:" i7

UNCLASSIFIED

SAl) 4 4 91

I.

DVFENSE DOCUMENTATION CENTERFOR

CiENTIFIC AND TECHNICAL INFORMATION

CAMERqON STATION. ALEXANDRIA. VIRGINIA

UNCLASSIFIED

tleaticas or otbar 4&Uan used for mW purpmeotber tban In cannectio with a 4et1fitey relUtdawtam&. pocuzmt -penations the U. S.Gocriimset thezmby incw o responaftlbltry nor' mWcilist~ou *ati0V6? £3a the fact that fte Goym-~mot mW hav fogpltc4 tfazuise4, or in my may

data Is zot to be reewde by ±±Im~1em or other-vie as ln any a?? 11censin the ba~ftw or aiyotber putou or cc Ciatiom, or caumim =7i rl ~tsor pea±ssion to uawnfoctumr, use or W1~1 mWpatentod inveitioa that Ow In am7 iWy be m1atedtberoto.

VCAMBRIDGE

ACOUSTICAL

ASSOCIATES, INC.

C0 SULTANTS IN ENGINEERING

PHYSC

MWIUAL A1bD SRLF-RADIAMO~ifU T)A'Mkne IN An ARRAYOF Fr-FODW, M~A, SPACM RONG TRM1WCMI

Tecemetcl Report U-17e--4Prepd red P~r Office of Novam %seaerchAcousticls Progrms Code5

M2 MOUNT AUSMt SMhUT CAMIMtDM, MM$SACHUSETTS

NM AND SUFR!1M~ADUZc DVic= W AN AN9l

0W mg-mm~Wm, 03AZIAL, SPAM RU. m~"

-I 16 March i964&

Technical Plepcrt U-178-48tPrepaed fr Office af rN.a1PResearch

contract X=a -2Y39(O0)TI'vk n. 165-301

I Reproduction in Whbole or is Part, is .0-rmitted,for any Parpos3 of tbe- United Statft 0overmnt~

IIc

L721 ýbuxt Auburnr St.-mtC&zbidge, Nassachuaotto C2138

II

i r

1 2'The belp oari= vae- of tbi firm,, in

particular of Mr. Wilhi~a Thoqnsom, Jr., vto checkedthe Qebra, and of Dr. Alexalnar Silbigr, is

1 0I Rag~mering Research Associat~es, BaltIkare,iWryland, bas pit the soluticm Into practice by evsl-uating the Green's f-mctioms and cbtaxningquantitative results.

1

it

I :

|I-

Rzpressions are ftsrivt f')r the a=tA*i- tad self-

kme~i8 is~o radial vibralm! . Fluidd cfrcu-latioa throu~h the ga~a betý~ elementsand4 araurd. thearray extreadties Is accounted for in a formulatioa of thisproblem uhich van presented in a ratent study of singleOsquirters", (Q4CL.o6.r3*y4 m44& An integral equation iseonstructed vhich def ines the unkawm radial velocity die-tributica cCz) betveen elements aen on thee two semi-infinite W-ind-&-cal aw-ftces Aildi brar~wt the array. Itis ohmw that the radiation Impedance is'-- ams of two coL-?cnents: (1) the jV;eUnoe evclasted by means of the math-ematical model ozig~±8ied by Robey, vhich assumes that the

Four methods ane presented for obtaisning the xunctton ca(z):

tio ir les ttrcti in hecase of the array problemtha inth cae o te Tinge squ:1,e?" oID-*use *all tboselfraditio !z~~sne X"be btained directly ftom thestatinaryfluntimalin trms f 4ch this aprproach is

fos~~te- Te mtue 1qedances =ut be eftaiunsted tvom the=mdetermined coefficients in terms of uhich the trial ftne-tionsa C(z) are' expres,,ed.

Dr. J. 2. Greenepor, of z GA Englneerixa, Resaerchlamooiatas, ait4the izverse Yourier trens.fora init~ras of uhich the volution Is prmiz-ute; ftn6 Obtainedquantitative recsults b;.b for than sirg~e cylinier ams brtize azrray. He viii present this ~mter-lal in a separatertporz entiiued "Axially Symwtric Green's Fuacteionu for..Jiiders.

r~~o CAl~ool=

i Ackno•eaveient. ..•.... . ..... ........................ I

Absrct... t .... .... c t .... . U

L ist oa Figures ........................ vl

fI Scope and Backgroind of Mis.Stud ... ... . . .

II Formuaation of the IToblem ............... 3

Ml Fadatim lapedane. .. .. .. ..... . ................. 7

IV P•-,rlation of the Problem in Wraw of Syietric (•rev)and Artisyinetric (Odd) Velocity Distributions . . . . . 8

V I-erturbation and Iteration Solution of t he IntegralEquatio ........................ 12

VI Finite-Dlfference Solution of the Integral Eqution. . . 13

VII Variational Solution of the I.Mtegral Squat ion ...... 15

&ftcrencco . 27

7..

! i

,] ,

III

(,U3tervea1v subscripts, vi. a, are used to condemse two equations

into one, the upper subscript on the left side of the eq,;ation beingsasociated vith the upper signs and 5ubscripts on the right side of theequmtion, and vice versa.)

mean radius of "suir-terr (Fig. 2)

als0 inner aia oui.r rcidius of 'squirter,ý respectiveiy (RSg.b half-width of gap between elements (Fig. I)

c soimd velocity in fluid iaedium

G 1 1 % o;U13o,Ysi8;g ,ri- r 8 ,iY4As and Aa Green's functions and related- functions defined In Table 1

H Hankel function of the first kind, of order m. Thi'ht function repre-%etsoutgoing waves for the assuted time dependence exp(-Imt)3

h half wall thickness of "squirter" (Fig. 2)

"J Bessel Nanction of order m

k Wave inmber. equat.L -.- /cS_idial wave m .. , equal to (2k2)

:r

k_ axial wave nuwber

L half-length of array element (Figs. 1 end 2)1 half-length of array (Fig. 1)

P sound pressure

U r•.ial velocity amplitude of active array element (Figs. 3 and 4)

uýz),ui(z),u (W) radial velocity distribution over cylindrical surfaces (r'a),(rSi ) and (r--ao) respectively, positive outward (Oigs. 2 Rtnd 3)

U S(z) 7 u5 (z) sy~ratric (even) and antisr~tric (odd) component ofveloc'ity distribution over cylkndrical surface (r-a) (Fig. 4)

Z self-radiation Lapedence of array elm•cnt j, in units of force/velocity,zj equal to [(Zt)r + (z )

Z., cutual radiation impedsane experienced by array element k as a resultof vibration of array element J, in units of force/velocity, "ý_-ae to[(Z .r + (Zj.)a]

(Z jk)r radiation impedance obtained from Robey's mathbentical codel for whichc(z).O (transducer elementz located on Irfinite, rigid, cyindrical baffle)

(z) correction factor associated with velocity distribution a(z), to be-added to (Zjk)r

4Other sym•bols are defined in the text.

11r

a xtPSion of z-axis consaiting of tbe Me between arrar elawr ant i* ob~t• the tvo saee-Lifinite eyU)rical surfaces (z > C, z < -) -ich

bx*'dket the arraY' (ft. 3)X r£+ Pttive half of Za

&a location of midplp-* of "ps between nel•bborg array elements (Fig. 1)F- zj loc-t.-I of uidrIAM of active arMY eM2ent (Mg. 1)location of midlane of arrar elct (Fig. 1); in imeadance calcuu-3k tica-..location of inactive element (F. 41

Cl(z) radial velocity in the region z (Fl. 3) nurnalized to velocVtyamplitude U of active ele t et'I-S

c?(Z) ,Cz(Z) syinetric (even) and anuumsyetric (odd) cowmpoent of velocity"distribution In region zf, norelzed to U/2

Z St. 'o coefficients in the relation of the velocity U of the am transducersurface vith, respectively, ui and uo, defined In Eq. 11.3, f 1 for

sall h/aS ,(I-.') Dirac de.ea function, (') .- ) d•' - a(z)

SPoisson's ratio of transducer material

0 density of fluid medilm

4(r,z) velocity potential (outward velocity it - r#/&); tubscripts "I" and"o" reftr, respectively, to regions r :Saind r > a.; subscripts ",n

and "a' refer, respectively, to symmetric (even) and antisyimetric"•(odd) co;ý with regard to zM circula.r requency [with thin notatim, z massive reactance is negative;

harmonic time deptndence factor exp(-imt) which multiplies thevelocities and the potentials, has ben suppyts throughout this

f ~rt'port]

Vt

fI

I

! v

ii

IsI

'~0

uu Fa 1#40 0 0

,4I 0 $4 %4 0m W.a

jj3 04 3

0-4 $4 $

014 14

to0 4 3 0 3

I 42

tZ44

.0,

I

Li T oF FIGURES

I Page

1. -M.F-LODI1G AMWY OF S'A=D H3IM POW .• C.. ................ 22

2I . G ;= OF T01 ZR ..................... 23

4. pT~L'nTJ oy 'JswC1'DmmwTmw~fO mm smeehc (Emu) AYAAMM&a C32=-A a~uI~ l iSTRAM MRE M1V EM-~MAY. .25

0r

iI

II

= '• -'-' '- .•0 " ' ; ; • i • i P' i ii• i --m . . .= , -" ... . . . . . . .i n - n i mn n e un i

1 . Scope and Beckground of This t4

The purpose of this study Is to, eveamate the radiation Ipedances for the el-

ements of an array consisting of identical free-flooding coaxial ring transducers

r vibrating in their r•adial, axisymetrie ode (Figures 1 and 2). The transducer

elements ere separated by equal axial distances, thus permitting the acomstic medium

r to flow both axially at the ends of the array and radially bet-ame elements. Th'e* impedauces take into ue-at the combined rdliatiuc loadng acting on the inner and

outer fatces of individual transducer e~ements, when caay oc. element in the- arrayis active. The result of this study will thus be lu the faom of self- and mutual

( impedances of the array elennts. Coutatios' vVIl be performed in the resa•sat

frequency range of the transducer elements, the variable parameter being element

specirg. Even though this study vas stimulated by the promising experimental re-

sults recently obtained by the U.S. Naval Research Laboratory on an array of inge-

tostrictive ring transducers, the present an21,sis In not limited to meaetostric-tive transducers nor, in fact, to free-flooding, axially spaced elements. When

coinbtd with the analynis of individual free-flooding or solid cylindrical radia-tars presented in an earlier report, the present analysis can be applied to the

evaluation of zatual Impedances of ring and piston eloments on a cylindrical

baffle of finite length.

The array analysis can be followed independently of teferency 1 if the reader

is willing to accept without proof certain results derived there¢n. It vas showin this earlier atudy that the radiation impedances of an isolated transducer ele-

ment can be expressed as the sum of two ccmponents:

Z. Z (.1)

Zr is the impedance computed by lobey, 2 Greenspon,3,4 and SBrb.n4 from Robey's

mathematical zodel which is frequently used for c:lindrical transducers. This

model asznes that the radiating surface is extended to infinity by rigid cylin-

drical baffles vhich prevent circulation of the fluid a--l. t-h•• -'--ducer edge--.

This assumption, in combination with a Green's function constructed so as to have

its radial derivative dG/br' vanish on the outer transducer surface (r'ra),

Me radiation loading ca thc iwside of a free-flooding cylindrical trnducerwas also studied by Rabey by teens of two lifferent models, idiich unfortuntelyalso place restrictions oan the fluid flow aro*xd the transducer edges%.'

I

qlI

el~minst2a the need of solving an integrai equation to obtain the potentials.

With this zodel an exprwsion for the far fielt potc=itza n h obtAined ana-

lyticalUy in ciosed form.5 The imedsance Z in Bq. 1.1 is in the nature of a cor-

rection factcr associated with the radia: flow velocity distri~oution a(z) across

the eW2Lndrical surface extending the transducers. Z is in the form of an iute-

gral whose integrand Is proportional to a(s). fte function a(z) satisfies u non-

homogeneous Fredhola fntegral equition. Unfortunately a direct solution of this

equation is not practitcble. Te analytical effort is therefore mwtly directed

at !pproxiting the unknown function C(Z).

In ref. 1 a variationls tehmique was developed which parallels the Levine-

Scbrriger raristionsl procedure for ccqutIIg scattering cross sections iL dif-

fraction problems.* In this procedure, a funetical J(0) is constructed frosthe Inteeral equation. This functional can be shown tc be stationary with respect

to first orIer variations about the solution a(z) c- the integral equstion. If

= otio of transducer win thickness to radius is smll, the stationery va'5a

of jo I] is proportional to &C. A ftyleigb-Ritz-tyPe •ciculatic is used to com-

pute Z , starting with a trial function for 0(z). The moet comm use of the

Rayleigh-Ritz method is the evaluation of the natual frequencies of a vibratina

system. These rr.4uencies are tne roots of the determinant of the coefficients

in a set of s inultaneous homogenemus equati•ns derived from a variational prin-

ciple. 7u- natural frequencies CM therefore be evaluated vithbut b"Ing to

compure the un,4eternined coefficients it: the assumd modal configurstin of the

ribrating body. In the a•nl3si of the single, thin-walled "squirter," Z0 can be

similarly obtained without ;ceviouslj solving for the unknown coefficients It%

terms of which a(z) h", been e-Pressed. Unfortunately, Only its self-radiation

impedn.Ice can bv determined in this fathion when the transducer 1s an element in

an array. In o.W r tO ca..lute the =utual itPedances, the set or si alt anecs al-

gebraic equations derived frem the variational principle must be solrý4 fcr t1e

,ui*nov coefficients in the trial function a(z). Thus, vW.le in the tingle trans-

ducer problem the varinti•nl twlhOd is considerably mo•e efflaiv-t thou m-term-

tire 0ethods for solving th, integral W e I o-zi, +-hs iu sot the .'. I-• 4 ' € lY

proble=.We en1 therefore pretwnt several other tecbsiques for dealing with the

integral equation. E?;rie'ce with ne=rical ;alcujAtIOus will show u.tich

'A cmczprehenle bibliogriY of variational anal 0- of. diffract.:on problems

is of-ven in the bibliographY of ref. I.

v~amas eficent frm iucat -dfof i array aseter. Eveno

the omrtw= calcuaxti as at usmlwt t-f~~o a~ tiom Save

th eb * oat cu2tqiom ation freuai3y been iwd *ftw~,oeac uth rn8 tosovftptf4 hi oikto ft* rbaa he bii 7- &,aprsn

r.Ing relemmt 8M in th the semi-nfint~eio auml i a_- ise~n theara

i~edance tbwri does the uwcnov potential over the trsmditoer -surfaces In theabore-mg-tioned a=Cyaes. A compatwble ae-cy In the i1ebaoe calculetionincan therefore be expected from the present approach using a coarer finite-

difference spacing.Quant-Itative results vin~ be presented In a separate report by Dr. 3. 3.

Oreenapon, of J 0 Ragizeering Rmuareh Associates, entitled *AzIaW Sy 0)ltric

Green's Functions for Cylinders." Djý Greenapon Is obtslizang mubrical results

for the transducer perareters correspoadirag to tba 9Cý array.

n I. Formiation of the Problem

Following Robey's cxs2~le the potentials are expressed in term of two

Green's functicus whose radial derivatives vanish on owe or the other of the two

infinite concentric cylindrical surfaces on which the ine and outcer transducerfaces lie. Two 2otentials are thus constructed *0 , in the outer region r > a 0and #I. in the Inne, cylindrical region r < ei (Fig. 2). These potentials are ax-

* pressed in terms of the radial velocity distributions ui(z) and uof a) over thesetvo coaxial cylindrical boundaries :

egere and elsewhecre in this report alternative subscripts and signs bave beenused to condense two equations into one, the upper subscript an the left aide ofthLquto bein associae vith t upper signs 3nd sulbscz-pts on the righltsieo h qaioadvc es

Oef Tlocity Vanishes over 8'U c mtive tre<ao uee suface and eqtuhs a know

c> thine, say Wianothe A4sen trh e active traensduer- Is h 2afted.* ne aim-

laticel 0 iff1ctSco1 in t• e tact tute the r•dial velocities areý Te t o ier o

the trontsucer sufact (Ati. 3). rtefore deftnsd tuoe ns. Cfnactio it is

S00CMudes toe pas tbet th e s ray lt and t(z) tIVervs- of the vaeocItyuee of• the ooritce (r-&). the ethis prposf - a seat = e&an 4 nA t eoible

p9orti1 in the em s.p•hacues o< a • < are das i lies -that th, ratio -if van

thickams to acoust.c wavelength is negligible'.4* With thix aassmptisc, the bal-

ance or• es-• inflo ant outfleow yields the following reasticos betaen velocities:

u (z) uu(z)(1± b -IL for z In region ze. ('2

aer egion zf encyosses those portions or the annua &pat ce 4ae <rV <to)

which contain acoustic Men id, ret•her then transducer eltats. Mw region zr tb

includes the gaps between the array elements and the tw sea-Inf uite annul,

z <a and z > A, wich extend the array:

-t z +b

a~ ~ z. g -b

where z_ is the coordinate of the plane of symwtr4r of a gap between two neiah-

boring array elements. 2h* values of z5 are given in the Table in Fig. 1. She

coordinates zk of the plane of s3yinetry of the elements are also given In this

Table. The velocity of the- inner and outer traducer faces are related to the

velo-Ity if' the zen transduer surface as follows.,

u Z (ZOY(+9 for z -L < z < L (IL.b)

where the coefficients 8 embody the Poisson'a ratio of the transducor material:

a W i ~( ) for piezoelectric trajicora

9 01 ( NO f)i~) for Mmgetostricti-v. transducers W-13)

~dp~c exy(-Icst) bas been suppressed throuýx this report forthe sake of brevity.

*Olt was pointed ou.t In ref. 1 that caq~ressib1.lity of the fAnid in tbe annularregion c="l be incliedd ft. the sanlyris but that this tiu14 resUlt In twe- coupled

~atotritivarray, the -stio of vanl thickmess to acoustic wavelenath is 0.07.fte ascumptita of an incos~reasible potential is therefor* Justified within thelimts of accuracy required for design purposes.

a ~the suabscript v ill be u~sed for the actIT2 transuc-mr, Sat the subeacrit k Zor

the inctive MWadcga h origin af the Z-COMA40U tas located in the p3*neof syMet%7 of the &my.~~, i

expressed as follovs (see Fie- 3):

Iu(s) aU on active arrayelement; Z44< -Zz+

- U Q(4 for zin region Z.

- 0 on Inactive array eleawnts, zi-Li < z < zk*L (see Table

I __ un~known function CX s nsnral, eOMlg.

Th~e Green's IfznCtioDs for the tvo 11g Integrals.a1 Inth for* ofinverse Fourier trsncforn:

(r a I-' r(r,eak )exp~ikr.(z-z #)I dkz W-l5)

iihare the transform of the Green's function are

0(r,a Sr1 Zk-r) (7E

o (r,a,,k).L. Jkr)(.6)± krJi(krl-i)

It can bee verified that

Ir -at

Th peuias anbeobaiedbyevlatngth iv~ Furertrnso5aBI.U5 n impfrigtez-fegainIdctdi q :..Atrnaiey h re fitgaincn ervre.Ie q.n5a usi

Mw po•tenti lIn E. n t .en be vritten in the I of an inrae PaierV trin" foxr

As the develoyment of thi awlysis need not be spec•ia•ze to on or the other

of thgse procedurce, their relative couvenience can be compared aftcr wre ex-

p.rience has been obtained in the a•l~tio* of umerical resUtU.

Wha ve subetitute Eqs. 11.2 and 4 in Sq. MI.; ve can express the potentials

generated by twa. vibratious of the ith element of the arry as follros:

z +L

to r,) el Up (ra z-z')dz' + fcgz")JO(rs-u)dz' (31.10)Szj-L

be unknown velocity distribution Utz') in these Integrals U defined by the ra-

q'uiresent that at every location the ptressure 4ifferential between the concentric

cyl~indrical surfaces (r~a,) and (r~so) balance the raf1.al coegmet of the maer-

tie force exerted by the i2a.d in the anoular region between these tie surfaces.

The radial accelerstion averaged over the radial thickness of the ==1us is,

taken positive outward

- ini~zn) for z In region z 1.1

TMe recial component of the inertia force associated with a differential volume

eln•t PbaV. of the fluid annulus is therefore

d F2) 2 z•a(_)p bd d (11.12)

The outuard force agsociated vith tla pressure diffareatial is

dPqz) , ipi(.) - acpa(z)) dp dz (n.13a)

In tr= of the pattntials and of the mean radius Ind Shell thickwee: thi8

becowes

dFl(z) -4=0(- 1• (i,-) (z- 4 C'" ("-"13b)

When ve ket the sto of Eqs. .12 end 13b, equal to zero, as required by

d'*Alezert s princ'.ple, ve obtain the following reltin between the Potentials

and the vzim=o ve-locity it o(

J

(3-+ t) #0 (a 0 Z) - (I- !.)#1(a 1L,z) -2bWx(z for z in region zf (11.14)

[ For tvo of the configurationa anmlyed in reference 1, viz. the infinitely thin-

walled, cylindrical radiator, and the solid '!linder, the right hand side of this

equatica mvdes and Eq. 31.14, reduces to a coutinuitr requirement of i and #oon the surface (r=3). Vhen we tubstitute tlhe q;ýreaaq foýr viw Pt-e-au-,

[ Eq*. 32.10, in Eq. 31.14 we obtain the non-homogeneous Fredbolm equation which

defines tc unknown function a(z):[A(z-z-5 z for % in region zf (U.]5)

f

[ In this equation th sycls r and A have been used to reprement two linear cm-

binations of the Green's functions 0G. and G:

f

r(z-z) -1 r. )(aa 0,Z-z') + (I- G) 0~a,,a,.z-z') (I1b

Having thus derived the integral equations vhich ,x(z) must satisfy, we proceed

to express the impedances in terms of this sae velocity distributi- Cl(().

13:1. Radiation Le~dance

The radiation loading exerted by the jth element of the array on the kth

element to obtained by integrating over tbe surface of the latter the pressure

differential, Eq. I1.13b, where the potentinlz are associated exclusively viTh

the =tion of the jth transducer elerent:

k +L

jk U a

A r•di•Al].y inward directed radiation load is positive in this wtation. When we

substitute the =pression for the potentials, Eq. II.10, and a•king use of tae

ebbreviatioas for codbinstlo of -3reen's functions, Eqs. 11.16, we finally obtnin

7!

I

the foflalogi ezqpression for the radiation I1WeAMO.

If f + Ca(') r(%.z') dz'3 dz (M.2)

As in the cue of the self-U•edance or the zsrgie treaducer, the sAtin Imped-

sz-j~n to be tzae &= of two copponents: (1) the U~eaftee obtained fromabbey's mtbentica3l model:

z+L z +L

(jk~r f ~ 2 {~ A(z-z') dz' dz (M-13)- z-L

and (2) an impedance component associated with the uricnow velocity distributionC%(z):

(Zjk)CI -(2x&)Lf ae (131.4)

nY~L Zf

The problem Is thus to find the solution 0(:) of the Integra equation, sq. ft.35

and to subatitute this solution In the above expressions fzr the radiation

impedances. The velocities of the Individual transducer elments can then be

obtained fron the usual array analye s. Finally, the velocity distribution a(z)

can be used to obtain the far field potentials from the expressions given for the

single "sqrirter" in Sections V11 and VII of ref. 1.

IV. Fornulation of the Problen in Term of Symetric (Rven) and Antisymetric(OW) vel2ZoLt Distributions

In Robay'8 mathe•ntical mde3 the velocity dictribution, and hence the poten-

tial, are always symmetric about the plane of symetry of the active transducer

ring. In the present iodel, Adhch allows for fluid floy between elements, this

is not the case except vwen an active elemet is located at the center or the array.

In two of the four solutions presented In Sections V to VIX, it In advantageous to

consider the velocity U of the active array element r- the resultant of two cmpo-

nent velocity distributions, wne of vbich is '.. a- - bout the plane of symmetry

of the rry, (zsO), arA the other cntisymetric. .8 seen from Fig. 4, this amounts

|8

j a to considering the velocity U of tile active element,

u(z) -U for a, -L < z < + L (~

as zhe superposition of (i) the velocity distribution associated vith two ele-

ments located, rsspe--ively, at za excited in p~ae, vith balf the

or.li•gel amplitude

II W.( z L < % - <z t± +L (Lv.l)

and (2) the velocity d•stribw'zi of the $ame two elements excited out of phase

11(Z =1fori -L<:<zi+L

!U1(Z 1for-Z -L< z <-z, + L (I'V.1c)

For a centrally located element, the antisymetric velocity distributia does not

arise, A

U(z) -u (z =U for -LC< z<L (z -O) (IV.14)

Because of the linearity of the problem, we can solve Imlependently the symetric

case, Sq. ri.ib, which give rise to a qwvetric potential,

S(r,z) - , (r,-z) (V.2a)

a, the antisymetric case, Eq. IY.le, vhich gives rise to an anIisy-netric po-

tidtial,

O 5 (r,z) 4 *(r,-z) (IV.2b)

The unhtwn velocity distributionz corresponding to these two potentials

batisfy aiSlar s).Oetry relations:

a -(z)- -)I

%(:) =•(-:)(W-.3)

It is ,dvantageova to nor=%lize the two comenento of the unknown velocity dis-

tributeion to U/2. With this convention, nd using Eqn. IV-.3, the velocity die-

tits-ution in regon z f =an be expressed as

u(s) -, ~ (z) + a (z)) -,or z > 0

3 au() - [a (z) - q.(z) for z <0 in reginI

9 /

It is seen fii m Njy. 13=1 3 tht -if 4ld, Ia -satisfy the i ntg al eqnI BoNN nSSO. I.l aUA 15, In tM- PoonUVOe hal of the r*i= 1. they sutotically *at.Usfy

toese esations In tbe nsptive half of xr. te who a• lue, to is A a. V, CQn

therefore confie the tetal eqaution -to te positive brlf of the region Zf, Vhich

vil be designated heneforth by the symol ¢f*. ft advantae in splitting the po-

tential into syawt4 aA antisymtric cocm ta results from the fact tat, with

the finite-differeace and variational techniques •e•o-m ed below, It Is less lebo-

rims to satisfy two separate integral equations in the subregion z., than to sat-

isfy mne Integrel equation over a.U or the region z:.

2he resultant potential generated by the active tranduce.r located at zi 2A

* 8 f(rr) - #r,-) + *.(r,z) (W-.58)

which, in view of Sqs. ZV.2a, can be uritten as

o(r,-z) - f.(r,+Izl) ± .,(r,+Izl) (V.5b)

We need therefore only evaluate the two component potetials for positive Values

of z to obtain inrornation an the resultant potential over the vbole reap of z.

Restriction of the analysis to posit.ve values of z does not ivterfere with

obtini•zg ell of the d2Irem4 rW-lation imeaanca$. The •t•usl ima4Ance Is deter-

mined only by tv separatton lzj-zkI between the activ_ Sd inactive array elewnt.

The self-impedanc is the o'e for elevets s9 tricaly located at z and -z,

We can therefore rest-' ,. tie awl,.s to active elements for ,,ich z, > 0, but,

can c¢ course be negative as -nll as positive.

wc shall row derive the cotponent Greea'o functions %.tich are appijc&ble to

the symmetric and antislyetric potentials. The Green's Tuncti4os in Eq. ML.5 con

be exp.•essid in ters of an inver se Fourier cos•Ir transform by noting that the

Sim nary cOvocet of the expGiintial does not contribute to the VALe of the in-

tegcal, beceuse the Green's function !a even in kz;

G (r,a,,Z-a' f G 0 (r~ao k) cos jk(%-z') I dk2 IV? 1

brpanding the conine in the 1nte* ,ndp the Green's fun1ction cou be written as the

x=• or a -conpsent vhich is even, or s=atric about both z- end z'-0, and of a

zccqnee which in odd, or antisyonetrie:

0 AJ0(r,a Xz) cos(icz)cos(k~zz) Uz (ZV. 74)S-xf

i ')

JainJr (r aHBH k 4k.,a J Il l|H lII

0

r If oth the G'reen' function an the velocity distributiw . a the Integran ofEq. 11.1, are expressol as a sum of ",metric and aantisy tric terw, the croes

f terms are found not to contribute to the value of the integral:

yrz) 4-%2fI2z 1(•m,- u(,. ra~-.Iad* (V

S] •Th two producer8 in th• itegrand thu corres'•onW r•espectively to #8 and .

Since the contribution of the integral to these tva motentials is the some for

the positive and negative halves of the E -axds, tbhee tio comp0Cunts can finallybe written as

1a

'5(~z .4ar U (Z') g(Iv.9-z's)z

and

to(r,z), t ±xa 0 UOi~j ag(ra0Z-z' (3V.6101

The integration over z" can be perfornd befo e the kz-intcratioa i ated inEqs. ur v~n~8i1I.eateV .7 by ;aking ua- of the Fourier tranzforn of us and urespect

. 2(af) (r a ) u(x) oa k z dk (!V.10b)

S' z aZ. z z

j .Mhe linear embuatien of Gre-e's functims, Zqo. 11.16 used in the integral I

eqvuati=. Eq. 11-. cs- == --lz;o upl into symaetric and Ratisy~aetric coa-

ponents:

r +Li',l;- .14

A 4 -S (1 xV- a .' + llfX...k ,a8 6 2 .. 3 ') 1nAr.. a -o' o'0

The two rumoomit velocity distributions, 4. azd a satisfy the two uncouple&. In-

tegral *Amtiona 14Aicb tame the place of sq. n.15:-

flobey's radiation impedance, Bqs. -M-13, nov b-2cctS

The corretion. factor associated with the radial velocity dIstributien a(s) Is

42n)~ - 2fk f [(zl(i)r (z-z') + a.,(zl)r(.) dz'd fVfli

3:f either 7.k or z jaxe zero then the antisymmetric termsr anAa.d rpotf

the lintegranda in Sqs. IV.13 and '.If an1 the elemn~ts In the array are Aentl-

cal and driven with the a "ý sginal, the velocity Jistribuatiou sand hones the fAL.

field potential do not contain any satisymietric compoent$. We will now diacuso

four approaches to the solution of the Integral equ~tion.

V. Perturbsticn anid nteation Solution of the &LtýE 2Mal 3 oat~*

In U "-e two epproches nothing is gained by dealInG separately vith sYM-

metric and antisjmtric velocity cmqx~nts. Mhe crudest of the four approaches

* ~to be dercribed Is the porturbatical solution In which we actually circuwent the

need of solving the Integr-al C-T.Wttion This eppmcblc starts with the near field

potentials associated with aobey's aodel. They or'e computed from Eq. In.1i where

a(%") Is set equsl to zero. Desipamting these uaperturbed ptentials by a' nd

sQubsti-.tutig +then in Eq. na11 , ve can solve directly for the perturbation solution

for af:)

(Z) -- 2hU

12

Whene ve suistltft tba mnperturb potetiis as compt•t from If. MID.1, vbere

S| I a(z) has been set equal to zero, wad using the .k =:5 fimctio codmtation in

A(z-z') dz' (V.2)

YIn the ecment subregions of zf vich correspond to the pps between the array

elements, a somewhat more refined solution for C(z) can be obtained if one defines

an effective radial annular uidth equal to the transducer well thickness (2h) in-

em-tsed byu length Ah emodying the accessio to Inertia of a slit In a baffle.

The introduction of an accession to inertia Is noet usetl In any of the other ap-

proaches, where the potentials to and th Ievea esbody this reactive i•ed-

ance c om-tunt.

Since, for design purposes, ve probably need a ware accurate expression than

the perturbation solu-tion, ve u~e the value 0), 0. V.2, as a start for an

iteration procedure. The first step consists in computinf, values for I by

substituting the perturbation solution 4•(0) in Eq. I1.10. Wvoen we subs ,Itute the

p-tentla•s tbw computed in the integrsl equatica, Z4. 11.15, we obtain the follov-

ing ruore rafined expression f=~ Q(z):

-z A[-' dz' + fr(z~zI)cj@)(zs) dz') (V.3)31 f

This iteration can be repeated ýmtil the fluctuations of successive soliuticiis

cr(z) are deemed sufficieet~y small. Convergent,- of this process c3n best be test-

ed capiricaily b7 performing t-be actuo'l- numerical calculations and coqmpring the

results of gee-.szive iterati-ags with rtsults obtained fros tho other approaches to

the solution or the integral e~quation.

V1. Finite-Difference &4olutlon or the Intemra Luauation

In this apprcseb, the region z,. is divided Into a nuaber Q of subregions.

Each subresicn 1z M4cntIMA4 by the coordinateatz of its p:lane of symetry. It i~s

wssumed thiat rc(z) has a constant, geners1.v~ cmqaex, valsl- aof a.q in Ptar-h one of

these subregions. Ttý axial diaension V,% of these subregiow) can be taken emaller

13

3

it

In the vidlnity of the active trwdtucer to account fom the more rapid

decor of a(z) rear the radieting surface. Mw zefate1 in the in-egriaequation, Sq. 3.15, Is nov replaoit by a aumation over the Q subregiow. fteintegral eqoaticn •wt be satisfie& at dinsrcte points z. in the region zf

p-q + f f &

vhex- 6. is the Kronecker delta function vhich equsls 1 when plq and zero vhen

pfq. If p is successively set eTAI to all valtvs of q, frts 1 to Q, a set oe

sinultaneous linear equations in a is obtained, In this particular .oproachqthere is an cbvious advantage at evalustiag sqsratay the symmetric and anti-

symmetric velocity couxonmn. The two uncoupled integral equations, Iqs. IV-l2

now give rise to tvo uncouled sets of suitauncous i.ner equatios. Aoiceve±,

since the tvo In,,egra equations, 24a. IV.2., only extend over the positive hal-of the region sf, the sam Unite-difference *pacig results In Q/2 (Instead oX o)similtaneous eqntions for (as) 3 and in the saw rnuber of equations for fP IThuM c rable accuracY is ef leved by solvig two matrlx e-uatl•s of order

"J2, instead or the sinsl. mtrix equation of order q Vhch ua• be solved if

the velocity distributimo s not • spji im n i f&ashion. TP-4se watri equations

are of the form:

h +2d (o 2d %- - 2vj(Z.zx (Cgs (21

. + (.o.. ............

................ ................. .. ... . .. . ....... + 2%F1(0) (as) ~ (

inher*

z .+L

zII

14

Like the, Orion's funatiow of ifideb tWe aike caod the functionsi r (z-2%')

diji a siguu sinc the am quisleest to an into&-*I ok r(z.!). =vr zuh~ch, life the potentiasIs 1 a we11-btbave4 Awtion.

When this matrix is solvedl for the valuesa of a the raeieatio impedances Za

[ ame obtained In the form or a sufwtioai

(ZkGa-2(2ia)fa* 13jdf I z-rs(.s) 4 aadezm (V1.4&)

VII. Varietiow1 Solutica Or the XAteul U~tun

As In the f-Saite-diffarence soluation, It iLl, he t~oua advantageo"z In the

tof~ th uniow veocity istributioan. Our pupste ~ ~In 1 this seaction s hu t

parall1el those pke~ented. in raf, 1., Section V. Both sides or the Integral equza-

One thus obtains a relation between twao functiocals

Bla I I A~a a3I (Vfl.i)

Alaz I I 1- cts(z)[f a (:-Z') clz'I Ate (VIL.2a)

,(a''r rf-" + -L. a(rzt4)Na(z') dz' clz (Vn.2bB[a I f f a ~ Z

2x

Wtien we divide both aides of 1q. V31.1 byr is 2a] end take their reciprocal, weobtain the following ralntiozi

-W55--ritsa and a4ilb11 dtdi sbeun

Uwe IM ft-Lnsl -ifeb~ div~y~sp the desired -,tationery aucterictics is defined iby the left -*Ift of the above equaticei

1S I 2IA

441 Aemn 8&;3j *ssociated iiith an incr4et 66 can be written ae3

JI3[E4 (BfCCl + Ajo]3eij z1 tdz'j~.) ' ftvf.5f fL

Tb badt~l, fthstransaftr on, tereader i eerd toef. 1,

verile-tbatUleawmts to requiring azhat the fUnctionals Aj[Q]ndBf4satify S. VIA. tisequaltion is derived frm the lategral eziastiotxa,

Eqs.IV.2. I istherefore sstisfied by functionas which are solutiors of theintegral equxation. The functicas whiich satisfy the inteeral equstion ftbQreerrealso wke W vanish. We have +-3- sifowa that the functiosal de*~.nd in Wt. TJ1.iis station=7r with r-espeat to variations aa about the zolution ct(z) oX -the orig-inial integral equation.3

This variational principle can be used to obt~ain approxiamte vailtr of the

radiation imped4ance in 5mch the saw way that the EAylcigb-R'itz actiod ylalII ex.-prwassies for natural frequencitts of -h;atzng system. The procedura was prtý-

sented iij 4tI n ref. 1, Section VI and vill therefore only be brlfatfy re-.viewod. In the IRxyleigh-Eitz procedure a trial function fw- CS(z) Is cemtrt-.-ted

shown that in the far field, etf:) mimat te ef tl.. form

a(:z) .- iz for ,;hirge (vn.6)

In the near-field of the ai.Tay, for valves of =34,t slightly larger thrm 1 amore rapid decay vith inct'easing z can be assumed to simuante near fiel~d condi-tions. It is thus orpe-41ent to ltscinde a tern proportionAl to z -3. When thegmps betveexz s aylenents are narrow, tne radial velocity throagh these gapscan be teIxcn equal to a CCoAtant, xwhich of course differs from apy to gap.Phase shifts between the radial ve!=Ities in different gaps are taken into

16

aceot by the fact that the coefcius x6 aze :In 408"al, Cccs. Since

L splIttiMg tJe velocity distribution into smitric W1 8fltIns'*%trIc coqiceentsconfloes the Integl equataon to the geu-rci zf+, the trial fmction need be

defne may3 for positive values ofr z. A trial fuct{.Un in tbe fooliovng form.is thus obtained

a(s) + ~ezp(Wk ) for z>* (4U.z

O( g for s-b < z < Z" with- =a (VlL7b)j(see Table I~n r.*. 3. for value ot A, refinement ittich vill be requirea vhen

S4ve Ma between the array elements ar large, i'aolves tqg Into account thechange of radial velocity within the width of each ip. Intead of Uq. M.7b,we can now assum

x( o + foz• z-b < z < z+b, with z > 0 (VI.7o)

Rxperience with numerical calculations, and In particular ecperism with the

approaches described in sections V and V1, will •ndicate whether the radiation

imedancE is sensitive to the sevle-ticn of the trial function a(Z). Since th*

resultant velocity distribution Is tta w-a of two trial functics za and a_, a

fai~rly modest numer of vr.-*=vn coefficients all'nwa considerable flexibility.

As an example, let us consider a three-elemet orray eb dying relatively

e_•-- gaps. The trial function in Zqs. VI.?7 now becomes

•(z). (--4 + p(i) for > 3L+ 2b

(z) x for L < z < L + 2b (V13.8)

B3y virtue of the variAtional principle derived above, the best values Of the uu-

knrown coefficients are thoe which give n stationary value to the fmctaonal Jail:

']/n - 0 Vith a 1,2, and 3 (Vn.9)

Wben ve sazbsttute the definition of J(ai, Eq. Vn.I4, ad nltr ly all terms by

B[Cg], Uthse equations be~

-_ A_ Act] 0 0, with a 1,2. and 3 (Vit.-oX

17

-z

When-Ahe ftmct±=*U AlczJ ae Ba]*, lqs. VX1.2 are animattut in tem ef

Xqi. VIi.8, the formr funetiona is foun to be a Ilea"r f9%tic.. ='-b ;t j a

cw oeftfie s in acz), and the latter a quadratic fnction:

33

3(a] •(bzx•+ 2 S'b xix)

m+ 2

VU-u w2e +L' ý b 5a

!be coefficients a%,N ar: bm ae kwou, ccow x comtants. When •q*. 7J1.1,

are substituted In the singtanecos variational equations, sq. VIM.10, a set

of linear equetionw in the unetermined coefficients x. is obtained:

A 3

(4ý -- *~jj~cn+ ~y -bneJjqz]z- 0 for a-l,a, and 3 (MZ.22)

"iTese equations being hmgeneous, they admt non-trivial solutions onl if the

determinant of their coefficients vanishes. One thus obtains a characteristic

equakice whihb can be solved for J[i0:

ab2 - b~9ta] Y3 - b13Jlaj

Yl2 - b1 2 j~J s2 a-VIjC)3 as 2 3Jlckl -0

aP& - b Jtc) s'983 - b2 3Jal *3 b3ýjcrj (V1-13)

la contreat to the %ayligh-ittz naturel frequency calculation, vwhch yie24 a

numter of roots eqtrl to the order of the matrix of the charcteriatic equa-

tions, Sq# Vn,13 adnit3 only me non-vanishitwg root, e.0, on* value for the

t nst a1 Jil. •/a reason is that a.. ta•me except those contalnu the two

higbhes poverr of J•[] cancel. Thls result Is tocsistent vit% the fact that the

am-hasogenems integral equation adaits only one noc-trivIal a.olution. fte

7alue of Jac , btai s 0o . VXLo13 Is given in ref. 1, . 3;j, Sq. V1.,2a.

rhe edwentage ofaalning the aatiaymmtria aad the apsetric coupoaente oft~evelcit dstrbutonseparately Is that If X iusoam coefficients are used in

each of the tvo trial functions *ich dencribe the resultant velocity distribution

(t4etwo determinants of =*x nt b ov.Ifthe velocity distributionisnot split In this fashion, adetezuiankt of order 2K utt be solvad to achiefe

asolution of cosparable accuracy. Wes, of course, in u Is'e3borioub.It is showan In reference 1 that when h/a, the twid.ucer 011l thickness to

radius ratio, is sufficient~ly small to mae the qTiautity

!( -V 11 « <1 in the case of Wma tostrictive transducers

the coefficients 01and 0 0 can be set equal to unity. ftia In turn aske" ttMGreen-s function ccabiratica r t.' *%-a -e in zp. mVl eqiwl. with

th~i assumption, and e~king =00'of the fact Ctthe tumetions r (z-z*) aresymmeri~c In toz invert the order ofitgain*tefunftlocal iEq. V2FI.2a can be written as

Ala~ I 3 ifc (.')r' (z-s') azl' dz (VZI15)

'3-L ~f+

When ae(z) satisfies the integral equation, Eq. VII.3 applies, and the functionalJ(oa], Eq. Vf.II- equals -Aci). If we nov compare Bqs. V11.15 and 17.1I4, we seethat (Zj3)0., i.e., the co~occent of the self-impedance associated with the radialvelocity distribution (q-a%4Qa), is proportional to the sam of the rornts, J(%i

anid Jja 1,of the characteristic equations, Eq. VII.13:

(Z) (3(CI,' 4 JEMJa) (VnI.16)

Thus ror arraj elenents 6bose ratio h/a is sufficiently small, the self-impedanceof any ele~net can I~e obtained directly from the characteristic c-tuatimn witbmioprerviously having to compute the undetermined coefficients in the trial functions.If the ratio h/a is too large for this approximation, the values of the functionals

*The expression "symmetric" zxAns that z and -z' can be interchanged, i.e., thatr(5~~)~r(1~).In Section IV, the tern "symetric" rofers to the fact that~

changing zinto -z does not cbange tbe value of rs. re is "'antisymmetric" ikn z,because chsuging the sign of z alters the sign of r,,, but it is symastric in (a-z')because interchanging these two variables do"s not alter re-.

19

_ mi

Jles mu s be substituted beck Into the correspoodg set of N heouineoo t seqwa-f B]qs. oM.320, each of vich Ii. yield (k-l) ratios of undetermined coeff-

iclents.

Feinally, t. e va3i of one i'l, r =nto urlulnm coef en sa is obtainedby a lbatet-iz %tuse ratios in the functicnal. A[ca as given by Eq. VU. oa. Asnoe ealir this fucioa equals -.4x], fer the taeue, of Cr(s) which satis-fies the Integral equation. 2herefore, settingr q. a1.11a equal to the aluneof J[(1] obtained fromt Sq. V3U.13, ve cin solve for the remsininZ undeterminedcoefficient. Th~us, for trial. functioni Involving these undeterainmd coefficient..,

ILI a(VU.17)

An alternative procedure Is to select the rcuin~ai undetermined coeffi-clents so azz to satisfy the original Integral equation itself. Unlesa the fune-tional dependence of the trial function, Sq. VI.8, on a is the correct one, thiscoefficient can not be selected so as to satisfy the Integral equation ower thewhole region z+. It is advantageous to select the coefficient so as to satisfythe equation for a value of z associated with a relatively larae value of a(€),i.e., a value of z which lies close to the active transducer element. In thecase of a tbree-element array, and for the trial function assumed in Sq. VI3.8,one =i.t elect x- so no to satisfy the integral equation at z-Leb. PraTIously,moe would substitute in the set of homogeneous equations, Eqs. V31.12, the valueof J Cij as obtained from Eq. V31.13. One can then solve for the ratios x1/23The Integral equation, Eq. IV..a, then yields

-j A(L4.b..') dz' ~.)XJ

$~~~~~ - + 'z~b-z1)d-z1+ r(Lg~b-7')[;- - .+--atiz~z

3!*2b 5 3 W~)3 r3 (),Jx2 k%)z

Once again, the subscripts "a *ad "a" have been omitted in this cqutio.Having thus deterazin.e the coefficients in term of w1ich the trial functions

are expressed, the self- and mutual impedance correction factors, (Z a~ nd(Zkcam computed by substitutltng the trial functions in Eq. IV.l!.. 7he far

field potentials can also be obtained, by using the expressions given in ref. 1,Sections VII and VIMI.

20

LiiSwas s m nioned In the introduction that the applicatioa of the variational

prInciple to the ar.-e Is less advantapow then its alpLieation to the single* squizter." - e roasoc Is pevelsely t"t, in contrast to the self-imedance1 the

mutual Imedances cannot toe obtalned d.etly fnm the value of J,.-] coute fromrq. vU.3Z, but requizr tt Bqs. v.i.2 be solved for tue undeterxied coem-

cents x%. It is possible that a iore sohbstIcated var)Lstloal arprcach, mberebythe mutual, as ven as the self-radietion impeft nmeft be fownd. d•rctly frM

stationeary fnctiomils, csn be evolved by formilating the an-ay analysls as a aul-

tlple scattering problem. In this formuat$.on the scattering cross s'fictols of

the array elements would be expressed in term of stationary funettovols constructed

in accordance with the Levine-S&bw1f.er vsriatiorml forculStIon of diffraction"pzebless. The radiatiot Je&Ances would them be relateA to 1he scattering cross

sections.

S[ 21

S|1

if

Fg1.M -TING AM OF SPMR TA=

NN

F~g 1. F +1'... A ' . 0?0 5PA) 2 MW

Ihibejo ZniNe roa( m f lneofSmt

Eent ~ of tS(N.1)/2 ofFP ! -1102 of)Se2e

elmets k bews~a@2innt, j el22,

2[

Raia Veoi"•ty tLe ai&Fig 2.hEM" FTAWE DF%_

I -

-- I

__-2_I

I I

Radial Velocity at Oute Radius a: U

Radiel Velocity et lImer Radius a1 : u1

23

J -_• II I I-_-:

k Aciee a:izu

Uactie elsmta: (ZI4U~m eact- $z-~.

Cyin)doricalSurfc roe

F~g- Vst® VAftimno GdmedTt: (sMM® ~ M Inactive lit A MY~z.

~~OV 2 h e.ct~ ~:aXE

z~zL-b

zi

® Active elearait

02 native elowent.

a;Resultanit v'elocity d_4si LIioGb 2L~b

sywretric cpCVZF lts

I i~~~~ Antisy~rntric co~ponents l:;. b(b) Transd-.zear Velocity Distritution

Pig. 4. SPLfl'M~ OF VEWCITY D=STRWIU703( ZM SMW#VMC WMV) AND~A.VMLRe C (CM)) WOS$M MMM TAM FOR I.O M&M-APJMY

25

2. D. C. Juager, AO the MIatOionImpedean. of an~t eAra e-Lof Finite lzI4udrst , JOm&bltr A out PSad. an* bpt. 7o. (.1770)0

3.J. R. Gr~eensym fh a i the flhmitdr cf Noin's amective

[ 4. J. 2. Greensapn arnd C. H. Sbarmn, "*tiul LpO~maee OWd Noar FpldI,Pressure for Pisam~w on a Cy11zmderz" rr. Acciat. goo. An. £ Ji l 6J4

t 5. D. 2!. laird mei B. Cohw, "DWrecticcaUty Patbern for Acouastic Aldiationfrom a Source cc a RigidA Cylinder,," J. Acont. Boa. An. a;. 46 (1952).

I: 6. D. H. Robey, "on the Cm-textutatic of a Ccctainsd TIsoou Liqgld to theAcoustic ft~edenca of a hEslany VIbratIng Tube," J.. Aeoust. Smc. An.

L 7. D. H. Mbbey, "On the %diation I~edence of a Liqrid-Filled Squirtingcyliudter," J. Acoust. Soc. Aa. ?I, 711 (1955).

8. N. L. Baron, A. TD. Hatthms, and 1. H. Meiecb, ced lbratics of an

[ ecosl MR Cotat .or3-3(O M Te ,ch& mpt. No *

[1 9. L. H. Chta und D. 0. Schuelkert, "Soma~ hWdltion frou an ArbitraryBody,' J. Acoust. Soc. Am. ,1.626 (1963).

I 27

Chief of Inval Beseazch P.b.-Na~val Electronics TAboratoryI ~ Dtuamt or ib NayS~De o 5, liforniavislinitos25, V). C. AtuSor.LJ Christensen

V'.~e .8 f2 -a&pofehip

Commanivz Officer Vashingtoa-.23, D. C.[ Office of ft-al Neseerch Attn: Cinpt. W. N. Cross, Code 1.03aisna Off ice495 5.Mr street W~S11 CfO~hCIittt

Boson Vaa 21 Woods Role, 11asachusette

OOfice±D of * Naa eeac hef, 3"'Qa of Naval Weaponsa

JobaLohhe Aircraftý uilin86 o adoii 62streetursPl AlttDr, Cali8ornTam RChcaod n 5, Souno riso U.S.t Naval ILr~c Test 1h ýtrti

Ceoe a ttatgon Thiilding SP*CTel PrJOmtS Offic

Office~~ura~ of Naval Reercetn:DToPiposBrahOffice of:_ Tecludaal SelceesetofteNavyh20 esat 24th tetDatet of Ccr'l the 25,Uvyc

Wonandingto 25,fi.cr Attn: LCD. W. N. Sayer, Cde40Office of Naval research (EeuJ v ereo,Bacofie Depirtuent of' the NavkrcylPoetsOfcNeavyMO le otOfc hington 25, i% C.(s.lI2ftew Dok NYrk (. O3.x Attn: Burea of SNosler eaons a 31

NavalResearyscs Laboratory Departzocnt of the Navy

Vashtto: Dr. C, 20,29Dr Washingtoni 25, D. C.Attn Cod 20M 2ec Wo t coies)Attn; x.P Weatueran coierar 6code62mK~baDiv

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Departum ofb cea f the Aray __U4

Attn: Rand D Dir'4y Cod& 34500OfieO h-Chief of Staffer Cle, bruofe Nav 112oDepartmat of the Ai~we Codrtex of U 3Wookington 25, D. C. D.68, m. *Co

Attn: Z-XL Lib. Dr., Ada.Ser Att~: XAAD Airfrome D"Lp

CO!mm~adfl, Officer-2 Chief- aclentist,

M34veer Research Dev4olomnt TAboratory UA -"A IAirframe Dr.Fotfior isnaW Au Division

UN, F&a.mkob P&.

comDA Officer ecl e' iefttetmu rmna Speial for Navel W=xce

ilatertoms, Maz O? Iep%--e. of thei wavyAttn: laborator Division cCommandlng Officer Atta: Cbde BP-001l, Chief 5clentintYhurak,%r Arsawi Cqde SP-20, Tech. Dhx--ctor

3. wmeibalrk Statics lMI Chief, Bureau of Yards and Dock,

Ph±ndephi 3, PnnslvaiaDertmee of the TavyAttr.: laboratory Division Va~ga 25, D. C.AlU.S. Army Research 01tice Attn: CoU3 70, 3q-archDuke Statica na r'redDrcoDair1am, North Caroluas Cdow~i O a an DirctoAttn: Di-v. of Erg. Sciences Xb e Ba. 20C)

Chidf of r~al-a 0peratioas A:Coa. 108, W. H. T. Mo~ldriekDepartwmnt cf the Navy Code E98, Dr. 14. StrasbargWasiieto D5 . C. Code .10 Tech. Info. Div.

-At:Cp 0?? Code 538, Dr. F. 'Adilei3miierC=wet ~I ep Code %3-, Mr. Ar 0. 5y~ez

U. 1. Y" r~ Corp 0ot -700, Dr. A. H. ZeilHeaqarttonr a Code 720, 14r. 3. -'. wooWasing-ju25,rk.Code- 731, Wr J7. 0. PMmcomoding ~TCrCode 7110, Dr. W. J7. gettauSWml Co" 762, Mr. Z. NocnnnKirtiss3- Air F~orce BaseeCd 761, Dr~. Z. BubArov,4aniqie, cvw Wxeioo Code 771, D~r. R. 7Idebowitz-6"tn: code 20 (Dr. -. X. Brna) Ceedn fcradmxtrDGUnderwater Expl.osion Res. Div. U.S. *-d Undervater Sowugd taborstorlNorfolk Naval Shiiyrd Fort 'fri*buUP -nomcuth, VirginiU Now L-ba'a~ciuAt=~n Mr. D). S. 03ben

Director Dr -- u

DAIretor-1,M .5 Dr. o X Street

Attn:. Divislau of 1kcbanics ~I5CU~t 2$

Natiuml Aeromiutica vA1 Space Aft. Profeascir A. C. ERdssom151t2 I. Streett, X. V. Dept. if, Acianutital BicaeerinVashincton 25', ib. C. - ?I2X4Im U1 iveO1tAttn: Cb$ef, Div. of Researcb Lafey-tt'), UUUMs~

I*2frmtti Dr. )kri11s Ga~dbeitDin".tor Special frojectsa. AWp34d xedfileWaticnu1 Aerwautiý* and Space Aft Or=-aAircrat 14teria COapI~SAa1y aesearch Centr Beb8,Ln I4*3NY-Inng1ty Field, VYfrgiisPofsorJ 1 &40Attn: Struetume Diviricai Nedawssor AT .Goe

B"eIn ardam of TM~tY s g~rsPaul Veldtza~r, Cottsulting Rngitaer Stanfora, Oklioriga779, Lexingto Dr.i~ Twh rk. cret3e~v York 21, NewYO* J G hMW xs Rtseiarcb Assoc.Profesaor B. H. Blieje 37V~ Oa23.vey.AvsmaDepartment of Civil Engin-eerIn Baltilmoe 12, b"MrID&Columbia Tlnireraity 13ip0Eos618 *Add Bidtwn oi~toNew York 27 ewYr 10 West 35th~ StreetProfessor B. A. Soley chicaso, mlizoisDcpbrt=ert Of Civil Enwineein Profooro W. PIV16r, (hairsornColw ial Univrsmity, F**clSine Im-i

Neu Y.~k 27, 31eu To:': onvlaeaiý, IsanProfessor Xiche-le '. Soff Pofessor Josiep&1 KempoerDept. of Aeronautticl Er~oneeriui et f eoote2.g~ernStanford University

Stanfrd, elifocand Ap~lied Reo14anicsStanfod, muarnIAPolytechni Irtsitute of Brook2.4M

Dr. C. W. Boxton 3133 Joy Street1~afese Researeb Laboratory Broodyn 1, 1ev YorkUniversity of Texas Prfso J l4 OosnerAustin 22, Te'%ss Dept. of t '±ca F riDr. M4. A. BWul and Appl~ad MhredinezEngiiieerIng Ycszc3anc Division Poelyckbcnielst itute of BrooklinVniversity of PerseYlianifl 333 Jay StreetThIlode2.phia 14, Pennsylvarda Brookjy2 1, New York

Dr. F. DMEiaglo Professor E. R. LeeDept. of Civil ftgineering Bro,= UlniversityColusbis Univers!,tr Diviioa of App-iled Mothcstic'a618 Mudd Bu~ilding Proridmene 22, RMode XAUlA.Ntv York 2-,, Isv York Prfso R. D.MdiProflessor D. C. Drucker Dept. of Civil EngineerinsDiVIS lea of Rhgineexring ColufslkivratBraum Uhiv*ri=ty 618 Mudd BizildingProvidence 12, Rxxde Islazd Rev Yoric 27, New York

mil

U7.S. Uava2. O ~~ Loartory U. S. Xavaý ProvivS GroundWh±~~k, ~r~~vn~1Dablgren, V~rginis

Ata V Tectws.tl Idrr Supervisoo of Shipbuilding, USI and

C-,: tXýAerxster 'deapc eealDnmcsCopoainES Azute ivsoEectric Boat-Division

Comondutg 0,*fi;cer (Irotou, Connecticu~t 063ý0U.S. gaval Mine Dcoc~e laboratory Co~andez.

pasa 01tV IRriavel Orduance Test Station, China lake,ComolerAttn: Thyaies Division California

U.S. Naval Air ]nave pmen$7 C~einter Machanics Divisiov

Dutetar Naval Oidneiwe Test St~tilonU.S. 1$.wy 1Vderwater Eh-Ad Ref. L4b. Undereter Ordwace Divi;IonOffice oa 14val Posefarth 3202 E~. rootIll BoulevardP. 0. Boz 83317 Pasodens 8, CaliforniaOr~azao, Flo. Ida Conod-Itg Wfictnx r DirA ectorCemanding Officer --. Dir~ector U.S. ?Zavnl 11neexTine bCpvhinent Sta.U.S. Navy Electroaies Xabkv'rto7Y Awaspolit?, IbrylandSen YDIi.So 52, 05ILfor~iis

Attn WýGm..,q S.CoLý-3a. ~e 323 U.S. Naval Postgradwate ShoC~otandr Monterey, Califor.-itP-resvmntb iiwal ShipV-rd C, re

r~1 ~ 9" ~ Air Material Comw42dco~wndarwrtF~bt-Fuattmron Air Force Base

~~ Struoturea Divis iemU-ev York 9sval S1tIpo1rd cm~auder, WADI)areookbi. New York 11251 Wright-uetterscn Air Forms Bose0 Icern-hxcNava -i-hm Attn: WW.P

U.S. raval Cons rxictio. Battaion~ centz~r~ IFurt- Humne-0, Calli1oruall i~rectxýr of Intelligene"

D~rectorHeadquýsrte=, U.S. Air lbz-'eDirectoAr aoOWaabSi~ton 25, P). C.Naval* Air Expnxeiw1Cntertti Attn: P. V. Brano1; (Aix- Tvrgets DivSuval Bsee, Philadelphia l'.. Peins CononderAttn; Structures laborat~ry Air Force Offt cc; of Sclentli& Rest.

Ofrcrz-IzChsaar washin t'on 25, D. C.Wvi Tnhlr Model. Basin Attn: l~cionica Divizionthnclerwakter L'zp1esicn Research Divtaioin U.S. Atozzic EnerjW 0=issionNorfolk Naval Ship,"rd fiashingvtz 25, D. C.port-tnaczh Vizrguzia Attn: Director of' ResearchAttn: Dr. H. Xt. Scksawr

Professor P. X. hghd± Ordnance Research laboratoryUniversity of California UriversIty of Pennsylvania

* College of iingeering P. o. Box 30, SuZe College, N•. 16801Berkeley, Caiornia o-4371 ~| ]•ele¥ Calfor~ 0•-371Dr. Rict'rd Wat~rtouse

I Professor K. 14. NevWnek, Head American UniversityDeartsent. of Civil Engineering Physi• s DepartmentUniversity of Illinois Washington 16, D. C.Urbans, 3Iinois Mr. Stan LeM

Professor F. Poblo Chesapeake Instr~wnt CorporationDepartment of Hatbezatica 1badyside, ;-rylandAdelphi College Dr. M. X. Bckus

Garden City, New York Texas ts Incor rted

Woods Hole Oeano phe Insttuti 100 Ekxeange Park NorthWoods Hole, Yhsacusetts Dallas, Texas

Professor F. "!. Pcoano Dr. K. BasinDept. of Aeronautical Engineering Hughes Aircraft Comny

sd Applied Mechanics P. 0. Box 2097Polytechnic Institute of Brook.in Fullerton, California333 Jay StreetErooklya 1, New York Mr. John Mabobey

M Resident RapresentatIvePrfestiueor E.Matemaiss l Ha~~ rvard University

N•ew Y'k taiversity Cambridge, Mbes. 02138S25 Waverly Place

New York 3, New fork UNaval Ordnance taboratory

BDeen V. L. Salerno Silver Spring, MarylandCollege of Science and Engineering Attc: Dr. D. F. Bleil

Fairlei6h Dickinson UniversityTeaneck, New Jersey

Professor A. S. VeletsosDepartment of Civil EngineeringUniversity of 1linoisUrbana, llinois

Brown UhiversityResearch Analysis GroupProvidence, Enode Island

Hydrospace Research Corporation1749 Ro_-kvllle PikeRockvllle, )•Aryl•nd

Hudson LaboratoriesCo3lXnbia Ur.versity1i5 Palisades StreetDobbs Ferry, New York

Laront .eolsdl--l ObservatoryColumbia University

W-20 CliffsPalisades, Nev York

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