assuming that the showed that under appropriate conditions at leastdistorted drop was an ellipsoid of revolution about three different equilibrium configurations can exist.the direction of the electric field. They also dis- He predicted instability only if the dielectric con-cussed the possibility of enhancing the droplet's stant, E, of the drop is greater than 20.801.distortion by an alternating electric field, but failedto observe that the frequency of the electric field T Ooo, nd FS. Ha,. J Po,. h. 61(ndan). 11.

_____________7G G Gaon ,and Z. ascki. Pron, RoV. Sac. (London). A28O(19643. 211.

'Lord Rayleigh. Pet. Roy. Sac (London), 29 (1879. 71. 77e Thieon, of 8Sir Geoffra Ta,or. Po. Roy. Sac. (London). A280(1964 , 383: A291

Sound I1. MacMillan Co.. London (1896). Ch. XX t/9661 1592H. Lamb. Hrdrodrnamics. Dorer Pueblications. Inc.. X.- York (/12)

9C Sat. P por Soc. (London). A331 (1972). 263.

3Lord RaYleigh. Phil. Mag.. 14 (1882). 184. I0( E ,tosenkild. Proc Rot. Sac. (London), A312 (1969). 473.

4H. Thacher. Jr.. A Phys. Chem.. 56 (3952) 795. I5 ( Aandmsekhar. Hlrdmdrnamics and Htdromagnetic Stability. Clarn-5

C. T OlKonski and H. Thacher. Jr.. J. Phrs. Chem.. 57 (1953

). QS5 don .Ps. OrfodI9OAI)

iBL"iW% F

it~

Most of the above work was devoted to the the center of mass to a point on the surface of thedevelopment of equilibrium distortion of a droplet drop can be expanded in a Legendre series asunder the action of various forces. None of theseworkers attempted to develop their results using r(0,t) = a0(t) + 2' ak(t) Pk(COS 0) . (1)the original techniques introduced by Rayleigh.I In kfact, the technique used by Rayleigh in his dis-cussion of a charged drop3 was somewhat different The method then of solving any particular problem

than in his original paper. A detailed derivation of is to express the Lagrangian for that problem in

Rayleigh's result for a charged drop was given by terms of the variables a,(t) and treat these a(t)

Hendricks and Schneider 2 and later by Schneider 3 as generalized coordinates to obtain the equations

along with many applications of the results. Dis- of motion. Such an expansion as equation (1) issipation by viscosity was not considered in either possible if we assume that the droplet is symmetricof these investigations. about the z-axis. At this point, the z-axis can be

chosen in any direction; but later, when we includeIn our consideration here, we follow the spirit the electromagnetic energy, the electric field will

of the original investigation by Rayleigh. In section be assumed to be along this axis. We assume this2.1, we find the energy due to the surface tension, symmetry throughout the discussion. The prime onthe kinetic energy of the fluid in the drop, and the the sum in equation (1) indicates that the lowestRayleigh dissipation function. From these quan- value of k in the sum is one.tities, we derive the generalized equations ofmotion by using the Lagrangian, as given by With the assumption of incompressibility,Rayleigh, which includes the results of Lamb. the volume of the drop acts as a constraint on the

ak(t) in equation (1). The volume of the drop isIn section 2.2, we include the terms involving given by

the energy due to the electric field in theLagrangian. We discuss this aspect in consider- , ff7 0 r(.t)

able detail since we have not been able to find any V = d j0 sin 0 dO x2 dx . (2)reference to an extension of Rayleigh's result thatincludes an electric field. Since we are assuming that the drop is symmetric

about the z-axis, letting jt = cos 0, we haveThe remainder of the report is devoted to a

discussion of the result of section 2.2. The resultsare used in the investigation of instability under the V = 27 l r3(/i) d. (3)action of an electrostatic field as well as the 3 J-idynamical behavior of drops under the action of a Using equation (1) and the relationtime varying field.

2. THEORY f Pp(11 ) d = 28ee,/(21 + 1)

2.1 Derivation of Rayleigh's Result we obtain

In the dynamical theory of the oscillation V n- 2al + 3a0 Y,' 2ak (4)of a liquid drop, we assume that the radius, r, from

L.TIf we assume that a is the radius of the equilibrium, ~~~~~Lord Ro,. eigh. Pro Ro,. Soc. (London,. 29 (1879). 71. r, Theory of p ee( n 33,teSound II. MacMillan Co.. London (1896). Ch. XX. sphere (V 4 4fa 3/3), then

3Lord Rarleigh. Phil. Mag.. 14 (1882). 184.

2C .Hendricks and J. M. Schneider. Am. J. Phrs.. 31 (1963), 450 (jak()

13John Matthew Schneider. Thbe Stabd" of Electr~6ed Liquid Surfaces. a0 2 a (5)Charged Particle Research Laboraron'. Uniw, O,-affnlinois. Urbana. CPRL- 2-64 a k 2k + 1(5 March 1964. ONTIS 6044311.

i6

which holds for terms through order a with V v =0 (11)k 2! 1. Since we are interested only in terms ofsecond order in ak, we treat equation (5) and If further we assume thatcorresponding subsequent results as equalities.

The potential energy, U, of the drop V 0 (12)

due to the surface tension, y, is the area of thedrop multiplied by y or then V can be derived from a potential function,

01, such that27rUs = ' do rsin~ds , (6) V = -v~ 1 , (13)

where ds is the arc length in the surface given by and from equation (11) we obtain

ds2 = dr2 r2 d 2 V 2' 10 (14)

or Thus, the velocity potential satisfies Laplace's[r2 + (dr " 1/2 equation, and the solution appropriate for ourds = dO [2+ Q-] 12 (7) problem can be written

Using this result in equation (6), we can write = (n rn Pn(cos 9) (15)n

1l r 1/2 (Here r represents a point on the interior of theU, = 2 r I r + (1 - dg drop and becomes r(0,t) at the surface.) Rather

than use equation (15) in equation (10) directly,

we use equation (13) in equation (10) to convert(8) the volume integral into a surface integral as

The second term in the square root in equation (8) 1 "is smaller than r, and the square root can be T = -2-p f J 1 (Vol) "d .(16)expanded and integrated to give

We assume that the area element d'is approxi-+ Ia (k- 1)(k + 2)a" matelyalong t. (Actually,d&= (IrdO - 6dr)

s-= 41r 1) ' X r sin 0 do, where t and 0 are unit vectorsk 2(2k + 1 along r and 0, but the correction is of higher

where the constraint given in equation (5) has been order than we are considering here.) Under that

used to obtain this result. The result given in assumption, we have

equation (9) holds up to fourth order in the ak. T 2 p ka2k - i

, To calculate the kinetic energy, T, we k 2k 1* need to evaluate the volume integral

The /k in equations (15) and (17) can be evalu-

T d". , (10) ated by equating the velocity at the surface2 Tpv2 (-a 1 /car) with : calculated from equation (1)

with p the density (constant) and v the velocity so thatof the fluid within the drop. We assume that the ik = -kak k (18)fluid is incompressible and that there are nosources or sinks; then Then equation (17) becomes

7

.2.

T =2ipa i094npa3 + a(n-1)

k k(2k+ 1) (19) n(2n+ 1) n

This result, equation (19), is valid only through 4n-y(n - 1)(n + 2) an = 0 (22)terms of order 4. If the result given in equation + 2n + 1(18) is carefully inspected in the relation Thus, if the viscosity vanishes (I = 0), we have

Vr = - Y, k~krk-Pk(COS 0) Rayleigh's result,'

and the rk- , is expanded by using equation (1), o(OR = 0) " n(n - 1)(n + 2) (22a)correction terms occur in .k" These terms give pa3 ncorrections to k in the form akan, therebycoupling the equations of motion of ak in a com- If the viscosity is small enough, we obtain Lamb'sresult2 for the decay time of the nth modeplicated manner, which we ignore in the presentinvestigation. However, any serious investigation [ an(0) -trflof dynamic instability should include these terms pa 2

because they are of third order in the ak and can - (22b)be important. tj(n - 1X2n + 1)

To derive the effect of viscosity on the The decay, when present, shifts the resonant

equations of motion, we use Rayleigh's dissipative frequency so that

function,' 4 R, instead of the procedure used by 2Lamb.2 As will become apparent in the develop- 2 = 2( 0) (22c)ment, this method is much more consistent with the n

treatment presented here. A convenient form of thedissipative function given by Landau and for wo > 0 otherwise, the drop does not os-Lifshitz5 is ciliate. For the mode n = 2, the frequency

1 1'0 = w 2/27r, r2, and the product V2r2 were calcu-R = 2- 17 f(V v2) • d' (20) lated by O'Konski and Thacher5 for 1-, 10-, 100-,and 1000-pmo water droplets, where they used

where rl is the viscosity. As in the kinetic energy, a 70dncm ad we t .se.

it is sufficient to assume that d is along . All of 72.0 dynes/cm and t = 0.884 X 10- 1 poise.

the quantities necessary to calculate equation (20) All of the results obtained above wereare given in equations (15) and (18) so that previously derived by various techniques, and we

R 4 a ,(k - 1) have shown that they can all be obtained byR k = 4 (21) methods compatible with Rayleigh's original ap-

k k proach to the problem. Also, as we have shown,corrections to Rayleigh's results are more trans-SThe equation of motion for an(t) can now parent by this approach.

be readily found by forming the Lagrangian fromthe results given in equations (9) and (19) 2.2 Electric Field(L, = T - U.) and the dissipative functionequation (21) to give The inclusion of an electric field into the

analysis presents a problem of considerable com-2H. Lamb. llydroadnamics. Dover Publications. Inc.. New York (1932).

14H Goldstein. Classical Mechanics. Addison-Weslev Publishing Co.. 'Lord Rayleikl Pro. Roa. Sac. (Londond. 29 (187 9

A 71: Th7e 7ean ofReading. ,A (1950). Ch. I Sound It MacMillan Co., London (18) Ch. XX.

1 5L. D. Landau and E. M. Lifshitz. Fluid Mechanics, Theoretical Plivsics. 6. 2H. Lamb. Itdrodnamics. Dover Publicationt. Inc.. Ne"' York (1932).

Addison- Wesley Publishing Co.. Reading. MA (1959). 54. 5C. T OKonski and H. Thacher. Jr.. J. P ys. Chem,.57 (1953) 955.

8

plexity. Therefore, we present the approach and for the exterior region andthe solution to the problem in more detail than inour previous discussion. To be consistent with our i _ r nBnPn(cos 0) (28) I-preceding analysis, we need to obtain an expres- nsion for the electromagnetic energy of the drop for the interior region of the drop. The last term inexpressed in terms of the an(t) in equation (1). This equation (27) represents the potential of the ex-energy can be added to the existing Lagrangian, ternal electric field, E, parallel to the z-axis.which can then be used to obtain the equations ofmotion. The applied electric field, E, can be

A convenient form for the electro- writtenmagnetic energy stored in the drop is given by E(t cos 0 + G sin 0) (29)Jackson:' 6

t and G are unit vectors in their respective direc-UE = - 1 I . d , (23) tion, and from equation (25)

81 T

where Eiis the electric field inside the drop,E is the field in the absence of the drop, e is the Then using equation (27), we have

dielectric constant of the drop, and the integral rcovers the volume of thedrop. Thus, our problem E .E' = -E Y Bnrn- r[n cos 0 Pn(cos 6)is to find a solution for E before proceeding. n L

For a dielectric body, we have from - sin 0 0 Pn(cOs 0)]. (3Maxwell's equations

From the recursion relation for Legendre poly-nomials, with t = cos 6, we have

(24)(1 - A2) On = nPn-i - nPn,7De7E 0 di_

The first of equation (24) implies that the electric where we have discontinued writing the argumentfield can be derived from a scalar potential, 0, or in the Legendre polynomials. Using this result in

equation (30), we have

= 7 , (25) EE i - -E nBnrn-'Pn_, (31)

and the second of equation (24) implies that and,t V2 = 0 (26) ,f2Tr f 0t).o _

,I do I sin 0 dO f•r E E drThe appropriate solutions to equation (26) for our

problem can be written n n2.,*= -21TE Y, Bn r- n -n I d1 .

=0 Pn(COS 0) -Ercos0 (27)n-0 Substituting this result into equation (23) we have

1"J D Jackno, Classical Eerrroatnamwcs. John Wilet and Sons. Netu York -- [2B anB(1975. 160 The denrattn of equation (23) iv not trivial, and Jackson c,,es UE - E 3 + n

,nsderahle attenton w the dertationo 4 Ln>1 n +2

9

X F l~+2o 1To evaluate Bn and A, in equations-f l-r" n-nmI d1l , (32) (27) and (28), we need to apply the boundary

conditions on the fields at the surface of the drop.

where we have separated the term n =1 from the These are that the tangential components of theremainder since this term is the integral over electric field are continuous, E' = E', and theju of r3, which is proportional to the volume and is normal components of the electric displacementthe constraining equation (3). are continuous, Din = Do, where the subscripts t and

n on the vector components denote tangential andTo evaluati the terms in the sum in normal, respectively. (The use of n here and in

equation (32), we use equation (1) and expand the the summing index need cause no confusion sincepowers of a0 through linear terms in ak or no sums are performed over the field components.)

The continuity of the tangential electric field cannr +2 be easily shown to be equivalent to the continuity

r r pn- d fI of the potential so that 0' = 0b0 at the surface.k dThe boundary condition on the normal componentS+ (n + 2)4 + ' Y , akPk Pn-, du of 5, however, requires some consideration. Since

by equation (25) E = -V 0b and since we are

2(n +2) ,+1 assuming that D' = 9E and 6 0 = E6 with_ a+an_ (33) fh a unit vector normal to the surface, we have

(2n - 1)

Vl V ° (35)for n > 1. The resulting equation (33) can beused in equation (32) to give at the boundary of the drop. To construct fi, we

note that the vector ds" = f dr + Or dO lies inthe surface if r is given by equation (1) and the

UE 2 E [- a3 unit vector in the 0 direction also lies in thesurface since we are assuming that the drop issymmetrical about the z-axis. A vector normal to

+ 2 n + I Bn+lan+2ar (34) the surface can be formed by the cross product ofI 2n + I these two vectors or

and we have shifted the summing index so that the n _ds" Xlower limit on n is 1. We have also replaced the Ids x Iproduct ana+ 2 by anan+ 2 since the correctionsto a0 are of order a2 so that corrections to this t r dO -0 drquantity would be of order a', which are ne- 2 (36)glected. Also, in equation (33), we have expanded ( r2 dO2 + dr2 )'2r only through terms linear in ak because Bn in hich can be used in equation (35) to obtain theequation (34) for n > 1 have to depend on ak wor a de us , t eeate, th ounarysince they vanish for undistorted spherical drops. normal derivative. Thus, to reiterate, the boundaryconditions on equations (27) and (28) are(That they vanish appears in the subsequent an-alysis.) Therefore, to obtain an expression valid 0 ,through terms of order ak, it is necessary to obtainB, through terms of order a' for contributions i' dr 1-O d i[ O' _I°_r dr N ae,"from the first term in equation (34), while we need r - r ] O r (L--)r

only first order terms for B, with n > 1. To [ r dr

proceed, we have to return to the problem of (37)

evaluating the B0 and consequently the An of evaluated at the boundary with r given byequations (27) and (28). equation (1).

10

To keep track of the order of the correc- M'(l2m' + 2m' - 18m + 10)ftions, it is convenient to rewrite etjuation (1) as 2 l 2 2 1)(2m + 3)1 E(m - 1) + ml

r = a + 8 akPkWhen the results given in equation (39) are

and expand the various powers of r in powers substituted into equation (34), we haveof 8; also, we assume that

An_=_____ 8A~n" + 8' 2 .. UE = (E -I)E 2 a a 2 ±+6E-I) a2

-n = ~o + 8Bgn" + 82B~~ .. (38) 6(E - 1) 2 G+aa+

Then in the final result, we let I since it + e2H a0serves only as an artifice to keep the terms in (40)order. When this is done in equation (38), To form the full Lagrangian for the problem,we obtain we need only add UE to Us so that L

T - Us- UE, and the equation of motion forAT = =(' 0 , n 54 1 ,an is given by the usual procedure,

A (10 I)E4, d aL _ __ =_ a (41)e+2 'dt aa, aan aa,

B _, 3E_ The result of doing this is to add to the right side of+ 3E equation (22) the term -aUEOaa so that

-3(c - l)E fn(n + lXe - l)a,+, 4Tfpa' an + 87i n -1)iA' E + 2 - (2n +3)[(E + )n +lI n( +) n

na,~ + 4"r(n - 1)(n + 2) _

+2 1 2n0

+ 1n-

3(E - IV'E-a 6,+ 3(e - 1)'E a

- 3(c - l)EI (n + 1)(2n + l)a~+ 8n.2 ++)e+)

r + 2 1( 2n +3)[(E +1)n + JaOJ X (Gn+ 2an+2 + Gnan- 2 + 2Hnan), (42)and

which are the equations of motion for a drop in anB11 2) - ( Eelectric field. The right hand side of equation (42)

+ )a; appears to be new.

X Gmmm2+Ha2(39) Several aspects of equation (42) are asM should be expected from the genera] symmetry of

where the problem. That is, the right hand side is depen-dent on the square of the electric field so that a

m(m - 1)(2m - 1)(E + 2) reversal of the field does not alter the results.Gm=(2m -3)(2m + 01[0m - 1) +mln Further,the n =2 mode is the only mode driven

directly by the electric field, and this couples toHm= n m-1)12'+IO -1m-I the n = 4 mode and consequently to higher

mje~ -)( 1m' l~i 2 -12m -l) modes with n even. Consequently if no further

(2m + 1)2(2m - IlX2m + 3)[Em - 1) + ml perturbation couples the odd and even modes, only

those modes for n even need to be considered. 3. DISCUSSIONSome of these results would have been immedi-ately obvious if we had used Maxwell's stress The result given in equation (42) is to this pointtensor to evaluate the forces on the drop, but then general in that no assumptions have been madethe normal mode result of equation (42) would concerning the nature of the electric field otherhave been lost. If we ignore the coupling to the than those implied in equations (24) and (25). Inhigher modes in equation (42), the resonant fre- this section, we discuss results from equation (42)quency of the nth mode is given by for three different types of fields: a static electric

field, an alternating electric field, and an amplitude2 _ n(n - 1) modulated high frequency field.

pa3

+2)- 6(E _ l) 2a(2n + 1 )Hn 3.1 Static Electric Field~n + 2) 6 En - 1)(n + 22 I In the case of a charged drop under the

(43) influence of its self-electric field, the modes ofoscillation are uncoupled so that a simple criterion,

This result is similar in appearance to the result = 0, is usually taken as the onset of insta-derived by Rayleigh for a charged drop, bility. In the present case given in equation (42),

2 -1) E 2 a we see that the modes are coupled so that the2 n(n - (n + 2) - condition for instability created by a large electric

pa3 L 4TJ ' field becomes complicated. If we assume thatiin and in in equation (42) vanish, then we can write

with the charge on the drop, Q = Ea2. In Ray-

leigh's result, the frequency decreases as thefield (charge) increases, but in equation (43) this Dnxn - Ay- n 2condition occurs only if Hn > 0. The condition 5for no shift in the frequency in equation (43) isHn = 0, which for the first few modes are + 3Xy(Gn+ 2Xn+ 2 + Gnxn 2 ) , (45)n = 2,e 2.846: n = 4 , E = 1.588; n 6,

= 1.346; and n = 8, e = 1. Perhaps a fair wherecomparison with Rayleigh's result is for a con-ducting drop, which can be obtained from equation Dn = An - 6XyHn(43) by letting the dielectric constant be infinite or An - 47r(n-1)(n+2)

( = ) n(n- 1) 2n+1

paa

2 an6Ea aX. 1 7(n + 2) - 4 7r - i 2

n(12n' + ]On 2 - 12n -1) (44

(n-lIX2n+ )(2n-l)(2n 3)] V E'a'Y

This resulting equation (44) shows the samebehavior as the result of Rayleigh. but with a more with Gn and H, given in equation (39). For lowcomplicated dependence on mode number, n. fields (y = 0), the coupling between modes can

12

I

generally be ignored; but for larger fields, which A4(y) = (A4 - 6XyH 4)would be required for the drop to become unstable,the coupling cannot be ignored. Thus to investi- X (A, - 6XyH 2) - (3XyG,) 2 (gate the instability in the system of equations in (50)equation (45), it is necessary to consider the H4A2 + H2A4coupled equations in detail. Y4 = 3441H, - G )

If we write equation (45) in matrix nota- w h:z 12 2tion, then [(H4A2 - H2A4) 2 + A2A4j '

3(4H4H 2 - GI)

Tx = F , (46)

where which are the simplest roots to obtain algebraic~expressions for. The ambiguity of sign in equationTn'n = Dn8n'n - 3XGn'8n',n+2 - 3XGn8n'.n-2 (50) was determined so that, Y4 given, there was

the smallest positive root in the range of dielectricand Z" and F are column matrices (vectors). The constants, 1 < e • The result for Y2 inmatrix F consists of the single element, (3/5)Xy. equation (49) is not physical (Y2 < 0) for - _

The formal solution to equation (46) is given by 52/37 and becomes infinite at r = 52/37 (H 2vanishes) and is, in general, not even a good first

x = T - F (47) estimate. This result is not surprising since Y2where T is the inverse of the matrix T. The totally ignores the coupling between modes. Theinverse of the matrix T contains the determinant result for y4 given in equation (50) is, on the other

of T in the denominator, and when the deter- hand, a good estimate even for small values of e.minant vanishes, the solution equation (47) be- The lowest values of y for which A2N(Y) = 0comes unstable. Thus the lowest value of electric have been calculated for a few values of N and anfield (smallest positive y) for which the deter- extended range of e and are given in table 1. The

minant vanishes gives the onset of instability of the value of y4 differs from the value of y, 0 only bydrop. Since the matrix T has single off-diagonal 16percentfor E = 1.1; forwater(r = 78.2), y4elements, it is a simple matter to obtain a recursion differs from y, 0 by 1.3 percent, and for larger rrelation for the determinant, A. If the matrix is the difference is approximately 1 percent.truncated to contain N terms (Tnn' 0, TABLE 1. CRITICAL VALUES OF y (E 2a/.Y)n or n' > 2N), then ARE LOWEST VALUES OF y FOR WHICH

N = N, Ak(Yk) = 0 FOR k = 4,6,8, AND 10~~~A N =D 2NA 2N-, - (3XYG 2N)2A2N_4 ,(48)

r Y4 Y6 Y8 Y10with A, = I and A 2N- 2 0 for N negative.The result given in equation (48) can be used to 1.1 7009 6109 6054 6049obtain the smallest positive value of y such that 1.3 817.6 728.2 720.5 720.0A2N(Y) = 0. The first two of these are from 1.5 310.9 280.4 278.2 278.1equation (48), 2.0 90.16 83.71 82.80 82.78

5.0 12.87 12.42 12.40 12.39A2(y) = A2 - 6XyH 2 A2(y) 0 gives 10.0 6.511 6.367 6.362 6.362

A2 (49) 50.0 3.603 3.556 3.555 3.555

Y2 =- 78.2 3.406 3.364 3.363 3.363100 3.332 3.302 3.291 3.291

and - 3.078 3.044 3.044 3.044

13

-- ., .. ..

Upon comparing the results in table Iwith the results obtained by Brazier-Smith et al, 7

we find a discrepancy. The value of E(a/y)" '2

= 1.625 (y = E'a/y) given by them (along with 'references to previous work) corresponds to ourvalue of 1.745 for e = -. For water, which is *used in their experiment, we obtain 1.834. Thisdiscrepancy is caused possibly by the assumptionthat the drop is 'constrained to an ellipsoid of "'

revolution in their treatment of the problem.ala

The amplitude of the x2 mode is easilyobtained by using equation (45) along with thedeterminant in equation (48) to give for the 2Nth "

approximate solution~P2N

x2(2N) - (51)A2N L . .1 1 . 2?I

where the P2N obey the same recursion relationas A2N given in equation (48). The initial values Figure I. Amplitude of second mode, x2, as functionof P2N are of y (E2a/y) for dielectric constant

F = 78.2 (A) and r' (B).P2 = 3Xy/5 (52)

and

P4 = 3XyD,/5 , (53)

which are sufficient to generate all P N fromequation (48). Having determined the value ofx2(2N), we can generate all the xn(2N) forn :S 2N by using equation (45) and the conditionxo 0. The variation of x2(10) asafunctionofy 0°s

is shown in figures 1 and 2 for two values ofdielectric constant, F = 78.2 and e = -. Theamplitudes of the xn mode are approximately an 0 *

order of magnitude larger than the x,+ 2 mode inthe moderate y region. All the modes, x., di- 00verge at the smallest value of y, satisfying theequation Ao(y) - 0. When the xn becomelarge, the results are questionable, and a possible 0

°I

measure of the range of xn that are reasonable isfrom equation (5) (with xn = an/a), a $1 .I 241

ao oo

o _ l(54)a n 2n + I Figure 2. Amplitude of fourth mode, x4, as function

17p R. Bftvwr Smith, Al. BrooA. J Laniltam. C.P.R. Saundes, and M. 11. of Y (E2a/Y) for dielectric constant

Smi,. Pith. R,,- S- ,.ordao.. A322 (171). 523. r' 78.2 (A) and r . (I.

14

When ao/a becomes small, the entire results be- soidal approximation, when viewed in this light,come questionable because all the expansions would not seem very good and perhaps wouldused in the derivation equation (45) assume indicate the source of the difference in the criticala0 > I ak implicitly, field values obtained here and elsewhere by earlier

workers. (See fig. 3 on p. 16.)If the drop were distorted in the shape of 3.2 Dynamical Equation for Varying

an ellipsoid of revolution along the direction of the Electric Fieldsfield, we could write

r2 si 2 6 r2 cos 2 0 The dynamical equation (42) is, to ouri + B - 1 (55) knowledge, a new result in the sense that the losses

due to viscosity and a finite dielectric constanthave been included in the analysis. Further, the

If we let 0 = n/2, we have from equation (1) entire equation of motion was derived in a con-sistent manner by using Rayleigh's original

A= r(- - = Y anPn(O) (56) method. For convenience, we can write equation\21 (42) in the form

or 3Mnxn + Mnrnin + Mnxn = "- XE2a8, 2

A- + X2n(_|)n 2n)a n-=1

(5 9)(59)

+ 3XE2a(Gn+2xn+ 2 ± Gnxn- 2)where A = A,/a

whereand (2 ) is a binomial coefficient.

%ai/ 41rpa'Ifwelet0 = 0 or ir in equation (55), then from Mn = n(2n + 1)

equation (1 )

0 = 0) = ao + Y' an (M, is the total mass of the drop),B =2rla(n - l)(2n + 1)=r( Tr) =ao + 1'(l)na. p 3

Then2XE'a

B + 2' +X 2 ,n (58) n W,(0)- M n(2n + I)Hn

awhere wtn(0) is the frequency in the absence of an

where B = B,/a. If the distortion were an ellip- electric field given in equation (22a) and H, is, soid of revolution as given in equation (55), the given in equation (39). The quantity X is defined, volume of the drop would be given by A2B 1. in equation (45). If the electric field is assumed in

For E = =. a,/a was computed from equation the form(54). A by equation (57), and B by equation(58) as a function of y, and the results are shown E E0 cos wt (60)in figure 3. A measure of how closely the dropapproximates an ellipsoid of revolution is thenI - A2B; for small values of y, this quantity isquite small. However, for y = 2.8, 1 - A2B- E2

= E + - cos 2t (61)0.28, and ao/a is still large (-0.9). The ellip- 2 2

15

'F

0.5

* 0.4

0.2

0

Figure 3. Quantities ao/a, a2/a, and I - A2B obtained from normal mode analysis versus field strength y.

Thus, the driving force in equation (59) for the andn = 2 mode varies at twice the frequency of theelectric field, and resonance for small amplitude of 3XEgaoscillation should occur near 2,= W1 4 2(t) = 1OM 2R,

If we ignore the coupling to higher modes where(xo = 0) in equation (59) and replace E2 byEj/2 in on, we have R2 = [(.2 - 40 + 4,2r]2

j 2 + r 2 i 2 + 2,x2 andi !3XE~a JXE~a _______

- IOM + 10M cos 2wot , (62) tan 62 - 2 -4w 2

where we have used the results of equation (61 ). Also, in the derivation of equation (63), we have, .The result given in equation (62) is a standard ignored terms of higher power than E' in the

linear equation, and the solution can be written electric field.x2(t) = x° + 42(t) with

Thus the results given in equation (63)3XE0a show that in small electric fields the drop, in the

X -1OM 2- fundamental mode, is similar to an ordinary reso-

16

Iv

nant circuit with damping; however, the driving that in certain cases the resonant frequency couldfield on the drop has a frequency double that of the be unshifted if the viscosity were small enough.applied field. The higher modes are driven by the Care must be exercised in extending equation (66)electric field indirectly by coupling through the to higher electric fields because the coupling tolower modes as shown in equation (59), and higher modes given in equation (59) can have moreapproximate solutions can be obtained by a pertur- important consequences. Also, for higher electricbation theory solution of the equations of motion, field strengths, the term in wn in equation (59)If the coupling is ignored, the amplitude of vibra- involving the electric field has a time dependenttion of the nth mode can be written part that could have some interesting conse-

quences in large electric fields.Xn(t) = ae - cos (Ont + an) (64)

The dynamical equation (62) or the morefor general equation (59) can be used in more compli-

cated situations where the electric field is of anr2/4 < w2 impulse nature such as that caused by the presence

of charged drops. A number of sources ofelectricaland and mechanical disturbances have been consid-

ered by Brook and Lantham in their study of

(to2 _ ), modulation of radar echo from rainstorms."

A second and perhaps a more interesting case

with to and F, given by equation (59). The re- of varying fields is the electric field as an amplitude

suit in equation (64) shows that the nth mode modulated high frequency wave. The carrier may

decays more rapidly than the fundamental (n = 2); be radar or an infrared laser, and the amplitude

that is, since modulation can be chosen near a resonance of thedrop. Such a field can be represented by

= (n -l)(2n + 1)E = E(l +mo cos wt) cos wot , (67)

the n = 4 mode decays approximately six timesfaster than the fundamental. The shift in frequency where w, >> w, and particular space depen-due to viscosity and electric field can be written dence has been ignored because we assume that

the drop radius is small compared with the wave-Aw0n = won(0) - f1 (65) length, X0, of the carrier (No = 21rc/woo). We

Assuming the electric field and the viscosity small, need the square of the field, which is given by2Ithis can be written E

Aw' + XE~aE 2 = E +1 + +c 2wt co8),* n(n(O)

"rrFn XE2a n(+nc+sI.)(68):[-8wo~(O) + Mn 2(0) (n+lHJ.2 wt

(66) If we ignore all the high frequency terms varying atfrequencies near 2w 0 and if the depth of modula-

In general, the shift in frequency given in equation tion mo is not large, the terms involving m4 can(66) is positive. However, H, given in equation(39)can be positive or negative depending on Eso 1

8M Brok and D.I LaMa,. kopk ,. Res. 73 19M). 713.

17

IV

be ignored. Substituting equation (68) into equa- equation (62) for n = 2, except that the fre-tion (59), we obtain quency here is at half the value used in equation

n + i n + n _ 3Eja (62).

10 The time varying displacement of the drops

X (1 + 2nm0 cos t)2 +" 3XEa could be studied19 by amplitude modulating a high2 powered laser and observing the modulation of the

reflected signal by a second low power laser. Someof the analysis given by Brook and Lantham8

" (1 + 2m, cos wt)(Gn+2xn+ 2 + Gnxn_2) , would apply to this case, except that the modula-

tion of the reflected signal from the low energy(69) laser would be at the frequency o of equation

where as in equation (62) we assume that the (60), rather than a distribution of different fre-term E2 in won is replaced by E2/2. Also ina- quencies. By observing the reflected signal of theplicit in the equation is the assumption that the low energy laser and sweeping through a range ofdielectric constant, e, wherever it appears, is to values of o, one could obtain some estimate ofbe evaluated at the carrier frequency, w0. Only in the distribution of particle size.cases of extreme dispersion is this invalid, and 18

M Brook ad A J. Lanhorn . Geophys. Res.. 73(196). 7137.such cases have to be treated quite differently. The 19

D. E. Woman,. Possible Use of Two Lar eamso Detemne Paricle-solutions to equation (69) are the same as in Size Distibution, Han, Diamond Laboratories HDL-TR-1878(Januarv 1979).

LA

[ , iIg

LITERATURE CITED

1. Lord Rayleigh, Proc. Roy. Soc. (London), 29 12. C. D. Hendricks and J. M. Schneider, Am. J.(1879), 71; The Theory of Sound, II, Mac- Phys., 31 (1963), 450.Millan Co., London (1896), Ch. XX.

13. John Matthew Schneider, The Stability of2. H. Lamb, Hydrodynamics, Dover Publi- Electrified Liquid Surfaces, Charged Particle

cations, Inc., New York (1932). Research Laboratory, University of Illinois,Urbana, CPRL-2-64 (5 March 1964). (NTIS

3. Lord Rayleigh, Phil. Mag., 14 (1882), 184. 604431)

4. H. Thacher, Jr., J. Phys. Chem., 56 (1952), 14. H.Goldstein, ClassicalMechanics, Addison-795. Wesley Publishing Co., Reading, MA (1950),

Ch. 1.5. C. T. O'Konski and H. Thacher, Jr., J. Phys.

Chem., 57(1953), 955. 15. L. D. Landau and E. M. Lifshitz, FluidMechanics, Theoretical Physics, 6, Addison-

6. C. T. O'Konski and F. E. Harris, J. Phys. Wesley Publishing Co., Reading, MAChem., 61 (1957), 1172. (1959), 54.

7. G. G. Garton and Z. Krasucki, Proc. Roy. 16. J. D. Jackson, Classical Electrodynamics,Soc. (London), A280 (1964). 211. John Wiley and Sons, New York (1975),

160.8. Sir Geoffrey Taylor, Proc. Roy. Soc. (Lon-

don, A280 (1964), 383: A291 (1966), 159. 17. P. R. Brazier-Smith, M. Brook, J. Lantham,C. P. R. Saunders, and M. H. Smith, Proc.

9. C. Sozou, Proc. Roy. Soc. (London), A331 Roy. Soc. (London), A322 (1971), 523.(1972), 263.

18. M. Brook and D. J. Lantham, J. Geophys.10. C. E. Rosenkilde, Proc. Roy. Soc. (London), Res., 73 (1968), 7137.

A312 (1969), 473.19. D. E. Wortman, Possible Use of Two Laser

11. S. Chandrasekhar, Hydrodynamics and Beams to Determine Particle-Size Distri-Hydromagnetic Stability, Clarendon Press, bution, Harry Diamond Laboratories HDL-Oxford (196 1). TR-1878 (January 1979).

,*1

19

f-l

4 j-

r71

SELECTED BIBLIOGRAPHY

During this work, a number of related papers were examined, but were not found to be essentialreferences to the reported analysis. Nevertheless, for one reason or another they were examined forpossible application in our investigation.

Ausman, E. L., and Brook, M., Distortion and Disintegration of Water Drops in Strong Electric Fields,J. Geophys. Res., 72 (1967), 6132.

Barber, P. W., and Wang, Dan-Sing, Rayleigh-Gans-Debye Applicability to Scattering by Non-spherical Particles, Appl. Optics, 17 (1978), 797.

Barber, P. W., and Yeh, C., Scattering of Electromagnetic Waves by Arbitrarily Shaped DielectricBodies, Appl. Optics, 14 (1975), 2864.

Bukatyi, V. I., Zuev, V. E., Kuzikovskii, A. V., and Khmelevtsov, S. S., Thermal Effect of Intense LightBeams on Droplet Aerosols, Soy. Phys. Doklady, 19 (1975), 425.

Bukatyi, V. I., Zuev, V.E., Kuzikovskii, A. V., Nebol'sin, M. F., and Khmelevtsov, S. S., ThermalAction of Continuous Radiation of a CO 2 Laser on Artificial Fog, Soy. Phys. Doklady, 19(1975), 605.

Chu, T. S., Attenuation by Precipitation of Laser Beams at 0.63 1, 3.5 g and 10.6 p, IEEE J. QuantumElectron., QE-3 (1967), 254.

Chu, T. S., and Hogg, D. C., Effects of Precipitation on Propagation at 0.63, 3.5, and 10.6 microns, BellSyst. Tech. J., 45 (1966), 723.

Chylek, P., Grams, G. W., and Pinnick, R. G., Light Scattering by Irregular Randomly OrientedParticles, Science, 193 (1976), 480.

Davis, E. J., and Ray, A. K., Determination of Diffusion Coefficients by Submicron Droplet

Evaporation, J. Chem. Phys., 67 (1977), 414.

Gallily, I., and Ailam (Volinez), G., On the Vapor Pressure of Electrically Charged Droplets, J. Chem.Phys., 36 (1962), 1781.

Garton, C. G., and Krasucki, Z., Bubbles in Insulating Liquids: Stability in an Electric Field, Proc. Roy.Soc. (London),A280(1964), 211.

Gel'fand, B. E., Gubin, S. A., Kogarko, S. M., and Komar, S. P., Breakdown of a Drop of CryogenicLiquid by Shock Waves, Sov. Phys. Doklady, 17 (1973), 970.

Glicker, S. L., Propagation of a 10.6 p Laser through a Cloud Including Droplet Vaporization, Appl.Optics, 10 (1971), 644.

Harney, R. C., Hole-Boring in Clouds by High-Intensity Laser Beams: Theory, Appl. Optics, 16(1977), 2974.

21

FiiLC"I4'G FAA BLa-Mno s L

SELECTED BIBLIOGRAPHY (Cont'd)

Matthews, J. Brian, Mass Loss and Distortion of Freely Falling Water Drops in an Electric Field, J.Geophys. Res., 72 (1967), 3007.

Mullaney, G. J., Christiansen, W. H., and Russell, D. A., Fog Dissipation Using a CO2 Laser, Appl.Phys. Lett., 13 (1968), 145.

Nolan, J. J., The Breaking of Water Drops by Electric Fields, Proc. Roy. Irish Acad., 37 (1926), 28.

O'Konski, C. T., and Gunther, R. L., Verification of the Free Energy Equation for Electrically PolarizedDroplets, J. Colloid Science, 10 (1955), 563.

Pozhidaev, V. N., Change in the Optical Thickness of an Aqueous Aerosol under the Actionof CO2 Laser Radiation Pulses, Sov. J. Quantum Electron., 17 (1977), 87.

Pozhidaev, V. N., and Novikov, V. I., Possible Dispersion of Mist Droplet Using Giant Laser Pulses,Opt. Spectrosc., 40 (1976), 325.

* Sinclair, D., and LaMer, V. K., Light Scattering as a Measure of Particle Size in Aerosols, Chem. Rev.,44 (1949), 245.

Stephens, J. J., and Gerhardt, J. R., Absorption Cross-Sections of Water Drops for Infrared Radiation,J. Meteor., 18 (1961), 818.

Sukhorukov, A. P., and Shumilov, E. N., Brightening of a Polydisperse Fog, Soy. Phys. - Tech. Phys.,18 (1973), 650.

Svetogorov, D. E., Rate of Transfer of Radiant Energy Accompanied by Evaporation of a DisperseMedium, Soy. J, Quantum Electron., 3 (1973), 33.

Taylor, Sir Geoffrey, Disintegration of Water Drops in an Electric Field, Proc. Roy. Soc. (London),A280 (1964), 383.

Torza, S., Cox, R. G., and Mason, S. G., Electrohydrodynamic Deformation and Burst of Liquid Drops,Phil. Trans. Roy. Soc. (London), A269 (1971), 295.

Williams, F. A., On Vaporization of Mist by Radiation, J. Heat Mass Transfer, 8 (1965), 575.

Wilson, C. T. R., and Taylor, G. I., The Bursting of Soap-Bubbles in a Uniform Electric Field, Proc.Camb. Phil. Soc., 22 (1925), 728.

Zuev, V. E., Kuzikovskii, A. V., Pogodaev, V. A., Khmelevtsov, S.S., and Christyakova, L. K., Effect ofHeating Water Droplets by Optical Radiation, Soy. Phys. Doklady, 17 (1973), 765.

22

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GENERAL THOMAS J. RODMAN LABORATORY CRANE, IN 47522ROCK ISLAND ARSENAL

ATTN SWERR-PL, TECH LIBRARY COMMANDERROCK ISLAND, IL 61201 NAVAL RESEARCH LABORATORY

ATTN CODE 5709, MR. W. E. HOWELL

COMMANDER 4555 OVERLOOK AVENUE, SW

US ARMY CHEMICAL CENTER & SCHOOL WASHINGTON, DC 20375

FORT MCCLELLAN, AL 36201

COMMANDER

COMMANDANT AF ELECTRONICS SYSTEMS DIV

US ARMY INFANTRY SCHOOL ATTN TECH LIBRARY

COMBAT SUPPORT & MAINTENANCE DEPT. L. G. HANSCOM AFB, MA 01730

ATTN NBC DIVISION

FORT BENNING, GA 31905 HQ, FOREIGN TECHNOLOGY DIVISION (AFSC)ATTN PDRR

COMMANDER WRIGHT-PATTERSON AFB, OH 45433

US ARMY LOGISTICS CENTER

ATTN ATCL-M4 COMMANDERFORT LEE, VA 23801 ARMAMENT DEVELOPMENT & TEST CENTER

ATTN OLOSL (TECHNICAL LIBRARY)

COMMANDER EGLIN AFB, FL 32542HQ, USA TRADOCATTN ATCD-TEC, DR. M. PASTEL DEPARTMENT OF COMMERCE

FORT MONROE, VA 23651 NATIONAL BUREAU OF STANDARDSATTN LIBRARY

COMMANDER WASHINGTON, DC 20234

NAVAL INTELLIGENCE SUPPORT CENTER

4301 SUITLAND ROAD NASA GODDARD SPACE FLIGHT CENTER

WASHINGTON, DC 20390 ATTN CODE 252, DOC SECT, LIBRARYGREENBELT, MD 20771

COMMANDER

NAVAL OCEAN SYSTEMS CENTER NATIONAL OCEANIC & ATMOSPHERIC ADM

ATTN TECH LIBRARY ENVIRO.'MENTAL RESEARCH LABORATORIESSSAN DIEGO, CA 92152 ATTN LIBRARY, R-51, TECH REPORTS

COMMNDERBOULDER, CO 80302

COMMANDER

NAVAL SURFACE WEAPONS CENTER DIRECTOR

DAHLGREN LABORATORY ADVISORY GROUP ON ELECTRON DEVICESATTN DX-21 ATTN SECTRY, WORKING GROUP D

* DAHLGREN, VA 22448 201 VARICK STREETNEW YORK, NY 10013

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DISTRIBUTION (Cont'd)

PROF. GERALD W. GRAMS MR. RICHARD HAHNSCHOOL OF GEOPHYSICAL SCIENCES SPACE ASTRONOMY LABORATORYGEORGIA TECH EXECUTIVE PARK ESTATLANTA, GA 30332 ALBANY, JY 12203

EUGENE A. POWER DR. GEORGE HESSSCHOOL OF AEROSPACE ENGINEERING THE BOEING COMPANYGEORGIA TECH P.O. BOX 3999ATLANTA, GA 30332 M/S 8C-23

DR.HENY PASKSEATTLE, WA 98124

DRDAR-LCE PROF. MILTON KERKER

AT NBS ARRADCOM CLARKSON COLLEGE OF TECHNOLOGYWASHINGTON, DC 20234 POTSDAM, NY 13676

DR. DONALD A. WIEGANG PROF. JAMES DAVISDRDAR-LCE CLARKSON COLLEGE OF TECHNOLOGYEMD BLDG 407 POTSDAM, NY 13676ARRADCOMDOVER, NJ 07801 PROF. RAYMOND MACKAY

DEPT OF CHEMISTRY

MR. WOLFRAM BLATTNER DREXEL UNIVERSITYRADIATION RESEARCH ASSOCS., INC. 32ND AND MARKET STREETS3550 HULEN STREET PHILADELPHIA, PA 19104

FT. WORTH, TX 76107PROF. AUGUST MILLER

PROF. PETER BARBER DEPT OF PHYSICSDEPT OF BIOENGINEERING NEW MEXICO STATE UNIVERSITY

UNIVERSITY OF UTAH P.O. BOX 3D

SALT LAKE CITY, UT 84112 LAS CRUCES, NM 88003

PROF. JOHN CARSTENS PROF. K. D. MOELLER

GRADUATE CENTER FOR CLOUD DEPT OF PHYSICS

PHYSICS RESEARCH FAIRLEIGH DICKINSON UNIVERSITY

109 NORWOOD HALL TEANECK, NH 07666UNIVERSITY OF MISSOURIROLLA, MO 65401 PROF. JOSEPH PODZIMEK

GRADUATE CENTER FOR CLOUD PHYSICS

DR. PETR CHYLEK RESEARCH

CENTER FOR EARTH AND 109 NORWOOD HALL

PLANETARY SCIENCE UNIVERSITY OF MISSOURIHARVARD UNIVERSITY ROLLA, MO r5401PIERCE HALLCAMBRIDGE, MA 02138 DR. T. 0. POEHLER

JOHNS HOPKINS APPLIED PHYSICS LABORATORY

MR. WILLIAM CURRY JOHNS HOPKINS ROADVKF/AP ARO, INC. LAUREL, MD 20810ARNOLD AIR FORCE STATION, TN 37389

! PROF. MARVIN QUERRY

DR. ROGER DAVIES DEPT OF PHYSICSDEPT OF METEOROLOGY UNIVERSITY OF MISSOURI1225 WEST DAYTON STREET KANSAS CITY, MO 64110MADISON, WI 53706

PROF. VINCENT TOMASELLIDR. JOHN F. EHERSOLE DEPT OF PHYSICS

AERODYNE RESEARCH, INC. FAIRLEIGH DICKINSON UNIVERSITY

BEDFORD RESEARCH PARK TEANECK, NJ 07666

CROSBY DRIVE

BEDFORD, MA 01730

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fo/

DISTRIBUTION (Cont'd)

DR. RU WANG PROF. DWIGHT LOOKSPACE ASTRONOMY LABORATORY DEPT OF MECHANICAL & AEROSPACEEXECUTIVE PARK EAST ENGINEERINGALBANY, NY 12203 UNIVERSITY OF MISSIOURI-ROLLA

ROLLA, MO 65401DR. ALAN WERTHEIMERLEEDS AND NORTHRUP COMPANY DR. FREDERICK VOLZDICKERSON ROAD AFGL (POA)NORTH WALES, PA 19454 BEDFORD, MA 01731

COMMANDER/DIRECTOR DR. RODNAY J. ANDERSONCHEMICAL SYSTEMS LABORATORY CALSPAN CORPORATION

ATTN DRDAR-CLB-PS BUFFALO, NY 14225ATTN DRDAR-CLCATTN DRDAR-CLJ-I (4 COPIES) US ARMY ELECTRONICS RESEARCH

ArTN DRDAR-CLN & DEVELOPMENT COMMANDATTN DRDAR-CLN-S ATTN TECHNICAL DIRECTOR, DRDEL-CT

ATTN MR. DAVID ANDERSON

ATTN ROBERT DOHERTY HARRY DIAMOND LABORATORIES

ATTN DR. JANON EMBURY ATTN CO/TD/TSO/DIVISION DIRECTORS

ATTN DR. JAMES SAVAGE ATTN RECORD COPY, 81200ATTN DRDAR-CLG ATTN HDL LIBRARY, (3 COPIES) 81100

ATTN DRDAR-CLY (4 COPIES) ATTN HDL LIBRARY, (WOODBRIDGE) 81100

ATTN DRDAR-CLB ATTN TECHNICAL REPORTS BRANCH, 81300

ATTN DRDAR-CL-B ATTN CHAIRMAN, EDITORIAL COMMITTEEATTN-DRDAR-CLB-P ATTN BERG, N. J., 13200ATTN DRDAR-CLB-T ATTN LEE, J. N., 13200ATTN DRDAR-PS, MR. VERVIER ATTN NEMARICH, J., 11130

ATTN DRDAR-CLB-PS, DR. E. STUEBING ATTN RIESSLER, W. A., 13200ATTN DRDAR-CLB-PS, DR. R. FRICKEL ATTN SATTLER, J., 13200

ABERDEEN PROVING GROUND, MD 21010 ATTN SIMONIS, G., 13200ATTN FURLANI, J., 48100

DR. ADARSH DEEPAK ATTN GIGLIO, D., 15300INSTITUTE FOR ATMOSPHERIC OPTICS & ATTN TOBIN M. S., 13200

REMOTE SENSING ATTN WEBER, B., 13200P.O. BOX P ATTN WILKINS D., 22100HAMPTON, VA 23666 ATTN WORCHESKY, T. L., 13200

ATTN WORTMAN, D., 13200 (10 COPIES)ATTN KARAYIANIS, N., 13200ATTN LEAVITT, R., 13200 (10 COPIES)ATTN MORRISON, C., 13230 (10 COPIES)

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