Q U A R T E R L Y
O F
A P P L IE D M A T H E M A T IC S
E D I T E D B Y
H . L , D R Y D E N T . C. F R Y T H . v . K A R M A N
J . M- L E S S E L L S W . F R A G E R I. S. SO K O L N IK O F F
J . L . S Y N G E
H . B A T E M A N J. P . D E N H A R T O G J . N . G O O D IE R R . V. SO U T H W E L L
W I T H T H E C O L L A B O R A T I O N O F
M . A . B IO T H . W . E M M O N S F . D . M U R N A G H A N G. I. T A Y L O R
L , N , B R I I X O U I N K . O. F R IE D R IC H S W . R . S E A R S S . P. T IM O S H E N K O
Vo l u m e I O C T O B E R : 1943 N u m b e r 3
Q U A R T E R L Y O F
A P P L I E D M A T H E M A T I C S
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•201
Q U A R T E R L Y O F A P P L I E D M A T H E M A T I C S
V o l. I O C T O B E R , 19 4 3 N o . 3
T H E A N T E N N A P R O B L E M * BY
L E O N B R IL L O U IN B row n U niversity
1. In tro d u c tio n . T h e re c e n t expansion of rad io to w ard s u ltra s h o rt w aves h as a ro u sed a new in te re st in th e o re tic a l pro b lem s of electro -m ag n etism a n d especially in th e pro blem of a n te n n a oscillations a n d ra d ia tio n p ro p e rtie s. T h e ty p e of ap p ro x i
m a te discussions used b y radio engineers for th e case of long w av e len g th s is of little p ra c tic a l v alue for u ltra sh o rt w aves, w here a m ore rigorous th e o ry is needed, because th e d ia m e te r of th e a n te n n a w ire can no longer b e considered as v e ry sm all w hen com pared to th e w ave length.
Som e older calcu latio n s on ra th e r th ic k a n te n n a s h av e a lre a d y been fo un d v ery useful. M . A b ra h a m ’s1 discussion of th e v ib ra tio n s of v e ry long ellipsoids h as o ften been referred to. A com plete discussion of p ro p e r v ib ra tio n s of ellipsoids of re v o lu tio n m a y b e fo und in M . B rillo u in ’s book P r o p a g a tio n de l ’électricité (H e rm an n , P aris, 1904, p p . 314-395) w ith num erical tab les for all eccentricities, from th e sph ere to ra th e r th in ellipsoids. M ore recen tly , L. P age a n d N . I. A dam s, a n d su b se q u e n tly R. M . R y d e r2 h av e discussed th e free a n d forced oscillations of all ty p e s of p ro la te ellipsoids of re v o lu tio n ; while B arro w, 3 S chelkunoff,4 an d o th ers h av e tre a te d th e p roblem of th e biconical a n te n n a a n d its free o r forced oscillations. M ie a n d D e b y e5 h a d form erly discussed th e free v ib ra tio n s of th e sphere. In m o st of th ese p ap ers, th e th e o ry w as b ased on a c o m p u ta tio n of th e w hole field d istrib u tio n a ro u n d th e a n te n n a w ith th e p ro p e r b o u n d a ry co nditions on th e surface of th e a n te n n a . F o r a p e rfe c t m etal, for in stan ce, th e electric field m u st be o rth o g o n al to th e m etal surface.
T h e aim of th e p re se n t p a p e r is to em phasize th e p ra ctical im p o rta n c e of a n o th e r m eth o d based on th e use of re ta rd e d p o ten tials. T h e p rinciple of th e p ro ced u re w as in d icated a long tim e ag o, 6 a n d th e m eth o d w as re c e n tly ap p lied b y H a llen an d
* R eceived M a y 3, 1943. Part of a research sponsored b y th e Federal T el. and R adio Laboratories, N ew York.
1 M . Abraham, Ann. d. P hysik, 66, 435 (1898); M ath. Ann. 52, 81 (1899).
! L. P age and N . I. A dam s, P h ys. R ev. 53, 819 (1938); R . M . Ryder, A ppl. P hys. 13, 327 (1942).
3 W . L. Barrow, L. J. Chu, J. J. Jansen, Proc. I.R .E ., 27, 769 (1939).
* S. A. Schelkunoff, T rans. A .I.E .E ., 57, 744 (1938); Proc. I.R .E ., 29, 493 (1941).
5 G. M ie, Ann. d. P hys., 25, 377 (1908); P. D ebye, Ann. d. P hysik, 30, 59 (1909).
* H . C. P ock lington , Proc. C am bridge Phil. S oc., 9, 324 (1897); Lord R ayleigh , Proc. R oy. Soc., Ser. A , 87, 193 (1912); C. W . O seen, Ark. f. M at. A str. F y sik , 9, N o. 12 (1913); L. Brillouin, R ad io-élec
tricité, 3, 147 (1922).
202 L E O N B R I L L O U I N [Vol. I, N o . 3
R onold K ing7 to th e a c tu a l co m p u ta tio n of an te n n a s. T h e finite c o n d u c tiv ity of a real m etal can b e ta k e n in to ac co u n t, b u t th e re are still a few basic q u estio n s to be discussed, a n d th ese will a p p e a r m ore clearly in th e pro blem of a p erfec t m e ta l w ith infinite c o n d u c tiv ity .
T h e principle of th e m eth o d is th e following: le t us first assum e a v ery th in wire a n d call 5 a d istan ce m easured along th e wire. T h e problem is to find th e c u rre n t d istrib u tio n , I ( s , t), along th e a n te n n a wire. T o such a cu rren t, I , th e re corresponds a charge d en sity , a(s, t), b y th e condition of con serv ation of electricity
da d l
dt os
or, if we assum e th e following tim e dependence I ( s , t) = I{ s )e iui, i d l
a(s, t) = — — e“ ‘. (2)
CO ds
H ere, real co m ean s su stain ed oscillations; while p ro p e r oscillations of th e a n te n n a a rra y will yield com plex p ro p e r values co, th e im ag in ary p a r t corresp on ding to ra d ia tiv e dam p in g .
A n a r b itra ry c u rre n t d istrib u tio n (1), creates an electro m ag n etic field in th e whole space w hich satisfies M axw ell’s eq u a tio n s. T h is field can be read ily com p u ted b y th e method o f retarded potentials. In p a rtic u la r, th e field on th e surface of th e m etal wire can be o b ta in e d in th is w a y ; a n d one m ay th e n w rite th e necessary b o u n d a ry condi
tio n , t h a t th is electric field shall b e o rtho gonal to th e surface. T h is yields an integro- differential e q u a tio n which is perfectly rigorous an d whose solution is th e a c tu a l c u r
re n t d istrib u tio n req uired.
U sing re ta rd e d p o ten tials, one is ce rtain to o b ta in , a t a large d istan ce, a field d istrib u tio n corresp onding to a w ave sp read in g o u t of th e a n te n n a . I t sho uld b e em phasized, how ever, t h a t th e sam e m eth o d can not a lw a ys be used for th e co m p u tatio n of oscillations inside a closed ta n k resonator, w here th e oscillations are of th e ty p e of sta n d in g w aves an d h a v e no outside ra d ia tio n (ad v an ced p o te n tia ls m a y som etim es be needed too).
T h e p ro p e r values of th is in teg ra l eq u a tio n give th e p ro p e r frequencies (including d am p in g ) of th e a n te n n a . T h e sam e m eth o d can be used to s tu d y forced v ib ratio n s, if one assum es an o u te r electric field a c tin g on th e a n te n n a (receiving a n te n n a ) o r a ce rtain electrom otive force in serted in th e circ u it (tra n s m ittin g a n te n n a ). In th e sec
ond case, one m u s t ta k e in to ac co u n t, for th e c o m p u ta tio n of th e re ta rd e d p o te n tia ls, th e field ra d ia te d from a dipole re p resen tin g th e electro m o tiv e force.
L e t us discuss th e free v ib ra tio n s of an a n te n n a . T h e field a t a p o in t P is given b y th e well know n form ulae:
f d V d F z 1 C <**ds
■ - ’ Í
d x dt J r
v ' MoH x = —--- — ) • • • > • • • ; F — mo J '
dFz dFv .. F i*ds
d y dz
7 E . H allen, U ppsala U n iv. Arsskrift 1930, N o. 1; N ova A cta, U ppsala, Ser. 4, 11, N o. 4 (1938);
L . V . K ing, T rans. R oy. Soc. London, 236, 381 (1937); R onold K in g and F . G. B lake, Proc. I.R .E ., 30, 335 (1942).
1 9 4 3 ] T H E A N T E N N A P R O B L E M 2 0 3
h, electric field; FI, m ag n etic field; V, scalar p o te n tia l; F, v ec to r p o te n tia l; r d istan ce from th e elem en t d s on th e circ u it to th e p o in t P w here th e field is o bserv ed; <r*, i*
charge an d th e c u rre n t a t th e tim e t — r / c \ e0, ¡j.0 d ielectric c o n s ta n t an d p e rm e a b ility in v acu u m , in n o n -ratio n alized u n its (rationalized u n its in tro d u c e a 1/47T fa cto r in th e form ulae for b o th p o te n tia ls). L e t us assum e an a n te n n a consisting of a s tra ig h t w ire along th e z axis, ex ten d in g from z = 0 to z = /. W e need th e z co m p o n en t, h„ of th e electric field along th e wire a n d m u s t w rite t h a t th is lo n g itu d in al co m po nen t v a n is h e s :
T h e field a t p o in t z is th e re su lt of in teg ra tio n o v er all th e points, z', of th e a n te n n a . F in ally , we o b ta in th e condition
T h is is o u r fu n d a m e n ta l integro-differential eq u a tio n for th e s tra ig h t a n te n n a . One difficulty a p p e a rs im m e d ia tely : G is infinite fo r r = 0, z — z ' . T h is m ean s th a t one m u s t ta k e in to ac c o u n t th e ra d iu s of th e w ire; b u t w hen th is rad ius, a, is explicitly in tro d u ce d in th e calcu latio n , th e re is an a d d itio n a l cond ition to b e w ritte n for b o th ends of th e wire. H ere m o st a u th o rs do n o t a tte m p t to w rite rigorous con d itio n s; th e y are satisfied w ith ap p ro x im a tio n s correspondin g to th e prob lem of v e ry th in wires.
T h e y neglect a / l b u t keep term s in Q-1, ft-2, • • • w here
Such a p ro ced u re is suggested b y th e sim ilar ap p ro x im a tio n s used b y M . A b ra h am in his discussion of ellipsoids. I t should w ork co rrec tly w hen Q > 1 4 , w hich m eans / / a > 1000, b u t could c e rta in ly n o t b e relied upon for th ic k e r wires.
F u rth e rm o re , Oseen a n d H allen b o th use th e following a ssu m p tio n s:
T h e first co ndition, A, is n o t q u ite correct, since th e re m u s t b e a sm all c u rre n t a t b o th ends in ord er to charge a n d discharge th e term in al capacities. I t is o nly for th e case of a hollow cy lin d er t h a t th e c u rre n t w ould be ex a ctly zero a t b o th en d s; an d th is hollow pipe is a v ery special case, as shall be seen later.
T h e second assu m p tio n , B, is explained d ifferen tly b y b o th w riters. Oseen a s
sum es a current flo w in g along the a x i s of th e cylin drical w ire a n d co m p u tes th e field hz, E q . (5), on th e surface. H en ce his b o u n d a ry cond ition (5) is rig h t, b u t th e assu m p
^ _ d V dFz
dz dt i* = I ( z ,) e '<-Ut kT\ k = — = co\/eoMo , c
(4)
d l ( z ') dG(r)
dz’ dz
e-ikr
- - ¿2/(z')G (r) J dz' = 0
r = | z — s ' | .
(5)
p u ttin g G(r) =
r
I
fi = 2 log — (6)
a
A) 7(0) = 0, 1(1) = 0, cu rren t zero a t bo th ends;
B) r = [(z - z ' Y + a2]1' 2. (7)
2 0 4 L E O N B R I L L O U I N [Vol. I, N o . 3
tio n a b o u t th e ax ial c u rre n t is c e rta in ly w rong. Indeed, ow ing to th e skin effect, th e a c tu a l electric^current, in a p erfec t co n d u cto r, flows along th e surface. Oseen assum es t h a t th e field created b y th is a c tu a l superficial c u rre n t could be o b tain ed b y a ficti
tio u s axial c u rre n t. T h is m ay be rig h t for v ery th in wires, b u t th e assu m p tio n is o b viously w rong for th ic k wires or for cylinders of large radius. M oreover, it c a n n o t be p rov ed t h a t th e fictitious axial c u rre n t satisfies th e first assu m p tio n A. So Oseen h a rd ly justifies th e use of b o th assu m p tio n s A a n d B.
H allen ta k e s a d ifferent p o in t of view . H e s ta r ts from th e w ell-know n p ro p e rty t h a t th e c u rre n t flows along th e surface; b u t in stead of co m p u tin g th e field on the surface of th e sam e cylinder, h e tak es th e hz field along the a xis. T h is field m u st c e rta in ly v an ish ; a n d from th is fact, E qs. (5) a n d (7 B) follow. T h is necessary co n d i
tion, how ever, is n o t sufficient. One m a y v ery well h av e no lon gitud in al field along th e axis an d still find a longitudinal field on th e surface of th e cylinder. T h ese ap p ro x i
m atio n s w ould p ro b a b ly be all rig h t for v e ry th in w ires; b u t th e y can ce rtain ly n o t b e used for th ic k w ires, w here B is w rong an d A m u st be replaced b y a m ore e lab o rate condition, in ord er to ta k e a c co u n t of th e electric c u rre n ts a n d charges on th e flat ends of th e cylinder.
2. C om plete s ta te m e n t fo r a cylindrical w ire of finite ra d iu s . T h e a n te n n a is a solid cy lin d er of ra d iu s a a n d h eig h t I. T h e oscillations stu d ie d are tho se of cylindrical
sy m m e try w here th e c u rre n t is eq u ally d is trib u te d aro u n d th e cylinder a n d flows along th e surface. 7 (z', t) is th e to ta l c u rre n t a t z ', a n d (1/27t) 7 (z', t)d<p is th e c u rre n t th ro u g h a sm all sec to r dtp (Fig. 1); hence,
<r(z', t ) d z ', E qs. (1), (2), is th e charge p er length d z ', all a ro u n d th e cylinder, an d (\ / 2tv) adz'dtp th e charge for a sm all angle dtp. F o r th e flat to p of th e cylin der (z = 0> we call /((p ) th e to ta l rad ial c u rre n t crossing a circle of ra d iu s p; w hile cri(p)dp rep resen ts th e electric charge betw een p an d p + d p :
dtti d l i i d l i
dt dp ic dp
S im ilar definitions a p p ly fo r th e b o tto m of th e cylin der (z = 0) w ith a c u rre n t 70(p) a n d ch arg e a 0(p)dp. T h e p o sitiv e signs co rrespo nd to th e d irectio n s in d icated b y arrow s in Fig. 1. T h e co n d itio n s fo r c o n tin u ity of th e c u rre n t aro u n d th e corners, a t z = 0 an d z — l read
I fia ) = - /( /) , Jo(o) = 7(0). (9) L e t us first s tu d y th e fields a n d p o te n tia ls at a p o in t P (z ) located on the cylindrical surface. T h e p o ten tials d u e to c u rre n ts a n d charges alo ng th e cy lin d er are th e following (e,at facto rs h av e been d ro p p e d ):
i r l r 2r d l{z') e~xkr dp
Fi g. 1 .
T h e c u rre n ts along th e cy lin d er flow v e rtic a lly ; hence, th e re are no h o rizo n ta l com p o n en ts F cx, F cy of th e v ec to r p o te n tia l. T h e d istan ce r is show n in Fig. 1.
On th e flat to p of th e cylinder, th e c u rre n t flows ra d ia lly in th e h o rizo n ta l p lan e;
hence, th e F tz co m p o n en t is zero, b u t we find h o rizo n tal com po nen ts, F u a n d F iy, of th e v e c to r p o te n tia l:
1943] T H E A N T E N N A P R O B L E M 205
/
. a p J T I / ¡ ( p ) g- , k rcos <p — dp,o ^ o f ' 2tt
/ .a ~2r e-ikr
I---/ ¡ ( p ) --- sin <p — dp = 0.
o J n T 2tt
(11)
T h e tra n sv e rse co m ponents F !v, for a p o in t P in th e x-s-plane, is ob viously zero b y sy m m e try :
i r r * * d l l e ~ ikT dip
V f a ) = ---- / ---^ d p (12)
€oCO J 0 J 0 d p r Z7T
a n d sim ilar form ulae for th e p o te n tia ls F 0x an d F0 due to c u rre n ts a n d charges on th e b o tto m of th e cylinder.
, T h e tp in teg ra ls are of tw o fu n d a m e n ta l ty p e s w hich will now j be explained in connection w ith Fig. 2.
r2= (z - s' ) 2 + P2 + p' 2 - 2 pp' cos <p, (13)
^ 2 t g—ikr Gk{p, p', z — s'
Fig. 2.
1 C e~x
' ) = - dip, (14)
2x J o f 1 r 2r e~ikr
C*(p, p', s — s') = — I cos ipe/p. (15) 27T J o r
Gk an d Ck are tw o fu nctions w hich will be discussed m ore fully in section 5. T h e y are sy m m etrical in p, p ' a n d even fu nctions of s —s '. W ith th ese fun ctio ns, o u r fo rm u lae (10, 11, 12) read
1 d l i F d l
F e(z) = ---- I — Gk(a, p , z — z')dz', ioCO J 0 oz
F „ (s) = po f I{z')Gk{a, p, z — s')<fz', v o
i r a d h
Vi(z) = --- - ~ - G k{ p ' , p , z - l ) d p ' ,
^ 0
F i3;(z) = po f h C k { p ', p, z — l)dp', J 0
(16)
° a /;
w here p = a for th e p o in t P (z ) on th e cylinder.
W e now are in a position to co m p u te th e lo ng itu dinal field lit (z) a t th e p o in t P , according to E qs. (4) an d (16).
f _ dVc dFcz d V o _ dVi
dz dt dz dz
2 0 6 L E O N B R I L L O U I N [Vol. I, N o . 3
€qo) r * r d i ^ n
ht (z) = I ---Gk(a, a, z — z') + F/(z')G i.(a, a, z — z') \iz'
i J o L dz' dz J
/
’° o — 7d l i d dp —dz Gk{p', c l, z — l)dp' + I Jr a dlo o — 7dp dz — dGk{p i ci, z)dp = 0.(17)
T h is is th e first integral eq u a tio n of th e problem which corresponds to E q. (5) for th e sim plified exam ple of a th in wire. I t should be noticed im m ed iately t h a t in th e first in teg ral
a a
— G k ( a ,a ,z — z') = Gk(a, a, s — s'). (17a)
dz dz'
T h is tra n sfo rm a tio n will be v ery useful, afte rw a rd s, in a p p ly in g in te g ra tio n b y p a rts.
A n o th e r in teg ra l e q u a tio n is o b ta in e d b y w ritin g th e fa c t t h a t th e h o rizo n tal field co m p o n en t is zero a t a p o in t P ( p ) on th e to p of th e cy lin der:
d V c dV i d V 0 d F ,x dF0z a a
l i x(p ) ---> — = — >
d x d x d x dt dt d x dp
1 d i a
(18)
€qü) r L d i d
h x{p) = — 7 —Gk(a, p, I — z')dz'
i J o dz dp
r a h a r a d i 0 a
+ I — 7 — Gk(pr, p, 0)dp' + I — ■— Gk(p , p, l)dp'
J 0 op op j 0 Op op
+ f * |/i(p ')c * (p ', p, 0) + h{p')C k{p', p, 1 ) W = 0. J o
A sim ilar eq u a tio n could be w ritte n for th e b o tto m of th e cy lin d er; b u t th is is a c tu a lly n o t needed, since it reduces to (18) b y reason of sy m m etry .
T h e p ro p e r oscillations of th e cy lin d er can b e d ivid ed in to tw o g ro u p s:
sy m m etrical oscillations I i ( p ' ) = I 0{p'), 1(1 — z) = — I ( z ) , d l ( l - z) _ a/(z )
dz dz ’
(19) a n tisy m m e tric a l oscillations J j(p ') = — Io ( p ') , I ( l — z ) —I ( z ) ,
d l ( l - z) a/(z )
dz dz
T h ese tw o ty p e s will be discussed to g e th e r in th e follow ing form ulae. T h e u p p e r sign corresponds to sy m m etrical a n d th e lower sign to a n tisy m m e tric a l v ib ratio n s.
1943] T H E A N T E N N A P R O B L E M 2 0 7
3. D iscu ssio n of th e first in te g ra l eq u a tio n (17). W ave p ro pagation along th e cylinder. E q u a tio n (17) can now be w ritte n in th e follow ing w ay :
r ‘ r d l d “I
—~G k(,a, a, z — z') + k 2I(z ')G k(a, a, z — z') dz'
J o L dz dz J
r a d h d r
= - 7 7 - [G*(p'. a , z - l ) ± Gk{p', a, z)]d P' = R ( h , z). (20) J o dp dz
T h e left h a n d in teg ral co n tain s o nly v ertic al c u rre n ts, I { z ' ) , alon g th e cylin d rical b o u n d a ry ; while th e rig h t h a n d term s, R , show th e cou pling betw een th ese v e rtic a l c u rre n ts an d th e c u rre n ts or charges on b o th flat ends of th e cylinder.
L e t us in te g ra te th e left h a n d in teg ral b y p a rts , s ta rtin g from d G k/ d z ' :
r l r d 2I 1 d l
—— + m { z ' ) \Gk(a, a, z — z')dz' = —~ G k(a, a, z — z')
d o Ldz 2 J dz'
z ' - l
+ R ( h , Z) z ' - 0
= (—y ) [Gk(a, a, z — I) + Gk(a, a, z)] + R ( h , z). (21)
\ d z / z'-i
T h is new fo rm ula h as been o b ta in e d w ith o u t an y ap p ro x im a tio n s. L e t us now m ak e a few sim plifying assu m p tio n s, in o rd e r to g e t a b e tte r u n d e rsta n d in g of th e m ean in g of th is e q u a tio n .
For a very th i n a n d long voire, we m ay neg lect th e i?(/;,z ) term , as b o th charges a n d c u rre n ts on th e flat term in als becom e v ery sm all. F u rth e rm o re , a t a ce rtain d is
tan ce from th e term in als, Gk(a, a, z — l) a n d G k(a, a, z) are also v e ry sm all, since G k decreases a p p ro x im a te ly like 1/ r for large distan ces. T h e only im p o rta n t te rm is th e one on th e left, w hich h a s th e obvious solution
d2I co 2T
— + k 2I{ z ') = 0, k = - = - • (22)
d z 2 c X
T h is show s w ave p ro p a g a tio n w ith th e velo city of lig h t along th e m a jo r p a r t of th e w ire. T h is re su lt is o b tain ed u n d er th e a ssu m p tio n /iS>a a n d w ith o u t a n y re stric tio n a b o u t th e w ave length X, w hich can b e of th e o rd e r a or even sm aller; b u t it holds only for th e m edium p a r t of th e w ire, fa r aw ay from b o th e n d s. 8
T h is show s th e connection w ith th e usual e le m e n ta ry th e o ry of a n te n n a s . T h e classical discussion8 s ta r ts from th e a ssu m p tio n of sinusoidal sta n d in g w aves alo ng th e w ire, w hich cancels o u t com pletely th e left h a n d in teg ra l in eq u a tio n (21). T h en , using th is c u rre n t d istrib u tio n , th e lo n g itu d in al field along th e w ire m a y b e c o m p u te d ; an d according to (17) a n d (21) it com es o u t as
h,(z) = i ( — [Gk{a, a, z - I) + Gk(a, a, z)] + R ( h , z ) \ . (23)
e0co ( \ o z )
8 It should be em phasized, here, th at our discussion is lim ited to th e case of oscillations w ith cylindri
cal sym m etry (see beginning of Section 2). V ibrations w ith nodal lines parallel to th e axis are not included.
* L. Brillouin, R adio-électricité, loc. cit.
J. A. S tratton, Electromagnetic theory, M cG raw -H ill, N ew Y ork, 1941, pp. 4 5 5 -4 6 0 . Stratton uses rational units, hence a 1 / 4 t factor before th e integrals, and he uses th e opp osite sign before i.
2 0 8 L E O N B R I L L O U I N [Vol. I, N o . 3
T h is p lay s th e role of a sm all a d d itio n a l av e rag e im pedance Z along th e a n te n n a , w hich can be defined b y
T h e real p a r t of Z is called th e ra d ia tio n resistance, Z r, a n d th e expression Z rI~ rep re
sen ts th e energy, W , ra d ia te d a t larg e d istan ce (see S tr a tto n , p. 458), from w hich th e d am p in g of th e a n te n n a oscillations m a y be com p u ted . F o r a v e ry th in wire, one m ay n eglect th e te rm R ( h , z), w hich re p resen ts th e role p lay ed b y th e c u rre n ts a n d charges on b o th flat te rm in a ls of th e w ire; an d one m ay ta k e for G k th e expression (1/ r ) e ~ ikT as in E q. (5). W ith these ap p ro x im atio n s, o u r eq u a tio n (23) becom es id en tical w ith S tr a tt o n ’s E q . (76a), p. 457.
I t should b e noticed th a t E q . (23) is physically w rong, as we know in ad v a n ce t h a t th e lon g itu d in al electric field along th e wire is zero. T h ese eq u a tio n s (23) a n d (2.3a) m erely re p resen t a second a p p ro x im a tio n in a sy stem of successive ap p ro x im a
tio n s s ta rtin g from (22). A n a tte m p t will b e m ade, in th e n e x t section, to b u ild u p a co n sisten t sy stem of a p p ro x im a tio n s of sim ilar s tru c tu re .
R e tu rn in g now to E q . (20), we m ay tr y a n o th e r in te g ra tio n b y p a rts , s ta rtin g from d l / d z ' , w hich yields
L e t us again discuss th is e q u a tio n for a v ery th in w ire. T h e term R ( I i , z) rep resen ts th e role of b o th term in als an d m a y b e neglected, I ( z ' ) is zero a t b o th ends (z' = 0, z ' = l ) , a n d co n seq u en tly all th e rig h t h a n d term s are zero. T h is tra n sfo rm a tio n is v ery closely connected w ith th e one used b y S chelkunoff an d F e ld m a n10 in a re cen t p ap e r. T hese a u th o rs discuss th e problem of forced v ib ra tio n s in a tran sm issio n a n te n n a , in stead of th e free v ib ra tio n s w hich we h av e in m ind. T h e y use b o th ap p ro x i
m a tio n s (7A ) an d (7B) of Oseen a n d H allen a n d ta k e fo r G th e sim plified expression (l / r ) e~ ikr, E q . (5). T hese ap p ro x im a tio n s m a y a p p ly for a v e ry th in w ire. F u r th e r m ore, th e y sp lit th e ( l / r ) e - iir function in to its real an d im ag in ary p a r ts before p e r
form ing th e in teg ra tio n b y p a rts . T h e ir final re su lt is a c tu a lly id en tical w ith th e one d erived from th e elem e n ta ry th e o ry a n d E q . (23). T h is is n o t su rprising , as b o th m eth o d s are v ery closely connected.
4. P rinciple of a m eth o d of successiv e ap prox im atio ns. As s ta te d in th e p receding section, i t seem s possible to b u ild up a m eth o d of successive ap p ro x im a tio n s in o rd e r to solve E q . (21) along a w ay ra th e r sim ilar to th e one followed in th e classical ele
m e n ta ry discussion.
F irs t of all, we m ay sp lit th e in tegro-differential e q u a tio n (21) in to a n integral e q u a tio n a n d a differential e q u a tio n , b y w ritin g :
o
(23a)
d
= 7(z') — Gk(a, a, z - s') + R ( h , z). (24)
10 S. A . Schelkunoff and C. B. Feldm an, Proc. I.R .E ., 30, 511 (1942).
1 9 4 3 ] T H E A N T E N N A P R O B L E M 2 0 9
f F(z')Gk(a, a, z — z')dz' = R '(z ), (25)
J o
w here R '{ z ) = ( d I / d z /) 1’„i[Gk(a, a, z — l ) + G k(a, a, z )]+ 2 ? (J i, z),
^ - + m ( z ' ) ^ F { z ' ) . (26)
dz 2
T h e first e q u a tio n is an in teg ral eq u a tio n of th e first kind, w ith th e kernel G*(z — z').
Its solutio n can be w ritte n w ith th e help of th e resolving kernel J J t ( z '—z"), which satisfies th e following conditions
f l Gk(z - z ' ) H k{z' - z")dz' = 3(z - z"), (27) j 0
F (z') = f ‘ R '(z" )H k (z' - z")dz", (28)
*7 o
w here 8 m ean s a d e lta function. H ence, th e first q uestion is to b u ild u p th e resolving kernel H k, a problem for which som e general m eth o d s h a v e been developed. T h is b eing done, we are left w ith E q . (26) to which we ap p ly th e u sual R ayleigh-S chrod- inger m eth o d of successive ap p ro x im atio n s. L e t us first n otice t h a t th e Gk function becom es v ery large for z = z ' w hich, according to (27), m eans t h a t I l k is sm all. T h u s we m ay re w rite (26) a n d s ta te explicitly b y an e coefficient th e sm allness of th e rig h t h a n d te rm :
a 2/ b k 2I{z') = eV{z'), F = tip. (26a)
dz' 2
T h en we use th e following expansions:
/( s ') = /„(z') + t h { z ' ) + 6 2/ 2 ( z ' )
2 2
k2 = ¿ 0 + e x i + « X 2 ■ • •
(29)
an d o b tain th e successive ap p ro x im a tio n s:
d2I a 2
— + .* o /° = °, c)2/
— ~ + k l h = - X x h + V , ( 3 0 )
dz -
d2I 2 2
+ ¿0^2 = — X 2 I0 — X \ I \ ‘ ‘ ‘ dz'2
Jo is a sinusoidal fu n ctio n , as in th e elem e n ta ry tre a tm e n t, Jo = A sin ¿0(s' + f)
w here th e f c o n sta n t is necessary in ord er to give a sm all b u t finite value for th e c u rre n t Jo a t th e b o tto m of th e cylinder (z' = 0). T h is is n eed ed for th e ju n c tio n w ith
210 L E O N B R I L L O U I N [Vol. I, N o . 3
th e c u rre n ts on th e low er fla t end of th e cylinder. B y sy m m etry , th e correction a t th e u p p er end m u st also b e f ; hence,
A 2t
ko (I + 2f) = mr, I -f- = n — , ko = — • (31)
2 Xo
T h e c o n sta n t f will be d eterm in e d b y m eans of th e second in teg ral e q u a tio n (18) for th e flat term inals. N ow le t us tu rn to th e second eq u a tio n (30). As is well know n, it is necessary for th e rig h t h a n d term to be o rtho go nal to th e solution of th e h o m o geneous e q u a tio n , which m eans
' i
sin k0(z' + f) [ — Xiko + <p\dz' = 0 o
o r ;
^X i f sin2ko(z' + l ) d z ' = j <p sin k 0(z' + i ) d z \ (32)
^ 0 ^ 0
/
j 0
T h is yields th e correction x i to th e p ro p e r value k 20. I t is read ily seen th a t eq u a tio n (32) is v ery sim ilar to th e relatio n (23a) used in th e elem e n ta ry th e o ry to o b ta in th e av erag e “ra d ia tio n re sista n c e ” of th e a n te n n a an d th en ce th e d am p in g coefficient in th e p ro p e r oscillations. T h e im p o rta n t p o in t, how ever, is t h a t e q u a tio n (32) co n tain s
<p, w hich is n o t R ' b u t is com p u ted from R ' b y m eans of (2 8 )-(2 6 a).
Once x i is o b tain ed , th e second eq u a tio n (30) can be solved; th en X2 is first com p u te d b y a sim ilar o rth o g o n a lity con dition, a n d so on. H ence, th e w hole p rocedu re should yield a solution along lines parallel to th e elem e n ta ry tre a tm e n t an d show how fa r th e usual form ulae can be tru s te d .
W e m ay alre ad y go one ste p fu rth e r an d w rite th e general expression of th e fu n c
tion F ( z ') on th e basis of E qs. (25) a n d (27):
* p V ) = F (s') = f 1R'{z")IIk{z' - z")dz"
J o
= [5(3' - I ) + 5(3') ] + f ‘ R(di, z " ) H k{z' - z")dz". (33)
W / . ' - i d o
T h e 5 fu nctio ns a p p e a r here a u to m a tic a lly , because Gk is an even fu nctio n of (s — z '), an d so is H k for z ' —z "; hence, th e integral in (28) comes o u t as
J
Gk{z" - l ) I I k{z' - z")dz" =J
Gk(l - z " ) H k{z'r - z')dz" 5 V - I) acco rd in g to (27).W e can use th e new expression (28) for th e discussion of som e sim plified exam ples.
L e t us s t a r t w ith th e wire o f va n ish in g radius. T h e whole R ( I i , z") term , w hich re p re sen ts th e term in al effect, d rops out ; a n d we are left w ith an eq u a tio n
~ + P I ( z ' ) = F{z') = ( ^ j ) [5(3' - I ) + 5( s ') ]. (34)
OZ L \OZ / z'„l
from (26) a n d (28). T h e condition on b o th term in als is o bv iou sly 7(0) = / ( / ) = 0;
hence f = 0 in (31), which resu lts in th e following e q u a tio n :
1 9 4 3 ] T H E A N T E N N A P R O B L E M 211
I a = A sin k 0z', k Q = mr/I. (35)
n = 2 m + l : sy m m etrical oscillation, s i g n — = ( —l ) n in b ra c k e t, n = 2m : an tisy m m e tric a l oscillation, sign + = ( —1)" in b ra ck et.
ep(z') = F (z') = A k o [( — l ) ’‘5(z' — ¿) + 5 (z')] as cos k 0l = ( — 1)" and E q. (32) reduces to 1
exi — = h 2
= ko [( l ) n sin ( k 0l) -f- sin (£0O)] = 0 (36) which gives no d am p in g a t all. T h e physical ex plan atio n is th e follow ing: a finite a m o u n t of en ergy is ra d ia te d p er second; b u t th is does n o t m ean a n y d am p in g of th e oscillations, because th e energy ac cu m u lated in th e field aro u n d th e wire is infinite.
As a m a tte r of fact, b o th electric an d m ag n etic fields are infinite as 1 /r n ea r th e wire of infinitely sm all radius. T h e sq u are of th e field is of th e ord er 1/ r 2; an d th e energy is o b tain ed b y m u ltip ly in g b y 2-irr dr a n d in te g ra tin g w ith resp ect to r, which gives lo garithm ic infinite term s. T h e situ a tio n is sim ilar to th e one o b ta in e d in a circu it w ith infinite L, zero cap acity , an d finite resistance R , w hich yields a negligible d a m p ing coefficient R / 2 L .
T h is shows th e difficulties involved in th e assu m p tio n (7A), as p u t fo rth b y Oseen a n d H allen. W hen such a condition is used in th e rigorous E qs. (25), (26), it leads d ire c tly to (36) an d yields p ra ctically no dam pin g.
Such is also th e case for a hollow cylinder. H ere again, th ere is no end effect, no term inals, no R term , an d condition (7A) holds good. T h e w hole p ro ced ure from (34) to (36) re p eats itself an d show s again no d am ping. Of course, th e G k a n d H k fu nction s w ould differ m a te ria lly in b o th cases; b u t these
fu nctions h av e been elim in ated from E q. (34) an d finally dro p out.
T h e ex p lan atio n is sim ilar to th e one given for th e th in w ire, b u t n o t q u ite so obvious. T h e p ro b lem of a hollow cy lind er of indefinitely sm all th ic k ness m u st be considered as th e lim it of a cylinder of finite wall thickness, as rep resen te d in Fig. 3.
On such a cylinder, one should ta k e in to ac co u n t, se p a ra te ly , a c u rre n t I i flowing along th e ex tern al surface of th e cy lin d er an d a n o th e r c u rre n t /,• along th e in te rn a l surface. A t th e lim it, th ese tw o c u r
re n ts m erge in to a single one, for w hich th e th e o ry indicates a sinusoidal d istrib u tio n . H ence, for a cy lin d er of finite thickness, th ere c e rta in ly is a
c u rre n t flowing aro u n d th e edge of th e cylinder, as show n in Fig. 3. On th is edge, one m u s t also consider th e electric ch arg e; an d th is resu lts in an ac cu m u latio n of electric fields an d of electro-m agnetic energy near th e cylinder, while th e en erg y ra d i
a te d p e r second a t large d istan ce rem ains finite. H ence th e d am p in g becom es negligible.
T h e re su lt is general a n d applies for a n y hollow cy lin d er of ind efinitely sm all th ic k ness, w h a te v e r th e sh ap e of th e cross-section m ig h t be. T h e field d is trib u tio n inside
\ \ / X "
E l e c t r i c
I« Ix Ii
Fi e l d
Ie
C U R R E N T S
Y \ \ s yV
Fi g. 3 .
f [ ( — l ) ’‘3(z' — l) + 5(z')] sin koz'dz' j 0
th e cylin der should correspond to a superpo sitio n of E 0 w aves (tran sv erse m ag n etic) an d should show a stro n g d ecay from b o th ends dow n to th e m iddle p a rt of th e cylinder, especially w hen th e d ia m e te r of th e cy lin d er is sm all com pared to th e w ave leng th.
T hese tw o sim ple exam ples show th e im p o rtan ce of th e role p lay ed b y th e sh a p e of b o th term in als an d th e d an g er of using assu m p tio n s like (7A) o r (7B).
5. S om e im p o rta n t fo rm u lae. W e h av e in tro d u ce d in (14), (15) tw o fu n d a m e n ta l fu n c tio n s:
1 r 2x e~ikr
Gk(p, p' , f ) = — I dip, f = s — z', 2ir J o r
• 2r p—ikr
21 2 L E O N B R I L L O U I N [Vol. I, N o . 3
1 /"• 2 r g - . t r
Ck(p, p\ f) = — I COS <pdip, 2ir J o r
(37) r2 = Î 2 + P2 + p'2 — 2pp' cos tp = q — 2p cos <p, } à 2p,
q = 2 + P2+ p '2, p = PP'.
F ro m these relations, we see t h a t G k a n d Ck d epend upon p, p', f on ly th ro u g h th e tw o com binations p a n d q. F u rth e rm o re , it is easily p ro v ed th a t
9Gk dCk 1 r 2x 1 d f e~ikr\
■— - = — 2 ---- = — — I — — ( --- ) cos ipdip. (38)
dp dq 2ir J o r d r \ r /
Ck an d Gk being b o th zero a t infin ity , th is can be w ritte n as dGk
2 J q dp
c * = ~~ f ~7~ dq- (39)
2 J 0
T hese in teg rals are closely co n n ected w ith th e com plete ellip tic in teg rals K an d D ,11 as is seen for a th in w ire w hen th e ra d iu s a is sm all co m p ared .w ith th e w ave length (ka sm all). T h e following expansions can b e used:
r = \ / q — 2p cos ip = \ / q + [ \ / q — 2p cos tp — v 7? ]
r , . (40)
e~'kr = e~ikVq{1 — iklxG } — 2p cos tp — \ /q\ ■ • ■J.
T h e b ra c k e t [ ] is of th e o rd e r of m a g n itu d e of a, a n d its p ro d u c t w hen m u ltip lied b y k is sm all:
- ( 1 p 2r 1 + i k \ / q )
Gk = e~ ik'/q<,— I --- ^— dtp — i k ■ • ■ >
i.27r J o [? — 2p cos <p]112 )
_ ( 1 r 1 + i k \ / q )
Ck = e ~ 'k q\ — I --- — cos ipdip — 0 + • • • z •
(2it J o [q — 2p cos ^>]1/2 J
W e m a y w rite
q — 2p cos tp = (q + 2 p ) ( l — k2 sin2yp)
4 p ip — ir
k2 = --- , yp = ---
q + 2p 2
H ence
(41)
( 4 2 )
11 E . Jahnke and F . Em de, Tables of functions, 2nd ed., Springer, Berlin, 1933, pp. 127-145.
1 9 4 3 ] T H E A N T E N N A P R O B L E M 2 1 3
f 2r d>p 2 Ç Tl2 -— # _ w ^
J o [g - [g - 2p2p cos *]«* _ [g + cos g»]1' 2 [g + 2 * ] 1/22/>]>'2 J _ r/2 [l - k73 * • “ 2 sin2 ^ ] 1' 2= r [g + ' 2p]'l* ^ a n d
L 7T
1 + iÆ\/g [? + 2 p ] 112
Gk = e - ik^ I — A(k) - ik (43)
W hen ¿'—>0, th e v ariables q an d p re ta in finite v alu es; b u t w hen a t th e sam e tim e p = p ' , q = 2p, th e n kis 1 an d K is lo g arith m ically infinite. T h is could easily b e fore
seen an d does n o t m ak e a n y special tro u b le in th e in teg ra tio n s. T h e second integral C k
is tran sfo rm ed in a sim ilar w ay:
/
■ cos >{>d<p 2 r 1/2 2 sin2p — 1o [g — 2p cos .p] 1' 2 ~ [g + 2¿ j1' 2
J
_ , /2 [l - k2 sin2* ] » '2^4
= 7--- r ~ 2 D M - # (* )]> (44)
[g + 2 p ] 1121 J
_ C 2 1 + i k y / q . . )
C , - *-**•■• { - • f i n « - * ( « ) ! • • • } .
T hese a p p ro x im a te form ulae should b e used for a th in wire an d re p re se n t.th e first tw o term s in an expansion w hen a / A is sm all b u t n o t negligible. F o r th e fu n d a m e n ta l v ib ra tio n , X is of th e o rd e r of 21 (tw ice th e length of th e a n te n n a ). H ence using th e expansions (43), (44), one should b e ab le to go one ste p fu rth e r th a n Oseen or H allen , w ho com pletely neglected a / l an d w ere satisfied w ith k eeping term s in 12-1 , 12~2, w here
12 = 2 log — • I (6a)
a
T h is p a ra m e te r com es in, w hen in te g ra tio n s a re p erform ed on D a n d K for kn e a r 1, 9 - 2 P f2 + (p - p') 2
g + 2p f2 + (p + p' ) 2
small; f = z — z'.
T h is h ap p e n s w hen z a n d s ' are n e a rly equal for tw o p o in ts on th e cylind rical surface p = p ' = a. I t h ap p e n s again for tw o p o in ts on one of th e flat term in als, w hen s = z ' = 0 or /, an d p is n e a rly p '. In such cases, K an d D are re p resen ted b y th e follow ing ex
pansions (Ja h n k e-E m d e, p. 145) A - 1
K = A - \---Kn • • • , D = A — 1 + f(A - £ )«'2 4
4 (z - z' ) 2 + (p - p' ) 2
A = log — = log 4 — i log k' 2 = log 4 — \ l o g --- (* - s' ) 2 + (p + p' ) 2
(45)
In te g ra tio n an d av erag in g process carried o u t on A will in tro d u ce th e p a ra m e te r 12.
F in ally , le t us discuss th e dependence on k of b o th fu n ctio n s Gk an d Ck• F rom th e definition itself (37), it is seen t h a t b o th fu n ctio n s can b e expressed in term s of
G u C i correspo nding to £ = 1,
214 L E O N B R I L L O U I N
k r e~ikT
Gk = — I d<p = kGi(kr) hence:
2ir J kr
Gk(p, p' , f) = kGi(kp, V , ¿f) = kGi(k~q, k 2p ) , C k(p , p ', f ) = « 7 i ( * p , V , * r ) = ¿ C r iU 2?, ¿ V ) .
T h e sam e decom position can be seen from th e expansions (41).
6. C onclusions. T h e preceding sections show clearly th e im p o rta n c e of th e role p lay ed b y b o th end-surfaces, w hose ex act sh ape sh ould b e ta k e n in to consideration v ery carefully. W e h av e show n, on th e exam ple of p lan e te rm in a tio n s, t h a t th e p ro b lem consists in finding two unknow n c u rre n t d istrib u tio n s, one for th e cylindrical surface an d one for th e (sym m etrical) term in al surfaces, an d th is requires solving two integ ral eq u atio n s. T h is is th e essential difference from th e problem s of th e p ro p e r oscillations of one closed algebraic surface, such as an ellipsoid. F o r p lan e te rm in a tions, a com plete stu d y of eq u a tio n s (17) a n d (18) should b e affected, an d th e su c
cessive ap p ro x im atio n s should b e w orked o u t sim u ltan e o u sly on b o th eq u atio n s.
O th er shapes of end-surfaces, like half spherical o r half ellipsoidal term in als, w ould ce rtain ly yield q u ite different results. A discussion of th is prob lem is n o t a tte m p te d in th e p re sen t pap er, th e aim of which w as m erely to offer a precise s ta te m e n t of th e m ath em atica l th eo ry of a n te n n a s an d to em phasize som e difficulties w hich seem ed to h ave been overlooked b y previous a u th o rs.
215
S T A B I L I T Y O F C O L U M N S A N D S T R I N G S U N D E R P E R I O D I C A L L Y V A R Y I N G F O R C E S *
BY
S. L U B K IN AND J. J. ST O K E R N ew York U niversity
1. In tro d u c tio n . I t is a well know n fa c t t h a t a rigid b o d y hinged a t one en d an d sta n d in g v ertic ally can be p u t in to sta b le equilib riu m b y ap p ly in g a v ertical periodic force of p ro p e r frequency a n d a m p litu d e a t th e low er end. T h e differential eq u a tio n for sm all oscillations of th e rod is a lin ear hom ogeneous eq u a tio n w ith a periodic co
efficient— it is a M a th ie u eq u a tio n if th e ap p lied force is a sim ple sine or cosine fu n c
tion of th e tim e. S ta b ility of th e rod would req uire t h a t all so lution s of th is eq u a tio n be b o u n d e d ; it is f ound t h a t th is is th e case if th e freq uen cy an d am p litu d e of the ap p lied force are p ro p erly chosen. A m ore co m plicated problem of th e sam e general ty p e in a system w ith m ore th a n one degree of freedom h as been considered b y G. H am el [4 ]1; lin ear differential eq u a tio n s w ith periodic coefficients p lay th e essen
tial role in th is case also.
W e shall be in te re ste d here in analogous problem s in elastic system s w ith infinitely m a n y degrees of freedom . One of these is th e problem of th e colum n u n d e r periodic com pressive forces F(t)' app lied a t th e ends of th e co lu m n.2 T h e an alog ue of th e problem s m entioned abo v e would be as follows: th e force F(t) consists of a c o n sta n t p a r t P plus a periodic p a r t I I cos cot. S uppose t h a t P w ere a com pressive force larger th a n th e low est com pressive load (th e E u ler load) for w hich th e colum n in th e o riginal u n b e n t position is in stab le. T h e questio n is, th en , w h e th e r o r n o t I I an d w can be chosen in such a w ay t h a t sm all m otion s in th e n eighborhood of th e u n d e flected position are sta b le ones. W e shall see t h a t th is can alw ays be done, th o u g h , as one would expect, th e q u a n tity I I m u s t b e chosen so t h a t th e to ta l force F(t) falls below th e E u ler value d u rin g a t least p a r t of th e tim e. H ow ever, th e tim e av erag e of F (over a cycle) m ay be v ery m uch larger th a n th e E u ler load. On th e o th e r h a n d , it is q u ite possible t h a t th e colum n m ay b e instable w hen P is a com pressive force sm aller th a n th e E u ler load or w hen P is a tension ra th e r th a n a com pression, if I I a n d w are p ro p erly chosen.3 F rom th e p o in t of view of th e p ra c tic a l ap p lica tio n s these la tte r possibilities are c e rta in ly th e m ore im p o rta n t ones. F o r th e case of th e colum n w ith pinned ends we give diagram s which m ake it possible to decide w h e th e r th e colum n is stab le or n o t u n d er a n y of these circu m stances. T h e s ta b ility of th e stre tc h e d strin g u n d er a tension which varies perio dically in tim e is also considered.
In all of th ese problem s th e M ath ieu eq u a tio n4 (m ore p ro p erly , a sequence of
* R eceived April 9, 1943.
1 N um bers in square brackets refer to the bibliography at the end.
3 A special case of this problem has been treated by I. U tida and K. Sezawa [16].
3 Analogous problem s for plates under loads in th e plane of the plate have been considered by R . E inau di [l ].
* W e consider alw ays th a t th e applied forces are sim ple harm onic fun ctions of th e tim e— otherw ise w e should have to deal w ith th e more general H ill’s equation.
2 1 6 S. L U B K I N A N D J . J . S T O K E R [Vol. I, N o . 3
M a th ie u eq u a tio n s in th e contin u o u s system s) play s a ce n tral rôle, since th e decision as to s ta b ility d epends upon th e c h a ra c te r of th e solution s of such eq u a tio n s. F o r th is reason a b rief su m m a ry of th e m ain facts concerning th e so lu tio n s of th e M ath ieu e q u a tio n is included here. A b rief tre a tm e n t of th e M a th ie u e q u a tio n w ith a viscous d a m p in g te rm ad d e d is also included because of its im p o rtan ce for th e s ta b ility problem .
2. T h e colum n u n d e r periodic axial fo rces a t its en d s. W e m ak e th e assu m p tio n s t h a t are cu sto m ary in dealing w ith th e tra n sv e rse oscillations of th in rods. Of these, th e p rincipal ones a re : 1) th e rod is a n in itially s tra ig h t uniform cylinder, 2) th e lateral deflection w (Fig. 1) an d th e cross sectional dim ensions of th e b eam are sm all in com parison w ith th e length I, 3) all stresses rem ain below th e p ro p o rtio n a l lim it,
F i t ) F ( t )
- X -
Fi g. 1.
4) th e effects of sh ea r an d ro ta ry in e rtia are negligible.6 In ad d itio n , we assum e t h a t th e colum n is su b jecte d to axial forces F d epending on th e tim e t an d a p p lie d a t th e ends of th e colum n; these forces are co u n ted p o sitiv e w hen th e y are tensions.
W ith these assu m p tio n s th e differential e q u a tio n for th e la te ra l deflection w (x , t) is well know n to be as follows:
d*w d2w d-w
e i F (t) — + m — = 0. (2.1)
d x 4 d x 2 dt2
In th is e q u a tio n E an d I are Y o u n g ’s m odulus of th e colum n a n d th e m o m en t of in e rtia of its cross section, a n d m is th e m ass p er u n it len g th . In w h a t follows we assum e alw ays t h a t F(t) is given b y
F(t) = P + I I cos 2tt//; (2.2 )
i.e., it consists of a c o n sta n t p a r t plus a h arm o n ic co m p o n en t of a m p litu d e H an d fr e q u e n c y /.
I t should be p o in ted o u t th a t th e d eriv a tio n of (2.1) involved a ta c it assu m p tio n n o t included am ong those e n u m e ra te d above. T h is w as t h a t th e forces F (t) ap p lied a t th e ends of th e colum n re su lt in forces th ro u g h o u t th e colum n w hich are, to a sufficiently close a p p ro x im a tio n , in d e p e n d e n t of x. W e proceed to show th a t th is assu m p tio n is w a rra n te d u n d er th e circum stances n o rm ally en c o u n tered in p ractice.
T h e d ifferen tial e q u a tio n for th e lo n g itu d in al d isp lace m e n t u ( x , t) of th e rod is
d2u d2u
E = p , (2.3)
d x 2 dl2
in w hich p is th e d e n sity of th e rod. T h e to ta l force F tra n s m itte d th ro u g h a n y cross section of th e rod of area A is given b y
5 T hese effects could be taken into account w ithout difficulty, bu t nothing new in principle would result.
1943] ST A BIL ITY U N D E R PER IO DICA LLY V A R Y IN G FORCES 217 du
F = A E — • (2.4)
d x We assume as boundary conditions
u = 0 at x = 0, (2 .5 )
and
du
F = A E — = P + I I cos 2ivft at x = 1/2, (2.6) d x
the origin of coordinates being taken at the midpoint of the rod in order to take advantage of sym m etry. W e seek the forced oscillation and neglect the free oscilla
tion. The result for the quantity F is readily found to be cos X*
F (x , t) = P + I I — — cos 2irft, (2.7)
cos (A//2) with
X = 2»f ( p / E y i \ (2.8)
It is convenient to introduce the fundam ental frequency /o of the free longitudinal vibration of the rod which has a single node at the center. T his is given b y
f0 = ( 1 / 2 0 ( £ /p) 1 / 2 . ( 2 . 9 )
Upon introducing this into (2.7) we obtain
cos ( r f x / f o l )
F ( x , t) = P + H - y cos 2 r ft. (2.10)
C O S ( i r / / 2 / o )
If / is small compared with f0 it is clear th at F will be nearly independent of x . For steel or alum inum ( E / p) 1/2 = 17000 ft./se c ., while for brass, concrete, stone, or wood this quantity is about 12000 ft./se c . For any column of usual length / 0 will therefore be of the order of 500 cycles/sec. or more. Hence if the applied axial force F ( t) is one of frequency below say 50 cycles/sec. it is reasonable to assume that the variation of the axial force with x m ay be neglected.
We introduce new independent variables replacing t and x in (2.1) b y the equations i? = 2wft and £ = irx/l. (2.1 1) In addition, it is convenient to introduce new parameters as follows:
P b = t2E I / 1 2, e„ = P e / E A , (2.12)
p = P / P E, h = H / Pe. (2.13)
T he quantity P E is the negative of the Euler load for the column and t0 is the tensile strain due to th at load. T he quantities p and h are the ratios of the constant part and of the am plitude of the oscillating part of the applied load to the negative Euler load. W ith these new quantities the differential equation (2.1) becom es
218 S. L U B K IN A N D J. J. STOKER [Vol. I, N o. 3 The q u an tity /o is the fundamental frequency of longitudinal vibration of the column given b y (2.9).
The general problem which we wish to investigate can now be stated: for given boundary conditions there are certain values of p, h, and / for which all solutions w(£, t?) of (2.14) remain bounded when arbitrary initial conditions are prescribed and other valdes of these quantities for which unbounded solutions exist. In the former case we say that the column is stable and refer to p, h, and / in this case as stable values. Our problem is to separate the stable from the instable values of p, h, a n d /.
We do not solve the problem in this generality; we choose rather a special case with regard to the boundary conditions to be imposed.
3. Formulation of the stability problem for the column with pinned ends. The boundary conditions we choose are those corresponding to the case of a column with pinned ends; that is, we assume th at the deflection w and bending mom ent M = E I ( d2w / d x 2) are both zero at x = 0 and x — l. We have, therefore, as boundary conditions for (2.14):
d2w
w = ---= 0 for £ = 0 and £ = r . (3.1)
<3£2
These boundary conditions can be satisfied b y taking for w a solution in the form of a Fourier sine series:
oo
w = F„(d) sin «£. ( 3 .2)
n*»l
The series (assuming that it converges properly) is a solution of (2.14) provided that the function F n(t?) satisfies the differential equation
d 2F n
+ (<*„ + j3„ cos d)F n = 0, n = 1, 2, 3, • ■ • , (3.3) at?2
in which
a» = n \ p j t0/ p ) ( n2 + p) (3.4) and
0« = n K P p o / f ) ( h ) . (3.5)
The quantities / , / 0, e0, p, and h have been defined b y equations (2.2), (2.9), (2.1 2), and (2.13) respectively. T he differential equation (3.3) is, of course, a M athieu equation.
We can now see w hy the choice of the boundary conditions (3.1) brings with it essential sim plifications. T o begin with, it is not possible to separate the variables in (2.14) in the usual w ay: if we insert for w in (2.14) an expression of the form w =/(£)F (t?) we do not obtain a pair of ordinary differential equations for / and F alone. B y assum ing for w the special form given in (3.2) we are able to satisfy (2.14) b y virtue of the fact th at only even ordered derivatives of w with respect to £ occur in it. This form of solution is, however, not useful for boundary conditions other than those given b y (3 .1 ) . 6 T he reason for this is-as follows: since w satisfies (2.14) we
6 The problem can be solved for other boundary conditions, but with much more difficulty. It is not possible, for example, to make use of the theory of the Mathieu equation in other cases. For a possible approach, see R. Einaudi [ l ], and S. Lubkin [8],
