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A1

REFLECTION OF FLEXURAL WAVES

AT THE EDGE OF A PLATE

BY

T. R. KANE

Office of Naval Research Project NR-064-I88

Contract Nonr-266(09)Technical Report No. 6

CU-7-53-ONR.266(09) .CE

February 1953

ABSTRACT

The re lection of straiEht-crested flenuarl wsves

at the edge of a semi=infinite p).ate is studied in terte

of a two-disemsial plate theory. It is found thats in

general, a flexuml wave propagated toward the edge at an

arbitrary angle of incidence gives rile to three reflected

waves: two flexural waves and a shear wave. A r-uber of

special casea, involving degenerate for;s of these motionsq

are investigated in detail.

I

In this paper Xlidlin's c)13 equationos of flexual mctionm of platee

are uend to study the re.OlectioL of a straight..oreted wn~ve at the edge of a

sei-n~iM~ plate. 'Me equations aacoodate three mces of motion: two

types of flexu.ral waves and a thickness-shear wave. It is found that, in

general, all three of these motions are excited upo= itoidence of any one of

them at a free edge, The shear motion, he,-* encountered, is of particular

intereatj for this motion is absent 4.n tb'3 o2.aesc i. eory of plates, wheot

t4e applicabiaity of that theory in restricted to a ran~ge of feuenciis enon-

Aidto -w~ tha~t corresponding to the first mode of thiolzs-ahtar vib~'a-

ii7~ y theswii token, the present thsom does not italude the higher modes

bT~o~io~er-to be found in ehe~~so~ laitioity theo.-Ij thuR$V%,i~ fot eovUyb expected to fuaigs -at- adequate deacrtps

'~ t~~o hi4ch do not materially exaeed that of the elrt§t thio!meaa-'

Mit 8hamcter of the reflected wavas is affeco.ed by 'ooth the antgle or

i=h'L-dee OAtf6&M the rhtio of plate thi-6msaa to wavs-lergth of the incident waves

'AO.6tint4 values of' these two parmet-era g±ie rine-to such speoial cases asI ~ Vwd whose aplituades decrease exponenti&Uly Atb -distae fr= the edge,vi :mtionst reactance, 6i-tparance of awe of the reflected waveal and, forI gvazing incidence, onplete ditappearance of t~ke motlot.

Following a resum of the plat-e equations in Section 21, traight-.oreated

waves axe consaidered in Section 3. Incident atd emergent waves are desoribed

in i"ection 4 and, in terms of these, a formal solution is reacohed in Seotion 5.

In Seotioto 6 and 7 various oases of normal and oblique incidence are discussed

in detail. Seotion 8 deals with grazing inoidence.

1. N;umrs in braoket.'s reftr to the Bibliography at the end or the pa-per.

-1-

Fox a pl~ate of thickn~essI or'iented as in~ Fig. 1, p2.at-e dtaplaoe-

.~em~ ~$/~) ~ a d 'x,l~ are given 'W (see

The functiona v(~y~~j~ and H4(ey)are goverred, respectively, by

equationB of ziotion

In the above,

j ,

-2-

D. ='-4'(6)

p is the oirculu1 frequency e saco ated with the wave motion. e are

r3apectively, Young's Modu.us, Poisson's Ratio, and the mas density of the

plate material.

Plata bading mwents A x M , shears a, d twisting toment/' A

are given by

14 is the mrdulus of rigidity o; the plate material. 7he definition of eK is

gi-Ve in []

Taking any one of the f'inotions W , or ,Y to be of the form

&rd letting the remai-ing two 7anish identioally we find that, in each aase

two of the equationj of motion are autonstica-lly aaisfied while the remaining

one requires that

:~ ~ V 0 o .~

Ltting/pL. ( )' ,

this leads to [see uations (3), (4), (5)] three possilble types of rlatio,

ohips between phase velocity C and wave number? 1

where

and C3 is the vmlocity of ahepar waves in an infinite med4ut. A plot of

versus AP (with ;. /4 ) for each of these cases is shown in ig. 3. It

for ; and P aves. In the sequel we shall call w, a "aiow" flexural wars,

a "fast" flexu--al wave and Iq a shear wave. (The fact that the preceding

disoussion involves waves propagated in the X -direction, does not resilt in

any loss of generality.)

-4-

T

! .. Incident ayn4 F nergt Wyes

Referring the semi-infinite plate to &.xes X, as shown in Fig. 2

we o anider a slow rlecral wve propagated tow.rds the edge of the plat-e,

where

yzV: Y,/,Acy XCOS CX'

We poetulate three eergent waveaq propgated respeotively towards P> ', ?'as ahown i Fig. 2. These are

a) a slow flexral wave:

b) a fa3t flavumal wave:

#. '-Iei'e "q (12)

o)a shea~r wavat

46-ra

and the c& are the aLglae of emergenue sbown in Fig. 2.

In accordance with Equations (9), these waves are propagated with

velocities &' which depend on their --aspective wave numbers

2. The ease of a faast inoident wave is similar in nAture and will not bediscuesed in the present paper.

Tha displacements correspor.ding to these .aves are obtained firo1

rquaticn (I), Denoting the incident wave by , and and callig

the displacerent componen-ts of the emergent waves Wi, V (/we observe that, since the oquacions of mot'on (2) are linear, the st-ate of

motion given by

Y

is a possible state of motio. 4 or an I.nftnite plate, For the semJ-infinite

plate under consideration, three boundary conditinns must be satisfied on the

edge X- aC,

?late stresses , / etc. corresponding to the motion defined by

Equations (%4) may be tcmputed from Equationt (7). To obtain a tractton-free

edge, we must have, on X-C)

in order that these equations be satisfied for all values of the time a (51

all values of the space variable , it is necessary that the circular fre-

quencies c and zassociated with the various waves be identical, i.e.,

a s, (16

and that the angles xe and NJ satisfy

I

Now, the relation btween C andJ i of the same forn as that between CI

atd? I both being slow floxural wavoc. Thus it follows fro= Equalions (26)

and (17) that the angle of aenrgence Lv/ of tha slow flaxural wave is equal

to the ongle of inoidenoe *' of the inoident wave, and that the wave numbers

ad phase velocities of these two wavs ar. identical. Equaions (16) and (17)

show, fu..'harmore, that the condition of vanishing tramtion Rt the edge of ths

plate% CEq. (15)]1 loads to a deermination of the phase velocity, wave nU.Mtr

and angle of emergence of each of the postulated emargent waves when the argle

of incidenie PnA w.ave number (or phase veloity) of the ircieent vsve are

specoifed.

The bounday conditions (15) also impose restrtioine on the amplitude

ratio As/A . ?ro= EquAtions (10)-(13), (14) and (7) we get, upon substitu-

tion ±itt 2uAtioe (15), a system of The no-lz.en 1ear algObT0 io

equtisna governing theae amp itude ratiocs

whera

-/I - -2-

-7-

with

'Me &nelog, in the present theor'y, to F, Neumr,=ns iniquenesn theorem

[2] gaamnteee that the above constitutes the unique solutlon of the problem.

We nay say then that a slow incident flex1ra wave produces, in general, three

reflected wves: a slow flexural, a fast fle^-arail and a shear wave.

We now proceed to study the character of the reflected waves in ta'm3

of the angle of incidence cp and the wave number p of the incident wave. We

begin by examining the aase of normal incidence, i.e., o-O,

Tha wave nmbers and of the fast reflected wave and of the shear

wave are given by C see Ens. (8) 4())

2 * (20)

j 5'

(21)

Botb ? and Jivw.ih when R51 , *ae. when jt'~~ For

?XP5< nd are imagi-nry while for >A 5- both are

real, The physical significance of % 5~ ill be discussed when the motion

correaponding to that value of the wavn number of the incident wave han been

dot-ermined, 'Me first consider

Fran Eq, (17) v'e have, for oe -go

//J

The amplitude ratios are found fro= Fq. (18)i

14A

where

- 0" 0-17 ~ 4f$~)

A ~ ,

n rid

so that is ral.

Me incident wave is given by [se Eqs. (1) and (20)]

. (23)

i -9-

The slow reflected wave [sq. (11)] becoces

sxd the fast rtflected wave [Eq. (12)] is given by

'7/

Zn the above t

'e see that the slow refleoted wave has the sane amplitude, wave-length anm

velocity as the incident gavej b. is out of phase with it. The fast reflected

"wave" is, in fact, a v - the amplitude of which decreases exponentially

as the distance from the edge of the plate itoreases. As b7 Eq. i2Z5)

no other waves are reflected.

Now consider

Squationa (22) are then replaced by

/4

-10-

ThG ircicdnt wave remains unahanged [Equations (23)], the slow refleted wave

bpeonee

'7' '/ -A e(26)X /

the rast reflected wave is given by

and the refleot-d shear wave again vanishe, (Te aml tudo ratios 14A ar4

A /A for this oase &re platted verffus Ain F'ig. 4.)

The inaident wave is seen to gi e rise to two reflected flex ral waves;

we ry exanine the manner ir, which the energy per unit length of wave-front,

per oy,'ole of the incident wave is distr.buted to the two refleoted waves. Using

the expressions for energy given in r ferenoe [439 we fmd

ereand are rspectively, the energy per cycle., per unit length of

wave front of the incident wavsq for the incident, slow reflected and fast

reflected waves. ae ratios ZF and plotted versus in Fig. 5.

-.il-

'IYe rmotio,, corresponding to

f.Jz5 " -:

may be fvarx by proceeding to the limit, as '/95-- , in either of

the preceding cases. By either method it may be verified that an incident

wave

.A '7 x¢ (28)

00

now gives rise to a 911. rflenta u,'avej

and to a thiok-ar _vibaton,

'AY

The circular £reqenoy /V for this vibrtion is [see EqS, (3), (5), (6), (8),

!cw the circular frequenoy; of the first antisyretric mode o thickness-

shear vibration of an infinite plateI according to the three-dLmensional tleory

-L2-

of elasticity, is

A

Thus it seems appropriate to lot

so that the thic ess-shear vibration in the present cast will occur at

the frequency predicted by exact theory for an infinite plate of the zame

thickness,

it is interesting to notice that the motion here being considered,

is one of the possible moes of motion of an infinite strip of thickness

and width / (see Refernoe C3), provided

!Nsmce, for at ,l51PR i e. f or /C z , the aemiinf inite plate nay

be regarded as consisting of an infinite number of independently vibrating

strips, each with its infinite dimension parallel to the edge of the plate.

To conclude the discussion of normal incidence, we consider two limit-

ing cases:

'hen the wave-length of the incident wave is large in comparison with

the thiokress of the plate, we find, by proceeding to the limit in Equations

(22)-(25), as

-13 -

II

is,

/1 a 0

These expressions show thaT the inoident wave is refleoted Vithout Ohange in

amplitude or phase velcdity, and that a vibration, confined primarily to a

region near the edge of the plate, takes place. The claasical theory of plates

(whose range of applicability is restricted to the limiting case under considera-

tion) predicts the eme results.

For wave-16gtha which are very small in am;prison with the plate thick-

ness 3 we let #i- approach infinity in Equaions (26), (27) with the result

3. The theoz7 is not expected to be good for very rhort -rvea. T i s I Lrnit -ing case is included for the aake of ccmpleteteses

-14-

'The total motion 'a

W~* 2A4eacp COe

which in~ a sezrcdng ,mve,

7.-Abligue Ingid-ongi

We have seen that f'undpAmetta2.1y different states of' motion obtain

accrding &a the circular frequency of the i-ncident wave is less than, equa

to, or greater than the oiroular frequency of thicaness-shear vibration of &n

irifinito plate. Hetce, for O~< 6e<~ r/A we shall agen eun~ina separately

the cases a/ / M

is-± aui.fvilnt to II we see from Equai'iot (20-) &rnd (21)

we pt, from Equation~ (17),

j# AV

whenc e

(For al ve.1uaes of -^,4 and , have, as notel eaxrl. j '

Substituting into te general o.preesics for the 7arioua rerlected vaves

[Equations (1I), (12), (13 )] we get fr=, Equations (1),

:- g '. vk'e)"s,.

""

jV7

where

Thease exprossions are valid for &U1 oc with the possible excaptiom of me

ie., "'9a'ing" inoidence. For that oase, the eolution of EquaTions (18) is

A. 'w- A i p ac o o

j anid '. gst, in pl.ace of Equations (31),

j -1.6-

so that t/ie entire notion 9 then [see Eq. (14

4/ ~ (32)

This ooplete disappearance of the motion is an£logous to the case of grazing

inoidenoe in th6 reflection of plane waves from the plane boundary of a semi-

infinite solid. A shall return to this cuestion later on.

The waves described by Equations (31) are

(1 " slow flexural wave, reflected at an anile equal to the anzle of

1"idnct f the iroident wsMve 9

&--. i -f a eI- wave of =Plitude decreasing 6X onentially with

~eue Tthet-edral srd Prpaogated in a dire oti= mpgil1 +-o tht edge",

e hear wiave of exponettisll.y deorea~ing eplitude, poapted stlqrg

Tulignow to the case

we note that, f£r Equations (20) and (21)

(Fig. 8 illustrates these facts for . /4,)

Equation (17) pernits us to construct a table which shows the relation-

ship between X and o'/ , N in terts of trigonometric funations of tbe e

argles.

-i '-

a: : * C a- :

<i

rrN ts '- 1 N , .- ' -, " ::' , IL I <- A* 1-N * I u~ f ':-a j Li Cz ArC c~z Ln

With the help of t.his table and Sqtitions (1O)-(J3) we can now describe the

notion assoolated with various angles of incidence.

For 01 S*Inx < 17 the ref'lcted waves are

a slow flexural wave,

a fast flexural waves

-a shear wa~ve,

These emerge at-azigles co< &x "with

as; ohovnm In Fig, 7. For Instsnce, 4wh t 5, wv& have I 4 .r2/ /

Taking 15'0! (so that simnk<,302) we get

S150arc sin - -29 59.1'

a srr sn t 2 259.593

For S, w-'a " , the ampli'.de ratios :see :s. (18)] are

Thus '.e have only two reflected -waves, (see Fi&. 9),

-1g

a slow flexural wav-e whose angle of emergence is equal to the angle

of irioidenoe of the incident wave,

C - a fast flexural wave propagated along the free edge.

tith e = 5, as in the numerical exa.nple abovej the critical anele,

o f for which the shear vave vanished, is

Letting o inorease further, until ?*< w <0 we find that

the fast flexmralw ave [Equations (12)] ahanges in character simce 4o is

tow iginary, The sheer wave reappeu:s aed we have (Fig. 10)

Q "-a low fl.exural wave:

0 an exponentially decaying fast fleriral wavef pro:-ated parallel

-4 shoa? Arve enereing at ai angle, 0 With :vd' C T/2,

Anothp ' ariticaa ralue of .g io reached when i'sa u'4. , 4e then

Sa slow Slaxumrl vavaj

0 n ae xp.Ientially deoaylnz eat flexu=rAl ave

CBoth a~C>re now propr.ated along the 5dge, itb 5 the aritiosl

value ford' -. 36L4m

Finally, for "J ( / , the ahear wave also acquires an ex-

ponentially decaying amplitude, giving us (Fig. 2.2)

- a slow flexural wavoe

- an exponentially deaying fast flexural waveg

(D _ an exponentially decaying shear wave,

Grazing incidence, i.e.j dcr= 90 , leads to vanishinS motion, as it

did for /040

To complete the diaussion of oblique inoidence, .,e consider the case

As the circul&r frequency of the iniident wvo approaches the thic'ress-

shear frequency, the amplitudes of the fast refleocted wave snd of the reflected

shear wave becoe infinite. The resonar.ce encoxuntered here is due to the fact

that reflection at the edge of the plate is equivalent to "forcirng the plate

at a frequon7 equal to a "natural" frequency. For, it nay eaeily be verified

that the vibration

• i s:motion w ohsch es the equatiors of notion (2) nai leaves 'tue edge

We'have ,eem [Eq. (12)1 that +he rave motion in a semi-infinite plate

disappeazr as the angle of incidence approach, 900 . A similar situation ob-

tains in the ease of waves reflected fro the plane '%nary of a semi-!nfinite

solid. By application of a suitable liiting process, wave motions for the

latter case have recently been found by Coodier and Bishop [41. A sirilar

lhrtiting process for the present case will now be oonsidered,

-Atting

and neglecting hieer powers of e , we have, from Equations (17),

-20-

Substituting into -quitione (le) tbs smptI*,udc ratio3 -e found to be of

the fox

A 33

Zxp&!vning the expmentials w-hich appear in the expressions for displacements,

wget Irm Equations (10)- (13) and Equation (1)

w 04

'14ex exsins£ are valid =31 when highr power of v x am. be "egleatc-,

t"-At lis fdt a rnge of distances tr= the edge whiah is sm&II in cos£is

F *= &;uations (14)t (32) and (34)) the total motion is &haracterized

b7 the expresimns

S(3)

-21

1: e now Derm.it A to beow infin~ite at lie approhea zero) and let the

approaah to infinity be sumah that the produot ,4-g z'eainb finite, we ace

that the firat tam~ of Equation (35) represents an Incident flexural wave

-while the aeoond~ torm corresponds to the wave found by Goodier and

Bishop. The third term a.nd Fouation (36) ±idiatel respectively, a flexural

ada shear wave, eadb propagated along the edge of the plate, &Mi eachphaving an eaponentially iroreasing or deoreasing amplitude, according as k

teautho? Vi'bhi-to theak Professor R, D, Hi.indlin of- Columbia

K~tr-~ ~t'u tis pf~n 6d--for -hill ho2lp Advics

r R.. Mhlt Itle~aof Rotatory Inertia w--4 Sheazon Fexunla

Motiots of IsotoPio, Eatapasf o~lo ple efncn

F2. A. E. H. Love$ Thaorry of Elasticit7, (Cambridge Univermity Press, London)

1927),J 4th WZ., p. 176.

3. R. D, Hi~dlin, "Thiknems-9iear and Flerura. Vibrations of Crystal

Plter,' Jounal of Applickd Physiopq VoL.22 (1951), p. 316-323.

4, J, N. Goodier and R. E. D Ydshop, "A Note on Critical Refleatiors cf

j Elaatio Waves at Free r-faces," Jouztal of Appliid Fnysios, Vol. 23 (1952),

-2 2-

Zoeo

Pe.2 g Incident mi.d Ergent Wave NorrAls.

1Al 9'0, 1 W/ .

________ W e/u

20 0 0 40

F-igi 3: Re~ationahip betva-en Ph.se Velooi.ty &nd ia Nuber',

I - A

I S/ow J~f/cc tea Wvo-e

fi lw dmov

I ~ f/e~'e~WYA

I1

I.

000

o,

7

Iv

0 10 2Fi. 1 Enrg arbtin o~l:nie/s

F ig. 61 Ob~liue ltaidmaji 1 3.

060

F'ig. 7: Obliqu.e Incidence, ppaosixc~

II

1 rT j -~11 I

'I-i

L I ____ I ____ I __ ____ -~ _____

0 30 40 50

Fig. 8: gave ~ RatioB,

0 r4

-Fg. IC Obiqi 7

Fle

iFlexural Y~

fI

IIII

ji

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Naval stati= oetoucsLbrtrFort Senew I Calif omis. (1) Foret ? setsimbort!

Superiintendent, U's, ava2. 'Post Ata L .~~~t(Graduate School National Advisory Oo~it-.e f~or

Annapoli, 9 Mry.and ()Aeronautics.1724 r Street, N.W.

*=Arzi -r Wahingtoj D.C.(1U.S. Naval Proing OroursD&hlgren, Vlirgina 1) National A~i'Aory Co~.ittee f'or

Coxm~ding cnera.U.S. Air Forces National Advisory Co~ittee f'or7hr, Fantagot AeromauticsWasington 25, D.C. Cleveland Municipal2 AirportAttn:. 14search & Development Cleve~nnd, Chic

Di~ision (1) Attnt J. H. Colings Jr, 1

Ccz m General1 US. Faxitime C~oAir, ,,-,orezi2 lcrmr Technical1 2ureau'igt- Aatterson Air Force Base 'Wash5ngton, D.C.

Lvtor., .*io Attn: M. 11. Rusaso()

I

Professor J. R~. A.Mdersen Dr,. F. Car-ierTovne School of ingirneerine Zra& Late Division of Appl.ied >VatYi-.tlcsUniversity of Pen~nsylvania Br own 'n ive r aI tyFhi2adephia, Fenneyl.vania () Providence, Rt-,de Island

Profmssor Lynn Beedl.e 1Y:e. Hilia L. CooperFritz Engineering !,&,om-tory 150 Ravine Avenu~e

fr Lehigh University Yonkers, New York ('5Gthlehenq Pennsylvania()

D~r, Antoine Z. I. CrayaProfessor C. 5. 6iezeno eriTeahn'sohe Hooge sehool Boite) Postale 52'Nieve Laanf '6 Grenoble, France

??Of~esor ?. L. Bsp~inPrf esstor J P.i Oniartog:irsorM .Bo ,asachusetts Inatitiite of Technologyou~i Uiest181.9 f B on-d a tol£i~ern 3 C-mrde 39 t Ystrehett )New~dg 39,k Nowt Yeork 2, e Yr

r. H~n~ , ~1ihDr, Perber Drr4eieicPoesrRL.hpgofDept. of Civil Engineering n~~ .tenc nttt,sahstsntiueo ehooyColunbia Un 'versit 7,NwYr 1

Unrde3tVsahasNw York 27 t'lo York() resoT.J oa

nF. H~s H, A.ic Dr.~of 0p. , of hr e~nd ApleFii 3u Unirtersity Instituite

De t f C v l E g n e i g Flest. at of 'h n oColumbia University ToNwYrNew York 27, N;.w York (I) Ilnis nttteo ohlo

F~~~ofeProfsso T, J. igen 1,Iinois~Poet. o J.A odnofist f hoeialv pleNarard Uftiversity Prof0hO D.cs rce

Laaete Idooaavirs!', o Iins1

Professor B. A.. Ulr'seyDept. of Civil Engineerin~g Dr.fso Lloy £cnnarClumbia Universit7 Det, of ScnicsCnuig.brtr

NoNw York 27 New York (1)os nttteo ecnlg

Presso F,~h .2 rdge= M10a Illinois iteof ecoog

Dept.of Fysic

?i~eri.~LirayDr. L. Z ,--a2ibia 'nivaraity :Dept, 0 CJivil 7ninerirng

Ne.' York 2.7, 'New York U1-nive'sity of Illinois

Prof~essor E. L, Erlksr.1:niversit.' of XMiohi~an Dr. R. J. HansenAnn Arbor, >-,ichigan (1) ehustt8 Inntitute of~ Tahr'o139

:ept, of iv. & Sarni sry Ingirneerintofeasor ., 0, Eringan Cnridga 39, Yag~ach%:satt5 W2

Illin~ois Inatitute of Teology2oTshnology Center Prolessor R., M, Her~es

D)r. 4. L. ei~erTechrnicohe Hoogenchoo1 Profeseor G. Hazt%&nNlieve Laan 76 Dept, of Civil Engieering

PrfsorA , ruenhlNew York 2.7, New York (.

Dept. of' Civil StieeringPrfso .xtti

New York 27 New Tork (J) The Tohnologca Inetittit

Profoon& B*,Feied

Wahigo ttetleProfessor s J. Magi.eThnxst o t~1Pu n &Dpt. of' i~1~~i Zake trlUa (1)f@riL

::~york 27,iversiyoDanig~ 8tr- J.r N. Jo.f Hr offt ea

Ur~v~rit ofplM I'nnay2.vani

N~d ~Mt F Fee~och P0Inerd ntitute hecd, wYo f (1)k'

Ka1nois Cmity a" 2, Nshnooogy 99) Liiesm Street

DPtofo AM.ie P. chRog=Prfeso J .Good ie on tp8n UniversityDept. of' civil~a2 Engnnoerr &1t more (1) taStan~ora University

iiw Yok2,'e okPrfsoV.L 4

I-nstttut de Mathet~atiqudis Profecior 0. T. (11. L4omyJrniversit4 Dept. of Civil giaginsertng,post. fah 55 Yale. UniversityScopje, Yugoslavia No)flw Hamn, Carnitiaut()

Professor L. S. Jacobean Dr. J. L. LubkinDept. of !imchsntea1 ?ngineerirg )'ldweot RJunrch la&titutoStard University 4049 Fenitneylvanis. AvvieStanfordq Californis (1 Kansas city 2, Mioal.(1

Pzlofssor euoe 0. Johnston Professor 3, F. I~1wSZUnivarsity od Michigan %chol of Aeronautic6Ann Arbor, Michigan () Now York University

NOV York , New4 York 0A.)Proessor K. MotterMtadford Univeruvi ty !'totesor J. N. ?A0aodrfS4anftrd, Califor-ria (1) snelaer Polytechnic Institute

Professor W. J, Kreteld ?oNwYr iDotdcivil Sncinuering rfso .W a~eo

lfttt uveraity Ute~t ~PnslaiZ16TV 7, Nov yakl Thlde.hi 9 nneyisvania (1)

1A Xsea Profeo,,or Lawrence E. lMlveme. ixAttals ftgineerinig Dept. of mteiegtiae

(1)Imp it~oug 13,ct penzv7ylvania

Miviirof-A e ah tisuiemt

froee , ieire L8) FvErvi s s ats Islan

XnsetsAer Fclrteohnic lti4tute Sae00up eryvriTroy, Nosv 7~rrk (

Professor 3. X. lAssells Ball Ttlephcs LbrtroV~s@achusatts Institute of' T8OhnoJ.OU t,4-r-ay Hill, 10snbidge 39, Ysssohusette (1)

Professor R. D. ttndlinLibnxry, En~tneering Founation Dept. of Ciw-il Lngineeriag29 West 39th Street Columbia UnivoruityNew York, Nov Thork (1) 632 West 125th Streeot

N1ew York 27, Now York (15)Professor Paul lieterDept. 010 £nsineertg or, A. rNadaiRoneselher PoJlytoehnio Institute 136 C.rry Valley RoadT-ry, New York (1) Pittsburgh 21, Ponnsylvania (1)

Dr. lieu Lo Professor Paul M. Nl.ediPurdue University !)opt. of Enneering MecheniosLa-fayatte, Indiana. (1) University of Micdlgcn

Ann 4rbor, M.iahigan (1)

ept 0f Civil nginering Amour Research F'dation(1

Professor Jesee Cmarodroyd ?rofeesor '-. H-eieeerUniversity of iohigen Dept. of MathematicsA=n Arbor, Mithigan (1 "kaausuetts Institute ov Technolof

Ccmbridge 39, Vasaohuetto 1rr. 4. CsgooiIllinois Institute of' Teohnology Professor H, RaissnerTechnology Center ?oirteohrio Institu-te t1' Srocklynchiogn 16, Illinois (1) 99 Uiviagetot Street

2'ocklyn 2f New York(1Dr. ®orge B, ?regnamC= ittee ch Uovement Aided Rastsareh Cr, Kenneth Robineon313 14w $etorial Libitry Nationa'l 91nAU of £tandardeCol=bia Unfive-rsit; aitnDC71New York 279 NeW York()

?rdfeaaw~ M. A. 8adowakyTr 1 , p tersmn 111inAm Vsettute of Teohnolog7

D~rnotor, Arplied flhysios flvielo Technology Cnter

Professor M, 3, &-andovi

;;etk&fl0R80 thb ftt~rnssd d Qnt~mbiUV Sty

Di'. A, Phillips - I.a.aa

-Sudod', CuroztA 2rokLyn 2, New York

flofeso Oenald Pitkett Dr. Dael T. Si lIept. of Meohtnios Appied IMysios o "

Uiivenity of Wiaowain JOY&n Hopkinls Vnt$M±BO't 6, Wionh 1261 Ge"CR~4 Avwnn

Silver Tprine, "l~'and )Dr. H, PotitikyQfteer Sleotrio Rargearcb Labt, Dr. 02. B. SnithSohoneotady, New York (1) Depatmenlt of M4athematios

'4alke r Ra1.Dr. loW, Pm gipr Uivrsity of Florid-,aduate Division of' Apliel M~themticq Gatnesvlle, Florida()

Srcvn UnivrsityProv7idence, Rhode Iular4i (1) Professor C. R, Soderberg

tsvachuootts Institute of TechnoloHAN,*D Corporatiorn Cmrde39qsah:aiSS O 4th StrottSanta !t nio-a) California Professor R, V, SouthwellAttni 2Dr. D. L. Judd (1 mprial College of Science and

To br lSouth Keninvton()

Lodn ,. 1inln

?rofeasor E. Steinberg Profe~sor :,- '1o1trraIllinois Institute of Teabr.:;logy Penaelaer ?olyle-chnic InstituteSaozhn olIo - ;ar Cetasr Tci e l

Chicago 16, Illinois toNwYr 1Mr. A. Z.a 1b

?rofeaor J. J. Stoker Wte stinghcuss Research Laboratoriesi:v w York UJniversity Ea4st ?ittsb-urgh, Pennaylvania ()4ashington Squ~reNow York, N ew York (1) oiessor C, , Wing

Dept. of Aeronautic~l SngineerirngNt', F,. A. Sykes 1New York UniveroityBell Te"leftone tabratoriee Umi-mriity Heights, Bronx,"Uray Hill, New Jersey (1) ew Yorkc, Nriv York()

Professor ?. 8, yrzonds Or. R, L, WgelHrovi Uiversity K70n 2Providero,1 M-Ode Island (1 Peekskill, NOW Yor'K(1

?rofesnor J. L. 3neP±'ofeear S, 3. -14eibelpibfln Itst itute ror Advnine4 Studies University of colom'ao$tocol or 1tOrtinal Fdysls Somfdert C4Sonado()

Nbun, :relaad (1 Tt eid r isteY3

Npt. §fASernutiocal mn~.nesr±g Goflege Pnrkg tln

New Yorkq New 7otk Yae1)irst

Professor S. re ?imoslonkoBoao2,l of gtg~ierIngStanford UaiversltyStanford, C&1±tonia W1

ZDr. C. A. Wi'usudeflOa4tf lnstitute for Applie 14thonias

Frofesoor Karl S. Van DyIkeDapirtaont of PhysiouSCott Laboratory

~uleynn Unieit

Dr. I, Vigness

Anacostia Stationdaobington, D.C.()

Dr. I~cra.Mo VJllenaOran Via J. Amtonio 6Matdlrid, Spain