Lord Rayleigh

Influence of Obstacles on the Prope1,ties ofa Medium. 481

limited to two particles only. Given the proper temperaturewith cprresponding conditions of mass, shape, and distribu"'tion of charge on the particles , and, as it seems to me, almostany amount of molecular complexity is possible. That I havenot taken this possibility into account does not, howevervitiate the results here brolight forward, as they do not pre-tend to greater accuracy than that of their order of magnitude.

It is the cumulative value of these results which wil, Ihope, be regarded as suffcient reason for the publication ofwhat is at best an incomplete piece of theory.

Univ. CoIl. Bristol.

LVI. On the Infuence of Obstacles arranged in RectangularOrder upon the Properties of a Medium. By Lord RAYLEIGHSec. R.HE remarkable formula, arrived at almost simultaneously. by L. Lorenz t and H. A. Lorentz:j, and expressing the

relation between refractive index and density, is well known;but the demonstrations are rather diffcult to follow, and thelimits of application are far from obvious. Indeed, in somediscussions the necessity for any limitation at all is ignored.I have thought that it might be worth while to consider theproblem in the more definite form which it assumes when theobstacles are supposed to be arranged in rectangular or squareorder, and to show how the approximation may be pursuedwhen the dimensions of the obstacles are no longer very smallin cOluparison with the distances between them.

Taking, first, the case of two dimensions, let us investigatethe conductivity for heat, or electricity, of an otherwise uniformmedium interrupted by cylindrical . obstacles which are ar-ranged in rectangular order. The sides of the rectangle wilbe denoted by IX, (3, and the radius of the cylinders by a. Thesimplpst cases would be obtained. by supposing the materialcomposing the cylinders to be either non-conducting or per-fectly conducting; but it will be suffcient to suppose that ithas a definite conductivity different from that ofthe remainderof the medium.

By the principle of superposition the conductivity of theinterrupted medium for a current in any direction can be de-duced from its conductivities in the three principal directions.

. Communicated by the Author.t Wied. Ann. xi. p. 70 (1880).t Wied. Ann. ix. p. 641 (1880).

- ""I'

JJprq &yligp.on tl!e Influence pi Obstacles

Sip-ce CQ1J,gnction pl!ralll tQ the ax()s of the cylinders pre-sl3nts nothing specilll for our consideration, we may limit011r attJltion to conduction parallel to one of the sides

(IX)

thQ rectangular strqcture. In this case lin13 parallel to

I;ig-. 1.

O't

o AI 0 symmetrically situated bet.ween the cylinders, such as ADBe, are lines, of flow, and the perpendicuh1r lines AB, CD arcequipotential

. If we take the centre of one of the cylinders P as origin ofpolar coordinates, the potential external to the cylinder maybe expanded in the l3eri

V ==Ao(AI1'+Blcos 0+ (Asr+Bs1') cos 30+.. " (nand at pointwithin the cylinder in the series

=CO+Cl1'CosO+Csrcos38+...(2)

being measured from the direction, of IX, The sines of and its multiples are excluded by the symmetry with respectto 0= , and the cosines of the even multiples by the. sym-metry with respect to o==!w. At the hounding surface

where r=q" we have the conditions V =V'vdV'/dr=dVjd1'

II denoting the condudiyity of the material composing thecylinders in terms of tha'l of tpe remainder reckoned as unity.The application of these conditions to the term in COB gives

" .

(3)

In the case where the cylinders are perfectly conducting,v=oo. If they are non-c?nducting, 11=0.;

II'

uodn puedep mM unooou OW! uaJI'B~ aq O~ S! se;).IOS JO rnaSASeHug.u! eqqo!lIU! JaUUl1m eq~ UBq~ ueAm:

Ef pUll)' O~ IeTI'BIt

AJ'BpUnoq .1'Bln:6uupaJ 'B aAuq: O~ s.mpUHAO aq~ Aq pe!dnooonO!~BJ (enuy!) eq~ eU!~'Bm! aM. J! SseUJ'Bep O~ eonpuoo TEPUOOBS Bq~ :6U!Jep!SUOO uI ,2'Aq pBuep eq A'Bm F'Bd ~s.eql. 'S,lBp1mq~ JO sex'B Bq~ UO pB'Bn~!S seo.IOS eldmnm O~PU'B (pesn'BO S! M.OF Bq~ qO!q M. Aq) AHug.U! ~'B seOJlOS pmJ:B

S'B pBp.1U~BJ eq AULII ~u!od AU'B u A rl1!~uB~od eql.PU'B neeM~eq uO!~'P!eJ ;WljJJ:J'B HJ!nbB.1 e,\\mn!pam

pmp'B eq~ JO BS'BO eq~ JOd: 'AHun S! AHAnOUpuoo eq~ qO!qM.

JO mn!pem pa~dUJ.IB~U!Un Bq~ uodn JIO'Bq TI'BJ eM '0 = fa JI(9)

. '

O== fij.!(; + IAFJ-or;snq~ S!

(v) JO uO!~1:o!Iddu eq~ mO.nnse.r l'Bug. eq; 'la.ug S! Iu.l~a~u! eqJO ~Jtld S!q~ 'Bq~ os

S09 (;-

y)

=1.p/APSOO(Ig +'QY)=A

aH.\,\\ AUUI e,\\ snq; SOD U! esoq~ eJ:'B nUseJeq~ O~ Bnqp~noo mMqO!qA\ (1) IT! I3mJa~ AluO aq~ snq~ PU'B

SO\)- =,1P/ilP- =uP/ilP 'SOD V=

Jno~nOO eq~ JO lJud J'BlnOJp eq~

av puuao ueeM~aq Su 'BI~uuwaJ; euo 01~U!puodsaJ.IOO I'B!Wfj~od U! II'BJeq~ S! 1 A eJBqM. 'if A S! sauH emus eq~ JBAO pu~e~u! eq~ JOJapU!'Bma.I aq~ JO BUI'B eql.

OXI snq~ S! ay 'ao .IaAO 1'B.I~e~u!aq~ JO ~.r'Bd s!1~ JO anI'BA eql. '0 Aq eouBp A'Bm aM qo!qM

u'B~oeJ eq1 SS0.10'B ua.rJ:o 1'B0~ eq1 s~uasaJdaJ 8P up/ APS

ao uo 'qS!l'BA q~oq up/ AP 'up/ ilP ay loa SBPP! aq~ .WAO

(!;). '

SOD .t=X'=

amnssu aM J! 'il II!A pSISU 'uo!'Bn1aao'Bld'B'1 says!~ua A np!:aa.1 S!q~ U!q~!.M . d JBpn!IAo aq~

PU'B aoay n'Bpa.I eq~ naaM.~aq UO!~a.1 aqJO Jno~nOO eq~ 01

tv)UP up

0=8p A"" ilP

QIe'IOaq~ suaBJ:!) Aldd'B Mon sn ~a'1

mUBJ~S aq~ uMop-19 gn sSBd aM S1: 'JapU!IAO O~ J;apU!IAO mOJJ sap'B '.1aAaMoqQy WJat ~S,Ig. Bql. 'Iu!~uaod JO eOJ:p.os eldmnm J'BI!l!S 'B S'B

p,1p,~a.1 eq A'B\l qouB puu 's;wPU!IAO aq~ TIJOJ am'BS Bq~ AI!J'BS"'saDtOJ'B ." y 'rs~na!oHJaoG aq~ JO sanluA aqL

\1Qt UInJpN v uoan .ttO ,1vZ;-l,fj1.V1af)

484 Lord Rayleigh on the Influence of Obstacles

the shape of the rectangle. The simplest case, which suffcesfor our purpose, is when we suppose the rectangular boundaryto be extended infinitely more parallel to IX than parallel to (3.It is then evident that the periodic difference V may bereckoned as due entirely to H,x and equated to HIX. For thedifference due to the sources upon the axes wil be equivalentto the addition of one extra column at + X) , and the removalof one at -00 , and in the case supposed such a transference isimmaterial*. Thus

=HIX . (7)

simply, and it remains to connect H with BI'This we may do by equating two forms of the expression,

for the potential at a point

, y

near P. The part of thepotential due to H,xand to the Iqultiple sources Q (P notincluded) is

Ao+AITCOS Of Asrs cos 30+,...;

, if we subtract H,x we may say that the potential at

, y

due to the multiple sources at Q is the real part of

+ (AI H)(,x+ iy) +As(x+iy)S +Ao iy)O

+ ...

(8)

But if

, y

are the coordinates of' the same point wh8n re-ferred to the centre of one of the Q' , the same potential maybe expressed by

-1 -S "" 1. ,x

%y

s,x + zy

+ . , ,

'J,

the summation being extended over all the Q'

the coordinates of a Q referred to P=x-E, y

=y-

'T;

(9)

If E, 'T

so that

(,x = B (,x iy-E- i'T)-n

Since (8) is the expansion of (9) in rising powers of(x+iy), we obtain, equating term to term

=Bl +3B +5B

+'" . -

=1.2. +3.

+'"

(10)

1 . 2 . 3 , 4, . 5 Ao = 1 ; 2 . 3 . 4 . 5 BI 6 + 3 . 4 . 5 , 6 . 7 Bo 8 + . . .

and so on, where2,, (g+i'T)-2n

the summation extending over all the Q'

. . It would be otherwise jf the infiite rectangle were 8upposed to beof another shape e, fl' to be square.

(11)

in Rectangular Order upon a :Medium. 485. By, (3) each B can be expressed in terms of the correspond-

ing A. For brevity, we wil write

' -

=va n'"(12)where

=(l +v)/(l-'v).. (13)Weare now prepared to find the approximate value of the

conductivity. From (6) the conductivity of the rectangle is

f!) 2'1Bll = i!) 1- 2'1Bl

IX 1. flVl.I IX IXf3H..sO that the specific conductivity of the actual medium forcurrents parallel to IX is

2'1BaflH' .

and the ratio of H to Bl is given approximately by (10)and (12).

In the first approximation we neglect I4' 6 . . . ., so thatAs, A5 . . . Bs, B5 .. .. vanish. In this case

H=Al + B+ I

and the conductivity is2 '1a

IXfl( v'

The second approximation gives

(17)

- +

- jT 4, and the series may be continued as far as desired,

The problem is thus reduced to the evaluation of the quan-tities I., I4' ' , . We wil consider first the important parti-cular case which arises when the cylinders are in squareorder, that is when fl=lX. E and 'f in (11) are then bothmultiples of IX, and we may write

n = -n 8

(14)

(15)

(16)

I l!

where(18)

= I +im)-n(19)the summation being extended to all integral values of

, m positive or negative, except the pair m=O, m!=0. The quantities 8 are thus purely numerical, and real.The next thing to be remarked is that, since , mare as

---

.u_.-

- ------ -


much positive as negative, 8" vanishes for every odd valueof n. This holds even when IX and /3 are unequal.

Again

" =

l +im) -2n

!( -

' +m)-

= (-

1)nim'+m)-

=(-

1)"8

Whenever is odd= - 8", or vanishes. Thus for

sqnare order

tilL

;11

= 810 = 8

= . , . =

O. . (20)

Ths argument does not, without I'ese\'vation, apply to 8In that case the sUm is not convergent; and the symmetrybetween and essential to the proof of evanescence, onlyholds under the restriction that the infinite region over whichthe summation takes place is symmetrical with respect to thetwo directions IX and /3-, for example, square or circular.On the contrary, we have supposed, and must of course continueto suppose, that the region in question is infinitely elongatediu the direction of IX.

The question of convergency may be tested by replacingthe parts of the sum relating to a great distance by the corre-

sponding integral. This

= rr os 2n()r dr dO

(.x+y)2n

JJ 1,21

"i-

md herein-2n+l dr -2n+2/( -2n+2) ;

'So that if 1 there is convergency, but if = 1 the integralcontains an infinite logarithm,

We have now to investigate the value of 8appropriate toour purpose; that is, when the summation extends over theregion bounded by.x= :!u, y= :!vwhere and are bothinfinite, but so that v/u=O. If we suppose that the tegion

of summation is that bounded by.x= :tvy= :!vthe sum

Fig,

vanishes by symmetry. We may therefore regard the sum-mation as extending over the region bonnded externally by.'v= :too

y=

:!vand internally by iV= :tv (fig. 2). 'Vhenis very great., the sum may be l"eplaced by the COl'esponding

o I"

integral.

in Rectangula1' Order upon a JJIedium.

Hence

487

d!Jdy- 2 (21)

.x +

the limits for being :: and those for !J being and 00 Ultimately is to be made infinite.

We have+v

v (.x+iy)2

==

.x + it. .x-and

'"

2v d!J= 2 tan00 ""2 tan -I l=i11, l! +

AccordiIgly(22) 7T.

In the case of square order, equations (10) (12) give

2-. 3 8'" 2 '"2

- ,

4 - -- '"8 -

.. , .

1 '

7Ta'l 7 a

= V

+ ,,

., 8 8

- -

,"l 8S2 -

. . .; .

IX VIX (23)

and by (14)

Conductivity27Ta

=1-IX

the proportional space

(24)

occupied by the If denotecylinders

p=7Ta/1X2 ;(25)and

. .

2p

' .

ConductIvIty = 1-

2 2E s 'l

+p-

4 - 8 8

(26)

Of the double summation indicated in (19) one part canbe effected without diffculty. Consider the roots of

sin

(g-

im'1)=O,They are all included in the form

t: s = 7T+zm7T

where is any integer, positive, negative, or zero. Hencewe see that sin (E irn7T) may be written in the form

A 1-

-;

1-

..,

m7T tm'1 zm7T - '1W17T + 27T

._----- --_.- .


in whichA '= -sin im7r.

Thus

log 1cos ,-cot im7r sin E=log 1- un7r

+log

+....

m7r + 7r

If we change the sign of and add the two equations, we

get sinlog == log 1-- SIn Zln7r m7r

+logJ1-

("

+Iogf1-

(, +...;

m7r+7r 1. m7r-7r

whence, on expansion of the logarithms

sin+ sinsinsinim7r 2 sin4im7r+ 3 sinim7r ....

1 l (im7r):im7r+ 7r?(im7r-7r):

+ l.t J

--

, 1

+ . . . .

l(zm7r)4 (zm7r+7r)4 (m7r-7r

H:6+ 3S 6 + 6 +

. . .. +....

Z1n7r un7r+7r

By expanding the sines on t,he left and equating the corre-sponding powers of E, we find

1 12 +

('

1)"2 +.... 2'(27)(1.m zm+ tm-zm+ sm zm7r

1 127r. 7r

' . ('

1)4

+ . . . =

3 . (28)un) Z'n+ SIl un7T sm m7T

1 27r

('

1)6

+'" =

15 . . (29)

(zm zm+ Sil zm7r SIn un7r Sil zm7r

In the summation with respect tv m required in (19) weare to take all positive and negative integral values. But inthe case ()f m=O we are to leave out the first term, corre-sponding to =O. When m=O

1 7rsinim7r (im)2= 3"'

. in Rectangular Order upon a Medium.

which, as is well known, is the value of

12+ 1)2+ 22+2)2

489

Hencem",'"

2'7in -2im'7 +!r m=l(30)and in like manner

m=c.2'7- i siim'7+sinim'7t, .

;)

m=1(31)

2-6 m=oo

27 . 352'7h\ sin-im'7

im'7+sinim'7t. .

We have seen alr8ady that 8, and that ='7. comparison of the latter with (30) gives

m=o:m'7=

- -

m=1

We wil now apply (31) to the numerical calculation of, We find:

(32)

The

(33)

- sinim1!.sin-4imlT.

'00749767'0000562150

1395

Sum00751165'0000562152

so that

='7x -03235020. . (34)In the same way we may verify (33), and that (32) =0. If we introduce this value into (26), taking for example the case where the cylinders are non-conductive

= 1), we get

1+p-'3058p(35)

From the above example it appears that in the summationwith respect to there isa high degree of convergency.

490 Lord Rayleigh on the Influence qf Obstacles

The reason for tbis wil appear more clearly if we considerthe nature of the fiTSt summation (with respect to

).

(19) we have to deal with the sum of (x+iy)-nwhere

for the moment regarded as constant, while receives thevalues x=m, instead of being concentrated at equidistantpoints, the values of were ulliformly distributed, the sumwould become

f A

+'" dx

-'"

(x+iy)Now, n being greater than 1, the value of this integral iszero, We see, then, that the finite value of the sum dependsentirely upon the discontinuity of its formation, and thus ahigh degree of convergency when increases may be ex-pected.The same mode of calculation may be applied without

diffculty to any particular case of a rectangular arraIlgement,

For example, in (11)a.+im(3)-=1X' +im(3/a)-

If be given, the summation with respect to leads, asbefore, to

7r +irn(3/a-2 = sin2(im7Tf3/a.),

and thusm=c.

=27T. sin(irn7T(3/IX) +tW. m=l

The numerical calculation would now proceed as beforeand the final approximate result for the conductivity is givenby (16). Since (36) is not symmetrical with respect to and (3, the conductivity of the medium is different in the twoprincipal directions,

When (3 = IX, we know that =7T, And since this doesnot differ much from !7T, it follows that the series on the rightof (36) contributes but little to the total. The same will betrue, even though 13 be not equal to provided the ratio ofthe tw.o quantities, be moderate. VIf e may then identify with 7T, or with !7T, if we are content with a very roughapproximation.

The question of the values of the sums denoted by 2n

intimately conllected with the theory of the O-functiolls *inasmnch as the roots of H(u), or (7T/2K), are of the form

2mK+ 2miK'

'" Qa.yley' Ellptic Functions;' p, 300, The notation is that of Jacobi.

(36)

in, Rectangula1' Order upon a Medium, 491The analytical question is accordingly that of the expansion

of Jog (.x) in ascending powers of .x. Now, Jacobi * has himself investigated the expansion in powers of ,'v

7J) :: 21 lf'1 sin IV-q9f4 sin 3.x 25/4

sin 5.x-

, , ,

(37)

- -

rrK'/K (38)

were

So far as the cube of .7J the result is

~~~

=1-3KE-(2-)K2

+"'

! (39)

D being a constant which it is not necessary further to specify.and E are the elliptic functions of usually so denoted.

By what has been stated above the roots of Bl(.x) are of theform

71 (m+miK'/K); .(40)so that

+i11=i13KE-(2-P)KL . (41)the summation on the left being extended to all integralvalues ofm and except m=O=O.

This is the general solution for 2' If K'k2=!, and

+im=212KE= 71

sinc.e in general t,

EK'+E'K-KK'=!71.In proceeding further it is convenient to use the form in

which an exponential factor is. removed from the series.This is

.J A3A5:J A7 (.x)=Ale-lAB"A.x-Sl3T +S2

51 -Sa

7T + .J'in which(42)

(43)A= B=

71 71 So fJ, Sl=()fJ, S2,=fJ(1X2fJ4=1X/3(a6fJ4

the law of formation of being

Son + 1 =2m(2m+ 1)fJ4Sdsm/dfJ-8fJ'dsm/dtX, (44)

'i 'Crelle' Bd. 54, p. 82,t Cayley' Ellptic Functions,' p. 49.

-----

492

while

Lord Rayleigh on the Influence of Obstacles

IX k'2(3= v(kk'

).

(45)

I have thought it worth while to quote these expres-

sions, as they. do not seem to be easily accessible; but Ipropose to apply them only to t.he case of square order, K' = Kk'2

=!.

Thus =(3,

AB=I/71IX=O

, s22(35

(3=I/v2; (46)

, S4

= -

36(39

and

20 J 4 A(m)

! + ' . ,

5" (47)

Hence. 0) (m) ,,;2 !1:

10g27116.35,

'"

(48)

If :ti\., :ti\2' .,. are the roots of (m)/x= , we have

2- 4-

"' -

271

' "' -

51 ,

"' -, "' - . - .

Now by (40) the roots in question are 71 (m+im

),

and thus

4 718 A 8

=71, 8, 870.

(49)

in which. 3. 3. 5

A = ?T

K = 1 + ' '2 + 22 . 42. 4 + 22 , 42 . 6' 8 + . . .

= 1'18034,

Leaving the two-dimensional problem, I wil now pass onto the case of a medium interrupted by spherical obstaclesarranged in rectangular order. As before, we may supposethat the side of the rpctangle in the direction of flow is IX, thetwo others being (3 and "1' The radius ofthe sphere is

The course of the investigation runs so nearly parallel that already given, that it wil suffce to indicate some of thesteps with brevity, In place of (1) ad (2) we have the

expansions V=Ao+ (AIr + B

+ ,..

+(A+B"-n-)Y,,

+..., .

(50)

Yl'r+... +

+ ,-.

(51)

in Rectangular Order upon a Medium.

denoting the spherical surface harmonic of order

from the surface conditions

493

And

V=V'vdV'/dndV/dnwe find

2n+I+v+

We must now consider the limitations to be imposed upon. In general

(52)

8=n Y,,e:, cos scp + Ksin scp),'=0(53)

where

l) = sinBO cos,,-sO

(2n 1) cOSn-S-20

+ . . . ,

(54)

being supposed to be measured from the axis of tJ (parallel

to IX), and cp

from the plane of tJz. In the present applicationsymmetry requires that should be even, and that (exceptwhen n=O) should be reversed when (71-0) is written for

Hence even values of are to be exclnded altogether.Further, no sines of scp are admissible. Thus we may take

=cos

=COS0-.g- cos O+Hsin0 cos cos 2cp,

Y 5 = COS- If cos+ ir cos

+ LsinB (COSO-t cos 0) cos 2cp

+ L4 sin0 cos cos 4cp.

In the case where /3=1 symmetry further requirls that, L=O, (58)

In applying Greens theorem (4) the only difference is thatwe must now understand by the area of the surface boundingthe region of integration, If C denote the total currentflowing across the faces /3ry, Vthe periodic difference ofpotential, the analogue of (6) is

IXC-/3yV+ 471B, =0. (59)We suppose, as before, that the system of obstacles, ex-

tended without limit in every direction, is yet infinitely moreextended in the direction of IX than in the directions of /3 and ry.

Phil, Mag. S, 5, Vol. 34, No. 211. Dec. 1892. 2

(55)

(56)

(57)

494 Lotd Rayleigh on the Influence of Obstacles

Then, if H.1; be the potential due to the sources at infinityother than the spheres, VI=HIX, and

f3ryf1- 471

IX L lXf3rH

so that the _specific conductivity of the compound mediumparallel to IX is

1- 471B

lXf3ry H .

We wil now show how the ratio BJ/H is to be calculatedapproximately, limiting ourselves, howevel', for the sake of

simplicity to the case of cubic orde,', where 1X=(3=ry. Thepotential round P, viz.

1"YAarYa+ ,. may be regarded as due to Hx and ,to the other spheres Qacting as sources of potential. Thus, if we revert to rectan-gular coordinates and denote the coordinates of a point rela-.tively to P by , y, zand relatively to one of the Q's by

, y

, zwe have

(60)

Ao+ (AH)x+Aa(J,

-g,) + . - ,

" x's

- %

l' (61 )

in which=x-g, y'T, z=z-',

if 'T, s be the coordinates of Q referred to P. The left sideof (61) is thus the expansion of the right in ascending powersof xy, z. Accordingly, A- H is found by taking did. v of

the right-hand member and then making x.y, z vanish, Iulike manner- 6Awil be found from the third differential

coeffcient, Now, at the origin

' - - ' - -

d ,,r- p

d.' 7- --'in which

=E2 + S2

It will be obsefved that we start with a harmonic of order1 and that the differentiation raises the order to 2. The lawthat each differentiation raises the order by unity is general;and. so far as we shall proceed, the harmonics are all zonal

, ,

. ;a

..-

JI::

' .'''-"' ''"': .

'7V

:".-:

in Rectangular Orde1' upon a Medium. 495.

and may be expressed in the usual way as functions

(fh) of fh, where fh=g/P. Thus

d 0/

- - =

-3 P7J 1,

1"

In like manner

tgp

= -

5 P(fh) '7 and

d" xdg"

p"

= - 24 P-" P(fh).

The comparison of terms in (61) thus gives

H=-

+" +,." ... - .....

In each of the quantities, such as the summation

is to be extended to all the points whose coordinates are ofthe form

(62)

lOl, mIX, nIX

where are any set of integers, positive or negativeexcept 0, 0, O. If we take IX= 1, and denote the correspondingsums by 8, 8, . . , . , these quantities wil be purely numerical, and

-"-

-n-

. .

(63)From (52) (62) we now obtain

2+v a3 321-2 a

:;

+28410(64) Ct

which with (60) gives the desired result for the conductivityof the medium.We now. proceed to the calculation of 8, We have

3fh1 2g2-rl-s2

= -

clg

By the symmetry of a cubical arrangement, it follows that(rnp(S2

so that if 8 were calculated for a space bounded by a. cubeit would necessarily vanish. But for our purpose 8

is tobe calculated over the space bounded by g=:t 00

?J :tv 2M 2


!;= :t where is finally to be made infinite; and, as we

have just seen, we may exclude the space bounded by

E= ::v, 'I= ::v

=::

so that !82 wil be obtained from the space bounded by

g=vE=oo'I= ::v

::.

Now when is suffciently great, tho summation may bereplaced by an integration; thus

In this

=- SSS)agd'Id!;.

00

a(j) dE=- +'l+'2

+u v d'I 2V-u +'l

1= (V) (2V+'2i'

and finally+u 2d()

-u (7,+'2(2V+'20"; (2+tan

rl/ "'2 2ds

Jo ";(2-8Thl:s

27T

= 3" (65)

, If we neglect a/lX, and write for the ratio of volumeslZ.

47Ta

3(.

we have by (1)0) for the conductivity

(2+v)/(1-v)-(2+v)j(1-v)+p

or in the particular case of non-conducting obstacles (v=O)

. +2P ,In order to carryon the approximation. we must calculate&c, Not seeing any general analytical method, such as

was available in the former problem, I have calculated an

.. Compare Maxwell'' Electricity,' 314,

(66)

(67) *

(68)

, \

---- --. ,

-.IP esaqJapu(l 'a.mp\l~S Bq~ JO

(/. '

Fl XI) po!.wd aq~ qHuospudmoo U! ~1Ja.

(~

S! qaAu\\ Bq~'Bq~ pamnss1I

n 'SB.Iaqds.1O SJapU!IAO pax!f pUB P!~!l A'q papn.qsqomn!pam snoas1l~ 'B U! UOnOm-BA'BM. JO marqold aqS! asaq~

JO ~sa!dm!s BqJ, 'UO!'BJq!A JO smarqold U!'BpBO O~ pa!TddBBq MOU Aum ABq~nq 'mn!pBm punodmo;) u JO AHA!Pl1P-nOG aq~ su a1In~uuI IU;)!lPBla UAH;)!Idm!s JO a1111s aq1J01 passa.Idxa uaaq aA'Bq UO!~B~!~SaAU! JUO JO sHl1saJ Bq.L

9800, +1.1'1 'I '1900, 1.1I: 'I' 'LL60'I' 'I' 'woo' + '1 '0 '1.600,I' 'I' '9600, -I: 'I' '1.61:0,I' 'I 'noo, -1:1I: 'I' '1090,I' 'I '

1'900. 1'(I' 'I' ''1601.I' '0 '9l.0. I: 'I '96(H.I 'I '1:171'0,I: 'I '17601:,I 'I 'ttIO, +0 '0009,1: +T '0 '

----'-----

(69)S! nnSBJ aqJ:

uO!~'Bmums oq~ JO A~!nU!~UOOS!P aq~ uodn~u!puadap Bnp!SaJ JO PU!11 U AIUO S! paJ!nbBJ mns Bq~uqos 'aJBqds a1BldmoG aq~ JaAO pa~u.Ia~u! uaq.\\ saqS!UBA d

1Iq~ ~ouJ aqJq pB.IlOA'BJ S! uO!1'Bm!xoJddu Bq~ JO ssaoons aql.

0 + gO+gg=

~1 +gg+~g=~SJ9 Bq~ ul 'uo!~!sodUiOO JO spuPI OM.~ S1uasaJdoJ 6 .10

dldm'Bxa JOJ 'asuo aq~ U! uom~BdaJ S!ql. '81 U!q~!"'10 a!I

rp!qM su!od II'B JOJ ~lnsaJaq~ su!u~uoo alq'B~U!AI.OnOJ BqJ,

PloJOM.~ 'qS!Ul!A OM.pUB 'PIOj-IlOJ S! UO!~!~BdeJ f)q~ 'qS!UUA sapm!p.IOOO BB.Iql Bq~

JO auo.H 'samn 8 s! ~'Bq~ '~UUGO qO'Ba U! pa~uadaJ s! a1!ug

BJl! J '/1'qO!qM. JOJ m.w~ LWArt 'J '/1'JO sanl'BA OJBZ

PU'B B.I.!Hsod JO uO!~'Bep!suoo Bq~ o~ saAIBsJno ~!m!I Aum B

dg + f,30ft-

puooas aq~ U! PU'B

Bq~ mOJJ'BlnmJoJ

uO!l'Bmmns pB.I!P Aq fS JO anI'BA a~umxoJddu

UlnJp&N v uodn .

~&p.

tO ,wznOUVliJ&'8 u L6v

u_-

498 Lord Rayleigh on the Infuence of Obstacles

cumstances the flow of gas round the obstacles follows thesame law as that of electricity, and the kinetic energy of themotion is at once given by the expressions already obtained.In fact the kinetic energy corresponding to a given total flowis increased by the obstacles in the same proportion as theelectrical resistances of the original problem, so that the

influence of tbe obstacles is taken into account if we supposethat the density of the gas is increased in the above ratioof resistances. In the case of cylinders in square order(35), the ratio is approximately (I+p)/(l-p), and in thecase of spheres in cubic order by (68) it is approximately(1+!p)j(l-

p),

But this i.; not the only effect of the obstacles which wemust take into account in considering the velocity of pro-pagation. The potential energy also undergoes a change.The space available for compression or rarefaction is now(I-p) only instead of 1; and in this proportion is increasedthe potential energy corresponding to a given accumulationof gas , For cylindrical obstruction the square of the velocityof propagation is thus altered in the ratio

I+p -7(I-

p)=

l+pso that if f- denote the refractive index,- referred to that ofthe unobstructed medium as unity, we find

I+p,(f- 1)/p=constant

, .

(70)

which shows that a medium thus constituted would followNewton s law as to the relation between refraction anddensity of obstructing matter, The same law (70) obtainsalso in the case of spherical obstacles; but reckoned abso-lutely the effect of spheres is only that of eylinders of halveddensity, It must be remembered , however, that while thevelocity in the last case is the same in all directions , in thecase of cylinders it is otherwise, For waves propagatedparallel to the cylinders the velocity is uninfluenced by theirpresence. The medium containing the cylinders has there-fore some of the properties which we are accustomed toassociate with double refraction , although here the refractionis necessarily single, To this paint we shall presently returnbut in the meantime it may be well to apply the formulre tothe mQre general case where the obstacles have the pro-perties of fluid, with finite density and compressibility,

*' Theory. of Sound 303,

, i

in Rectangula1' Ol'de1' upon a Medium. 499To deduce the formula for the kinetic energy we hav(lonly

to bear in mind that density cortesponds to electrical res.ist-ance, Hence, by (26), if denote t.he density of the cylin-drical obstacle, that of the remainder of the medium beingunity, the kinetic energy is altered by the obstacles in theapproximate ratio

(u+ 1)/(u-1) +p(u+ 1)/(a--1)-

(71)

The effect of this is the same as if the density of the wholemedium were increased in the like ratio. . The change in the potential energy depends upon the" compressibility" of the obstacles. If thmaterial com-posing them resists compression m times as -much as the re-mainder of the medium, the volume counts only as p/mand the whole space available maybe reckoned as p+p/minstead of 1. In this proportion is the potential energy of agiven accumulation reduced, Accordingly, if p, be the refrac-tiye index as altered by the obstacles

2 = (71) x (l-p+p/m).(72)

The compr6ssibilities of all actual gases are nearly the sameso that if we suppose ourselves to be thns limited, we mayset m= 1, and

u+ 1)/(u-1) +pP, -

(u+ 1)/(u-1)-(73)

, as it may also be written2 -

1 1

- =

constant, .

p, +

In the case of spherical obstacles of density we obtain inlikemanner (m = 1)

(74)

(2u+1)/(u-1)+p

p,

+l)/(u-l)-(75)

2 -1 1

" + !

constant.(76)

In the general case, where Tn is arbitrary, the equation ex-pressing in terms of p,2 is a quadratic, and there are nosimple formulre analogous to (74) and (76),

It must not be forgotten that the application of theseformulro is limited to moderately smull values of

p.

If it be

---

------. " ' "(..


desired to push the application as far as possible, we mustemploy closer approximations to (26) &c, It may be re-marked that however far we may go in this direction, thefinal formula wil always give p.

2 explicitly as a function of

For example, in the case of rigid cylindrical obstacles, wehave from (35)

+p-

'3058p

. ,

= (l-

p)

'3058(77)

p- + . .

It wil be evident that results such as these afford nofoundation for a theory by which the refmctive properties ofa mixture are to be deduced by addition from the corre-sponding properties of the components, Such theories re-quire formulre in which occurs in the first power only, asin (76),

If the obstacles are themselves elongated, or, even thoughtheir form be spherical, if they are disposed in a rectangularorder which is not cubic, the velocity of wave-propagationbecomes a function of the direction of the wavp,-normal. Asin Optics, we may regard the character of the refraction asdetermined by the form of the wave-surface.

The reolotropy of the structure wil not introduce any cor-responding property into the potential energy, which dependsonly upon the volumes and compressibilities concerned. Thepresent question, therefore, reduces itself to the considerationof the kinetic energy as influenced by the direction of wave-propagation. And this, as we have seen, is a matter of theelectrical resistance of certain compound conductors, on thesupposition, which we continue to make, that the wave-length is very large in comparison with the periods of the

structure, The theory of electrical conduction in generalhas been treated by Maxwell (' Electricity,297), Aparallel treatment of the present question shows that in allcases it is possible to assign a system of principal axeshaving the property that if the wave-normal coincide withanyone of them the direction of flow wil also lie in thesame direction, whereas in general there would be a diver-gence, To each principal axis corresponds an effcient" den-

sity," and the equations of motion, applicable to the mediumin the,gross, take the form

if!; do 17 do dt2=m1' U'=m1=m1

where 1;, 17, S are the displacements parallel to the axesthe compressibility, and

0: d!; d17 d' o -

.,"

in Rectangular 01'dm' upon a Medium. 501If XfL, v are the direction-cosines of the displacement

, m, n of the. wave-nOl'mal, we may take

g =

'T fLOwhere

i(lx+n1y+y.vt)Thus

do/dtV

= -

lB(lX+mfL+nv), &c.

and the equations become

l(lX+mfL+nv),JlVmlm(lX+mfL+

mln(lX+mfL+

from which, on elimination of X : fL we getp m

- + - + -

+r?n

if , b, c denote the velocities in the principal directions

.y,

The wave-surface after unit time is accordingly the ellp-soid whose axes are , b, c,

As an example: if the medium, otherwise uniform, be ob-structed by rigid cylinders occupying a moderate fraction

(p)

of the whole space, the velocity in the direction parallel to

the cylinders, is unaltered; so that= 1= 1/ (1 +

p)

In the application of our results to the electric theory oflight we contemplate a medium interrupted by spherical, 01'cylindrical, obstacles, whose inductive capacity is differentfrom that of the undisturbed medium. On the other handthe magnetic constant is supposed to retain its value un-broken. This being so, the kinetic energy of the electriccurrents for the same total flux is the same as if there wereno obstacles, at least if we regard the wave-length as in-finitely great *. And the potential energy of electric dis-placement is subject to the same mathematical laws as theresistance of our compound eleeirical conductor, specificinductive capacity in the one question corresponding toelectrical conductivity in the ot.her.

Accordingly, if denote the inductive capacity of the mate-rial composing the spherical obstacles, that of the undisturbedmedium being unity, then the approximate value of

* See Prof, Wilard Gibbs" Comparison of the Elastic and ElectricTheories of Light," Am, Journ, Sci, xxxv, (1888).

(78)

502 Influence of Obstacles on tlte Properties of a. jJfedium.

given at once by (67). The equation may also be written inthe form given by Lorentz

- = -

=constant; +2p v+2and, indeed, it appears to have been by the above argumentthat (79) was originally discovered.

The above. formula applies in strictness only when thespheres are arranged in cubic order *, and, further, when is moderate. The next approximation is

fk

= + . . - -

65 10/3I pv+4/3P

If the obstacles be cylindrical, and arranged in squareorder, the compound medium is doubly refracting, as in theusual leetlic theory of light, in which the medium is sup-posed to have an inductive capacity variable with thedirectiunof displacement, independently of any discontinuity in itsstructure. The double refraction is of course of the uniaxalkind, and the wave-surface is the sphere and ellipsoid ofHuygens,

For displacements parallel to the cylinders the resultantinductive capacity (analogons to conductivity in the conduc-tion problem) is clearly 1- vp; so that the value of

for the principal extraordinary index is=1+(v-l)p, (81)

law for the relation between index and

(79).1,

(80)

giving Newtondensity.

For the ordinary index we have

(26),

in which = (1 + v)/(l-v), while S4, S8' .,given by (49), If we omit &c. we get

fk

have the values

(82)

+ 1

,. -

+ 1(83)

The g('l1eral conclusion as regards the optical applicationis that, even if we may neglect dispersion, we must not ex-pect such f'ormulre as '(79) to be more than approximatelycorrect in the case of dense fluid and solid bodies, " An irregular isotropic arrangement would, doubtless, give the mmeresult,

f:!

Lord Rayleigh

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