1939 carslaw & jaeger - on green's functions in the theory of heat conduction

8/2/2019 1939 Carslaw & Jaeger - On Green's Functions in the Theory of Heat Conduction

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ON GREEN'S FUNCTIONS IN THE THEORYOF HEAT CONDUCTION*

H . S. CARSLAW AND J. C. JAEGER1. Introduction. In this Bulletin (vol. 44 (1938), p. 125) Lowan

discusses the Green's function for a line source at (r', 0') for the caseswhere the solid (i) is an infinite cylinder r = a, and (ii) is boundedinternally by r = a y radiation taking place at r = a into a medium atzero temperature.

He uses the method of the Laplace transformation. The solutionfor the first problem agrees with that already obtained by contour in-tegration, f There are some obvious mispr ints in his discussion of thesecond problem, and in his solution on page 133 he seems to have usedas boundary condition G = 0 on r = a, instead of dG/dr+hG = 0, in hisnotation. His result on that page should read

1 ^ f ff(1)(ofo)G = u + v = 2 j COS n(e 0o) I ae-ka l4 7 T n a, J - o o U n(oiO) ijn(ar)Un(aa) - Hn^ (ar) \a Jn{z ) + * ( * ) ] Ida,{ L d% Jz=aa)

whereUn(aa) = [ a Hn (s) + hHn ()] .L UZ J z = a a

Put in this form it can be reduced to the simpler form given below in(16) , except for the difference in the sign of h.In this paper we discuss this second problem, first by contour

integration, and second by the Laplace transform. In the latter weuse what appears to us a much simpler notation and a more rapidapproach to the solution.

We remark also that we have used this notation and method in anumber of other questions J and believe that it will be found in-creasingly useful and much simpler than the operational methods,

* Presented to the Society, September 6, 1938.t Carslaw, Conduction of Heat, 2d edition, 1921, 88, 89. This book will be re-ferred to below as C.H.% Cf. Carslaw, Operational methods in mathematical physics, Mathemat ica l Ga-zette, vol. 22 (1938), pp. 264-280; Carslaw and Jaeger, Some problems in the mathematical theory of the conduc tion of heat t Philosophical Magazine, (7), vol. 26 (1938),pp . 473-495.407


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4 0 8 H. S. CARSLAW AND J . C. JAEGER [Juned u e t o H e a v i s i d e a n d B r o m w i c h , e x p o u n d e d i n J e f f r e y s ' w e l l k n o w nt r a c t . *

2 . Solution using contour integration. W e t a k e t h e li ne s o u r c e a t( r ' , 0 ) , a n d w e r e q u i r e t h e s o l u t i o n o f t h e e q u a t i o ndv / d2v 1 dv 1 d 2v \(1) = kl 1 + ), * > 0, r > a,W dt \dr2 r dr r2 dB 2)w hic h sh a l l t e n d to ze ro wh en />0 in r>a, e x c e p t a t (V ', 0 ) , w h e r e

i t i s to be in f in i te as(2) (47T&/)-1 e xp { - (r2 + r'2 - 2rr ' cos d)/4kt}.A l s o , w h e n r*a, v i s to s a t i s f y

dv(3) h hv = 0 , / > 0 .drL e t

(4) u = (Awkt)-1 exp { - (r 2 + r'2 - 2rr ' cos 6)/4kt} .P u t v u-\-w. T h e n w i s to s a t i s fy

dw (d 2w 1 dw 1 d 2w\(5) = kl h - H J , / > 0 , r > a ,W dt \dr 2 r dr r2 dd 2J(6) lim w = 0 , r > a ,dw du(7) h hw = hu, r = a, t > 0 .d r o r

I t i s k n o w n f t h a t(8) ^ = X) c o s 0 I 0Le-ka hJ n{ar r)J n{ar)da

2T T W = _ O O / oi r(9) X co s (9 ae-ka2tJ n(ar')H nW (ar)da, r > r ' ,i r(10) = ^ cos nd \ ae-k2 tJ n{ar)IU (ar')da, r < r',47T n^-oo J

t h e i n t e g r a l s b e i n g t a k e n o v e r t h e p a t h P of F i g . 1 i n t h e a - p l a n e .* Operational Methods in Mathematical Physics, Cambridge Tracts in Mathematics and M athem atical Physics, no. 23.t C.H., p p . 184-185.


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1939] G R E E N 'S F U N C T IO N S A N D H E A T C O N D U C T IO N 4 0 9

F o r t h e p a t h a t i n f i n i t y o n t h e r i g h t t h e a r g u m e n t of 4 a i s l e ss t h a n 7r,a n d o n t h e le f t i t i s g r e a t e r t h a n Sir. L e t(11) 4ww = X ) COS7Z0 f ae-k2 tA v(a)H nM(ar')H nW(ar)da,n = - o o *w h e r e An(a ) i s t o b e c h o s e n s o t h a t w s a ti s fi e s ( 7 ) . T h u s , u s i n g ( 1 0 ) ,w e h a v e(12) A n{a)[aH nM '(aa) - hH n (aa)] = aJj (aa) - hJn(aa).A l s o(13) 4TTV = X) cosw0 f ae-k2 tH nW(ar')[An(a)H nM(ar ) - Jn(ar)]da,w h e n rr'.

OF I G . 1

T h i s v a l u e o f v s a t is f ie s ( 1) a n d ( 3 ) . W e s h a l l n o w s h o w t h a t i ts a ti s fi e s ( 2 ) ; i n o t h e r w o r d s , w e s h a l l sh o w t h a t lim * _ 0 ^ = 0 , w h e nr>a.T h e i n t e g r a l in ( 11 ) is c o n t i n u o u s w h e n / ^ 0 . W e s h o w t h a t i tv a n i s h e s w h e n / = 0 . T a k e t h e c l o s e d c i r c u i t of F i g . 2 , c o n s i s t i n g of

,y

+JL

F I G . 2t h e p a t h P a n d t h e p a r t o f a c i r c l e , c e n t e r a t t h e o r i g i n , l y i n g a b o v et h e p a t h P.T h e r e a r e n o z e r o s o f aH ^ '(aa)}iH^ (aa) f o r w h i c h t h e i m a g i n a r y p a r t of a i s p o s i t i v e o r z e r o .* F u r t h e r t h e a s y m p t o t i c e x p a n s i o n sf o r t h e B e s s e l f u n c t i o n s s h o w t h a t t h e i n t e g r a l

* See the footnote to 6.


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410 H. S. CARSLAW AND J . C. JAEGER [June(14 ) f affnU> (of )F w ( 1 ) (ar')A n(a)dao v e r t h e c i r c u l a r a r c a b o v e P t e n d s t o z er o w h e n t h e r a d i u s t e n d s t oi n f i n i t y . I t f o l l o w s t h a t t h e i n t e g r a l o v e r t h e p a t h P t e n d s t o z e r o .W e h a v e t h u s i n (1 4 ) o b t a i n e d a s o l u t i o n in t h e f o r m of a n i n t e g r a lo v e r t h e p a t h P .

3 . C o n s i d e r t h e c lo s e d c i r c u i t of F i g . 3 , f o r m e d b y t h e r e a l a x is ,t h e p a t h P , a n d t h e a r c s of a c i rc l e w i t h c e n t e r a t t h e o r ig i n a n dr a d i u s R ( w h i c h w i ll t e n d t o oo) j o i n i n g t h e r e a l a x i s a n d t h e p a t h P .

F I G . 3W h e n > 0 , t h e i n t e g r a l of 2 , ( 1 3 ) ,

f ae-k2 tH nM ( , ' ) [ 4 n( a)ffnU > (ar) - J n(ar)]dao v e r t h e c i r c u l a r a r c s t e n d s t o z e r o w h e n R > oo, r b e i n g le s s t h a n r ' .A l s o t h e r e a r e n o p o l e s o f t h e i n t e g r a n d i n s i d e t h i s c i r c u i t . I t f o l l o w st h a t t h e i n t e g r a l o v e r t h e p a t h P i s e q u a l t o m i n u s t h e i n t e g r a l f ro m oo t o oo o v e r t h e r e a l a x i s . T h u s w e h a v e(15) 4wv = X) cos 7*0 I ae-k2 tH nM (o r7) [J n(ar) - A n(a)H n (ar)]da

n c o ^ oow h e n r


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1939] GR EEN 'S FUNCTIONS AND HEAT CONDUCTION 4 1 1w h e r eUn(ar) = J n(ar)[aYn(aa) hY n(aa)] Yn(ar) [oJ' (aa) hJ n(aa)];a n d t h i s , b e i n g s y m m e t r i c a l i n r a n d r ' , h o l d s b o t h f o r r r'.

5. Solution using the Laplace transformation. W e s t a r t fro m e q u a t i o n s ( 1) t o ( 7 ) of 2 ; b y e q u a t i o n ( 8 ) , u i s g iven in the fo rm u =^2n^-ooUn c o s nd ; s o w e s e e k a s o l u t i o n o f t y p e

00

(17) V = U + W = ^2 (Un + W n) COS 0 ,n=oo

w h e r e(18) lim w n = 0 , r > a ,*->o

d 2w n 1 dwn n 2 1 d w n(19) + w n - = 0 ,dr 2 r dr r2 k dta n d t h e b o u n d a r y c o n d i t i o n ( 3) i s s a ti s fi e d b y (u n+wn). N o w a p p l yt h e L a p l a c e t r a n s f o r m a t i o n , u s i n g , v, f or t h e L a p l a c e t r a n s f o r m s o f u, v , a n d s o o n . T h u s , * w r i t i n g q (p/k)11 2 , w e h a v e

= f er**udt = { exp - pt (r2 + r'2 - 2rr ' cos d)11 2J o 4TkJ0 L U t J1

= K 0[q(r2 + r'2 - 2 r r ' c o s 0 ) 1 / 2]2irk

dtt

(20) = \ jf 1 : 2 3 In(qr)K n(qr') cos ^0 , r < r '27T& n ^ - o o

Z ) In(qr')K n(qr) cos 0, r > r ' ,Z 7 T / c w = o o

00

2 3 ^ n COS ^ 0 ,

w h e r e t h e n a r e g iv e n b y ( 2 0 ) . A l s o f ro m ( 1 8 ) a n d ( 1 9 )d 2w n 1 dw ndr2

1 dw n /n 2 \

* W e use successively th e resu lts, 6.22, (15), and 11.41, (8), of Watson, Theory ofBessel Functions.


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412 H. S. CARSLAW AND J . C. JAEGE R [JuneS o w n = B n(q)K n(qr), w h e r e Bn(q ) i s t o b e d e t e r m i n e d f r o m t h e s u r f a c ec o n d i t io n ( 3 ) . T h i s r e q u i r e s

dv (- hv = 0 , r == a .drH e n c e , f o r r


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1939] G R E E N 'S F UN C T I O N S A N D H E A T C O N D U C T IO N 4 1 3c o n t o u r . * T h e i n t e g r a l r o u n d y t e n d s t o z e r o a s e >0. A l s o , u s i n g t h ea s y m p t o t i c e x p a n s i o n s o f t h e B e s s e l f u n c t i o n s , w e f i n d t h a t t h e i n t e g r a l r o u n d T t e n d s t o z e r o a s R >oo . So in t h e l im i t R o, e>0,

F I G . 4v n e q u a l s t h e s u m o f t h e i n t e g r a l s a l o n g C D a n d EF . P u t t i n g X = ka 2e iri n t h e f o r m e r , a n d \ = ka2e~ iT i n t h e l a t t e r , w e o b t a i n , o n r e d u c t i o n ,( 1 6 ) .

7 . T h e p a t h P of F i g . 1 i n t h e c e -p la n e c o r r e s p o n d s t o a p a t h i n t h eX- plan e (X = ka 2) w h i c h s t a r t s a t i n f i n it y w h e r e a r g X l ie s b e t w e e n 7T a n d 7r/2, p a s s e s t o t h e r i g h t o f t h e o r i g i n , a n d e n d s a t i n f i n i t y ,w h e r e a r g X l ie s b e t w e e n TT/2 a n d T . T h i s i s t h e p a t h V of Je f f reys 't r a c t , Operat ional M ethods in M athemat ical Physics ( 2 d e d i t i o n , 1 9 3 1 ,p . 2 9 ) a n d n o t t h e p a t h L ( f rom c i t o c + i o o ) a s s t a t e d b yB r o m w i c h i n t h e P r o c e e d i n g s of t h e C a m b r i d g e P h i lo s o p h i c a l S o c i e ty ,v o l . 2 0 (1 9 2 1 ) , p . 4 1 2 . T h e c o n n e c t i o n b e t w e e n t h e t w o m e t h o d s e m p l o y e d in t h i s p a p e r is t h u s c le a r . T h e r e m a y b e a n a d v a n t a g e in t h ef o r m e r a s i t s e e m s t o g i v e a s i m p l e r v e r i f ic a t i o n t h a t t h e e x p r e s s io n a sf o u n d d o e s i n f a c t s a t i s f y a l l t h e c o n d i t i o n s o f t h e p r o b l e m .

T H E U N I V E R S I T Y O F S Y D N E Y A N DT H E U N I V E R S I T Y O F T A S M A N I A

* T o s h o w t h a t zKl (z)bK n(z), w h e r e b>0, h a s n o z e r o s f o r R(z) ^ 0 , w e m a y ,s i n c e K -n(z) K n(z), t a k e n^O. T h a t t h e r e a r e n o r e a l p o s i t i v e z e r o s f o ll ow s f ro m t h er e c u r r e n c e f o r m u l a a n d t h e f a c t t h a t K n(x)>0 f o r r e a l p o s i t i v e x. T h a t t h e r ea r e n o c o m p l e x z e r o s f o ll ow s fr o m t h e f o r m u l a ( G r a y a n d M a t t h e w s , Treatiseon Bessel Functions, 2 d e d i t i o n , 1 9 2 2 , p . 7 0 , ( 3 0 ) ) : (\*-ix*)f?xKn(\x)K n(ixx)dx=a[iJLKn(^a)Kn (tJLa) \K n(fxa)Kn ( X a )] , - R ( ^ + / 0 > 0 . T o sh o w t h e r e a r e n o p u r e i m a g i n a r y z e r o s z = iy , w e h a v e iyK.1 (iy) bK n(iy) %ivieZnivl2 [ yJl (y) -\-bJn(y) -\-iy F (y )i bYn(y)], a n d t h e r e a l a n d i m a g i n a r y p a r t s o f t h i s m u s t v a n i s h . T h i s r e q u i r e sM y) Yr! (y) - Yn(y)Jn (y ) = 0 , b u t i t i s 2/iry.

1939 carslaw & jaeger - on green's functions in the theory of heat conduction

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