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Masters Theses Student Theses and Dissertations

1950

The reflection coefficient at the truncated corner of a rectangular The reflection coefficient at the truncated corner of a rectangular

wave guide wave guide

John Raymond Barcroft

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THE REFLECTION COEFFICIENT

AT THE TRUNCATED CORNER OF A RECTA~GULAR WAVE GUIDE

BY

J OFJ~ RAYMOND BARCROFT

A

THESIS

submitted to the faculty of the

SCHOOL OF MINES AND METALLURGY OF THE UNIVERSITY OF MISSOURI

in partial fulfillment of the work required for the

Degree of

MASTER OF SCIENCE, PHYSICS MAJOR

Rolla, Missouri

1950

(-.

Approved by - Ccfl.-t. '7J.'Yc!) 6~Asaoc.Professor of Physics

ACKNOI'iLEDGMENTS

The author wishes to express his appreciatiorr to

Dr. Edward Fisher for the direction of this thesis.

The author a.lso 1I<Tishes to express his appreciation for

the interest shoi'\m by Dr. Harold Q,. Fuller, Professor

I. H. Lovett, and Professor Gabriel Skitek.

ii

iii

CONTENTS

Acknowledgments .•••••••••••••••••••••••••••••••••

Page

ii

List of illustrations............................ iv

Introduction•••••••••••••••••••••••••.•••••••..••

Physical interpretation of propagation within the

guide ••••••••••••••••••••••••••••••.•••••••••••••

Mathematical consideration of the TE wa~es in the

1

4

guide............................................. 15Derivation of equations for the truncated corner. 24

Conclusion•••••••.•••••••..••••. • • • • • • • • • • • • • • • • • 69

Appendix A.. . • • • . . . . • . . . . . . . . . • . . . . . . . . . . . . . • • • • . 70-'

Appendix B......................... . . . . . . . . . . . . . . 72

Bibliography•••••••••••••••••••••••

Vita.•.............................

..............

..... ... ......74

75

Figure

LIST OF ILLUSTRATIONS

Page

iv

1. Rectangular wave guide showing choice of axes.. 5

2. Typical field arrangement existing in a

rectangular wave guide (TEl 0 wave)...... 6,

3. Paths followed by waves traveling back and

forth between the walls of a wave guide.. 7

4. Wave front corresponding to the situation

illustrated in figure 3.................. 9

5. Second-order mode in: a rectangular wave guide

(TE2,0 we.ve ) • • • . . . • . • . . . . . . . . . . . . . . . . . • • • 12

6. Truncated corner of rectangular wave guide ••••• 25

7. Diagram of procedure used in mathematical

der1vation. . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . 27

1

INTRODUCTION

The term wave guide refers to a hollow conducting

tube (usually ot rectangular or circular cross section)

filled with a dielectric (or vacuum) and used for the

transmission of electromagnetic i.1Taves. We shall con-

sider the guide to be bounded by a perfect conductor

and to be filled with a lossless homogeneous isotropic

dielectric (or vacuum).

The electromagnetic wave in the guide must of,

course satisfy Maxwell's equations. It must also sat-

lsfy the boundary condition that there be no tangential

component of electric field at the surface of the con­

ductor. (1)

(1) For a discussion of the general theo~y of 't'lave

gUides see: Slater, J. C. Microwave Electronics. Re-

views of Modern Physics. Vol. 18, pp 459-467 (1946)

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

If the electromagnetic Haves tr2.veling elong the

guide meet a discontinUity or irregularity in the guide

then there itlTill be a part of the energy reflected be_ck

in the direction from 'Hhich it CEl.me and a pEtrt of the

energy transmitted on do~m the guide. The reflection

coe~cient, R, is the ratio of the emplitude of the re-

2

flected wave at the discontinuity to the amplitude of

the incident wave there. The transmission coefficient,

T, is the ratio of the amplitude of the transmitted

wave at the discontinuity to the amplitude of the in­

cident wave there. Obviously the incident energy (pro-

portional to the square of the magnitude of the inten­

sity) must be eoual to the energy transmitted plUS the

energy reflected.

S. O. Rice of Bell Telephone Laboratories hes used

a conformal transformation method in an attempt to ob-

tain expressions for the reflection coefficients of

corners in rects.ngular -v.rave guides. (2) The transform-

(2) Rice, S. O. Reflection from Corners in Rectang-

ular Wave Guides - - Conformal Transformation. Bell

System Technical Journal. Vol. 28, PP 104-135 (1949)

ation c2rries the bent guide into R straight guide

filled with a non-uniform medium. The reflection coef-

ficient of the transformed system can be expressed in

terms of the solution of an integral eQuation which may

be solved approximately by successive substitutions.

No numerical results are given by Rice.

The present paper satisfies the bound2ry conditions

3

at the truncl:tted corner of the rect2.ngular u-Jave gUide

by essuming the existance of waves of higher modes in

the region of the corner. These higher modes are not

propagated along the gUide, but are attenuated rapidly

in either direction from the corner, because the guide

dimensions in practice are such that the frequency of

the wave is below the cut-off frequency of the guide

for these higher modes. All the energy of the inci-

dent wave must then appear as the sum of the energies

of the reflected and transmitted waves of the original

mode. The higher order modes serve merely to satisfy

the boundary cOrl<1itions at the corner.

The present paper considers the TEl 0 mode with,the electric vector normal to the plane of the corner.

Equations are derived involving the amplitudes of the

reflected 8.nd transmitted 1'laves c=md B.n infinite series

of consta.nts. It is hoped thE.t in the cOllsider2.tion

of specific cases it \-Jill be Dossible to obtain 2pDrox­

imate numeric2,1 v2.lues of the C'r:lplitudes of the ref'lect-

ed and trensrnitted waves by using only a finite and

practical numoer of terms in the infinite series, al-

though time has not permitted a.n. 8ttempt to carry out

8.ny ?ctual numerical computations in this thesis.

An advantage of the method used in this thesis

lies in the rels.ti vely simDle m2.them2.tics involved.

PHYSICAL INTERPRETATION OF PROPAGATION WITHIN THE GUIDE

Before entering upon a mathematical consideration

of the situation, let us consider 8. physic8.l picture of

the nhenomenon involved in the transmission of energy

along the rectangular wave gUide. (3),(4) Figure 1 shows

(3) Terman, F. E. Radio Engineers 1 Handbook. N. Y.,

McGraw-Hill, 1943. pp. 252-255.

(4) Sarbacher, R. I. and Edson, W. A. Hyper and Ultra­

high F,·requency Engineering. N. Y., John 't'l11ey & Sons,

1943. pp. 207-214.

the choice of axes for our coordinate system. Figure 2

shows the distribution of electric and magnetic fields

for a typical kind of wave propagation along a rectang­

ular guide. This field distribution may be considered

the resul t8.n.t of ordinary plane electroIDe.gfletic waves

tr2.veling back find forth betl,reen t~e sides of the guide

with ~he velocity of light, following p~ths such 2S

those shown in figure 3. These plene electromagnetic

l'l8VeS 'l"lhen reflected from the conciuctor surface suffer

a complete reversfll in direction of the tangenti2.l elec­

tric field component. Thus the ~otal tangential compon-

ent resulting from the sum of the incident and reflected

waves is zero B.t the conductor surface, as is required

4

NON-OONDVC rING

. DIELECTRIC (OR

VRCUUM) /NTFR­

lOR} OF "PER /1­FABILITY fA- RNJ)

PERMITTIVITY e:

CON'PUCT/N6­

VlRLL5

5

Figure 1. Re(rtangll1ar WRTe guide showing choioe o't axes.

6

y.. m

t'

T~~ <f) EB,~ '~

1, ,

II, I

OlE II -4-_

i

i ,I

I I II

, I I , I-- :$ I 0,

0 0 0

5 E C T ION 'rYJ-)'l

y

t

ELECTRIC FLUX

__ -- - /1RG;V£T/C FL U)f..

Figure 2. Typioal ~ield arrangement existing in Q

reotangular wave guide (TEl 0 wave).,

7

(ct.) FR E:Q.UI=N C Y &R.. r::RTLY IN E ""I.G!; S 5

OF c.ur-OFF.

(b) FRi:QUE:IVc-Y I1tJOc=R/1T£LY IN

E-x.C,i:5S of cur-OFF,

(c.) F R t='Qu J:NG Y CI-OS E: ,0 cuT- OF F.

Figure 3. Paths followed by waves traveling back and

forth between the walls or a wave guide for frequenoies

exceeding the cutorffrequency by varying amounts.

by the boundpry cormi tion. The we.ve fronts of' the in-

cident 2nd reflected "raves Are of cou.rse normp.1 to the

direction of pro:p8.gation end. ere illustrated in figure

4. It m.gy be shovrn th2.t, 't\ri th the vlave fronts travel-

ing back End forth ~cross the ~uide as illustreted in

figure 4, the resultant field distribution is th2.t

illustrated in f'igure 2 which trClvels along the gUide

2.nd represents prop8.gation of energy.

As illustrated in figure 4, the angle e between

the direction of propaga.tion of the cor.rrponent waves

(?nd the normal to the sides of the 1fEve guide is given

by the expression

8

cos e = A I 2 Yo (1),

where Yo is the 'Vlidth of the guide as illustrated in

figu.re 4 and A is the vJ'2velength of the component

waves. Since the largest value that cos e can h&ve is

cos e = 1, it follO\"ls that the largest wavelength

'V>Thich can give pro:p8.gation dO'tffl the guide is given by

the expression

AC = 2 Yo •

This is knOh'!l e.s the cut-off v,ravelength and the cor-

responding frequency as the cut-off frequency. At

this frequency the component "Taves travel b2Ck 8nd

forth across the guide with no component of velocity

along the length of the guide B.nd hence no propagation

,'/'-.. .. ..

... ;'.... ./.... ,'>{..."- ..... ;'

... ,", ....

'~,/

/ ,.... ...

9

__- rOSITIVE" CKE5T

__ ---/VI:C,RTIVEr C/?GsT

"

\'.

I

I

,'\,

I \ 1';;t'I \

I, I

" ' -'4--1>/\I ,

~"

~I \/ ,

Figure 4. Wave tront corresponding to the situation

1l1ustrated in Figures Sa and 3b.

of energy elone the guide t2.lces pla.ce. The guide acts

as R high-pass filter.

Since the component waves travel with the velocity

of light in directions other tr~n along the axis of the

guide, it follows that the r8te at ,{hich energy is pro-

pagated a.long the axis of the guide will be less than

the velocity of light. In fact it may be seen from

figure 4 that the component of velocity(5) along the

(5) For an excellent discussion of group velocity and

phase velocity see: Skilling, H. H. Fundament8ls of

Electric Waves. N. Y., John Wiley & Sons, 1948. pp.

200-204.

axis of the guide is

(Group velocity) = (Velocity of light) sin e(3)

so thpt, from (1),

(Group velocity) = [ 1 _ (~/2Y )2 ]1/2 (4)(Velocity of light) 0

It may also be seen from figure 3 that the component

waves ivill combine to give an apparent y,lavelength in

the gUide, shown as ~g in the figure, 't'lhich is gre2:ter

than the wavelength ~ of the component waves according

to the expression

10

~g I A = 1 / sin e (5)

Since there is no change in frequency, it follows that

the apparent or "phase velocity" (6) 't'!i thin the guide

(6) Ibid.

II

must be greater than the velocity of light according to

the expression

(Phase velocity in guide)(Velocity of light)

(6)

= 1 1 sin e (7)

= [ 1 - ( A/2Yo)2 ]-1/2 (8)

It is seen immediately from e'q~ations (3) and (7) the.t

(Phase velocity in guide) (Group velocity)= (Velocity of light)2 (9)

We might also note that as the wavelength A is increas­

ed and approaches the cut-off wave length AC the phase

velocity increases and approaches infinity, while the

group velocity (the velocity at which energy is propa­

gated along the guide) decreases and approaches zero.

If the wavelength is much less than the cut-off

wavelength then it is possible for higher modes to be

transmitted. One possibility is that illustrated in

figure 5 (TE2 , 0 wave). The component vT8.VeS are re­

flected back and forth as in figure 5b and the resUlt­

and fields are as in figure 5a.

-- - - /'1 RG/Yt: TieI/VTE"/V$/T y

- GLSC.TRIC

IN T E /II :?Ii Y I I I I I I II

, I

12

~/VE

f---.---------l" ,'" - : ==.... " ,." -;:: : ,- ........... ,. .".-" " ~ -, - "'"', " , .. ..."'" /.... ," ." ,',- .. '" ,,', ,,- ... \ t" I ~,

" , ',' t .' \'..' f '. I & ~; l' t' " ....."It' \' __ ', ,.' , ..... _- ,I,' \ _

_" I,~\,' - .. "t'" ;~f",'_.. , 11'_.. " __ ." ," __" : _" , _ :-:

- -------

END vIew

--... ..---- .. , .. ,-""','",-=-\\-, ',\, I , ... _ "', '\''\ \ • \ I I ", I I

... ' , I" I I I I I \.~ I" \ ... ~.

' .. ' ~\" ..... - ,.',-- , ......... ===_.TOP

" ... ;.-: : " , -~ -I' ," - , \ I ~ tI'" -,..

, \ , I ,

I ' " • -" I \ I I , I, I , I I I I".,_.. ;/,', " ...." ' ...... _ ' til J '\ ' ....

\,',-_ .. ~", , -' ... _--- ~

V 1l3"\V

(a) Seoond-order TE2 0 wave 1n reotangular guide.,

"-"-

"

-

--- - /VEG-RTII./E

C Rt= ~T

(b) Wave fronts corresponding to (a).

Figure 5. Seoond-order mode ln a reotangular wave

gulde (TE2 ,O wave).

13

It may be Shown(7) that all the possible modes of

(7) S8.rbacher, R. I., p.nd Edson, W. A., Ope cit., pp.

179, 185, 197.

transmission may be divided into two groups: those hav­

ing no axial component of electric field, and those hav­

ing no axial component of me.gnetic field. The former

are known as transverse electric, (8) or TE, 'tlTaves 2..nd

(8) The transverse electric we-ves ere sometimes known

as magnetic or H waves.

the latter are known as transverse magnetic, (9) or TM,

(9) The trensverse rnagneGic It!aves 8re sometimes kno'tlTn

2S electric or E It!8Ves.

1-raves. Subscripts pre used to indicC",te the order of a

wave. In the case of a rectengule.r 't'!8Ve guide two sub-

scripts ere needed. If the x-axis is tRken p2..rpllel to

the axis of the wave guide in the direction of prope-

g'p.tion, then the first subscript, n, refers to t~1.e num-

ber of half sinusoids or maximE'. of the tra.nsverse field

in the y-direction between the walls of ~he guide. (10)

14

(10) Sarbacher, R. I. and Edson, W. A., Ope cit., pp.

185-186.

The second SUbscript, m, similarly indica.tes the number

of half-sinu8oids, or maxima, of the transverse field ': .~.

occurring in- the z-direction. (11) Thus we see from the

eLl) Ibid.

end view of the wave guide in figure 2 that we have in

this guide the TEl 0 mode. Similarly figure 5 shows theJ

TE Z o mode.,

15

HATHEMA'rICAL CONSIDERATION OF THE 'rE WAVES IN THE GUIDE

We shsJ.l prooeed to obta1.n the expressions tor the

difrerent eomponen-ts or the '!'E waves traveling a10ng the

guide shown in rigure 1. (12) The rield intensities must

(12) Sarbaoher, R. I., and Edson, w. A., Ope cit., pp.

177-187.

or coarse satiet'y Maxwell's equations as stated below in

di~erentIa1 form t'or a.homogeneous, Isotropic, noncon­

ducting medium ot perm11itivity e and permeabil1t1' J...L.

(ID)

(c)

16

dE~ d Cl- ?J H~=-jA-

d¥- ~c ~ t

;" E 7'- d E~ d Iff=-;-t "J tdl: #~

dE~ "'0 E~ aHa- -~-

a~ ;;~;)t

(1/)

(c.)

I~ we assume ~hat the e1eotrio ~le1d intensit7. E. and

the magnetio field intensit:r. H. inTolT8 ~1me 0D17 1h

the ~orJI e1e»t and d1stance onl.y in-~ tora e-n:. where

CD 18 the angular trequeno:r and If' 18 the pr-opagatlon

oon8~ant. then we haTe

( 12)

where E' and H' are ~o.notlO118 o-r 7 and z on17. 51noe

we are going to consider anI:r the If'E If&Ye8, we will I'et

Ex· 0.' Substitut1ng eqo.ations (12) and (13) into

equations (10) and (11) gives:

17

c>H~ CJ JI~ - 0 (~)-CJ.f 'de

"0 H~ I I (/4)+ o H~ - L.t.uf E'{- (b)-()e:

,, a H"X-

I [; t_)( 1-1 ~ - (.; '-U t::. i! (c.)

o~

(a..)

(c)

We must now solve these equations 'for the varlous com­

ponents o'f E' and H' and thus obtain E and H. Differ­

entiating equation (14b) with respect to z gives

,;l.H~ oH~ "dE'+ ~ = i. /..,.t.) 6 ~

"2-a2: C> ~ aZ;

18

Difrerentiating equation (14c) with respect to y gives

=iwc. (17)

(17) !'rom equ.ation (16) g1ves

(JH; ~H']+ lS' + 'f ::::d c 2)+

Subtracting equation, ,

() 'Z. H" o'L H~+

&1c'L ~12..

Differentiating equation (15b) with respeot to '7 gives

Differentiating equation (150) with respect to z giTes

(~D)

Subtraoting equation (20) from (19) gives

+

SUbstituting equation

~ H''1-

19

(15a) into equation (21) giTesI

-aHl:~ c (.2 2)

SabstItat1ng equa~lona (15a) and (22) into equation

(18) gives

We now have a differentIal. equation in Hx ' alone. '1'0

solve equation (23) we shall assume that

H; = Y l (~4-)

where y. Y(y) and z· Z(z). Substituting the

proposed solution giT8n by equation (24) into equa­

tion (23) yields

,-yEqua"tlon (25) can be true only if each term or the

lett member or the equation equals a constant inde-

pendant of y and z. Thus, we have

I ~~ YR,- -y -d ¥ z.

I (j "l. eR'l..- - --Z:. d i!=2-

20

where

The solutions of the ordinary dlrferentlal equations

(26) and (27) are of course

(~8)

y = (, ~'{ii: '+ = C2. Cc-3- VR7-f.

l = C3~ '{ii~ ~ =: elf~ VRz. t

One possible solution of equation (23) is then, ac­

cording to equation (24)

where A - °2.°4 • Equation (31) is the only one of

the possible solutions of equation (23) that will sat-

1sfy our boundary condit1ons. We may now prooeed to

find the remaining components of the fields by substi­

tuting equation (31) into equations (14) and (16).

For examp1e, eliminating Ez' from equations (140) and

(15b) gives

SUbstituting equation (31) into equation (32) gives

21

Simil.arly. we :find

Let us now use the boundary conditions for our rect-

angular guide to determlne the oonstants ~ and AZ

'

Sinoe there can be no tangential oomponent of elec-

tric field at the surfaoe of the ,guide, it follows

that Ey- I :: 0 when z :a 0 and when r = zo' and also

that Zz' • 0 when y =: 0 and when y • Yo. We see that

equation (36) will vanish a t ~ == 0 and also at z = zo

it

where m is an integer. We see that equation (14)

will vanish at y a 0 and also at y == Yo if

Where n is an integer.

Let us colleot together the expressions that we

now have for the components of the transverse electric

(TE) waves. Remembering equations (12) and (l~) and

the values just found for the constants VFf: and I{R~,

we haTe the following:

(4.)

22

(d.)

From equation (36)

-==-A ~~He ~~ +IA./~t<- 6 ~)e--(T.~'

~r~: t)Q..i"'t-y"..,~(C)

have TE waY8S

23

From equations (28), (37), and (38) it follows that '6 J

the propagation constant, is

If the quantity under the radical is positive then ~

is real and attenuation takes place and propagation

does not. If the quantity under the radical is nega­

tive then (( is imaginary and propagation takes plaoe

without attenuation. When it is imaginary ~ is usual­

ly referred to as i~.

As a matter of interest we notice that equation

(40) gives for the cut-off frequency of the TEIO mode

the expression

(ff2-- We. fA- t::-

V~ t::! -fe:--;z '-/-lj -- -V-/-f- c

'/-11 -;2..

::<. if lJ= A.c..

which is recognized as the same expression found

earlier when considering a physical interpretation

o'f the phenomenon.

24

DERIVATIO OF EQUATIONS FOR THE TRU CATED CORNER

At 8 corner in E ~<Jave gUide some of the incident

energy will be reflected back in the direction from

1-{hich it CE'.me. \ e shell consider only the case -here

the wave incident upon the corner is the lowest, that is

the TEI,O' mode. We shall further assume thet the si~e

of the guide is such that no hig2er mode can be propa-

gated 8.10ng the guide. Thus all the energy reflected

at the corner and all the energy trcmsmi tted at the co!"-

ner must be in the form of a TEl 0 wave. Consider the,truncated corner shov-m in figure 6. The z-axis is per-

pena.icular to the plane of "CUe figu.re BUo. extends to-

ward the ree.der. In order to satisfy the bowll1.ary CO£1-

ditions at the surfs.ce of the guide we shall Bssume the

presence of itm.ves of higher mode s at the corner, spec-

ifically TE -VJB.ves "There n te.kes on all values frann,O2 to infinity. \ve have already said thet the dimensions

of the guide itlOuld not permit these higher modes to be

propagated but that "Ghey 't1Tould be rCl.pid..ly attenuated 8S

we move away from the corner in either ~irec"Gion.

Assuming unit amplitude for the incident 1:113Ve, we

obtain from equations (39) the expressions i'or~he field

intonei ties in guide 1 and in guide 2. Vie let Xl eau8.1 1

and m equal 0, and we include the higher modes for i:[hich

x,=Xc-e-ee-f~8

:j' =x.~ e + ¥-~ e yx~::::x.~e+~~e

!/-.,. == -X-~ f) +:t~ e

~1>-'b

"{o1;~~

J j)lE' J • X

~

Figure 6. Trunoated oorner ot reotangular wave guide.f\)\.11

n is greater than I and m equals D. Since the second

subscript, m, is always zero in this paper we shall

simplify the notation by not ~ITiting the second sub-

26

script.

e.nd the

Thus the TEI,O mode will be designated as TEl'

TE mode will be designated as TEn. Also,n,Dsince for the TEl mode the value of gamma is a pure

imaginary we shall write i~ for gamma in the expres-

sions for the TEl mode. For the higher modea the value

of gamma ia real and gamma ia the attenuation constcmt.

In order to help the reader follow the procedure

used in the mathematical d.evelopment "t'Jhich follovrs I've

refer the reader to figure 7, a block diagran of that

procedure.

(42) Fields in'guide 1.

(~)(43) Fields in gUide 2.

I(45)

(46) Hxj(48) Hy at x:-.0

(50) Ez

(47) Hx)(49) Hy at x:'..(511)' Ez

I(52) E~ at y=d. - (53) Ez at y-dL . I 2 .

~5)"'V

Equate to zero andadd and sUbtract.

~

,. "t

l\,)-..J

~ Add & subtract..~

truncation.(77)J L-.'~ -:--

~---- -. I(78)

I(79) 0-0

r--- - - - - -~

(91) (92)

L (~13)--..r1Combine.

(94)I

(95)I

(96)I

(97)I

(98) Equa-tion in''''

,. - -- --- -,(64) (65)

(6~) J"T'1Combine.

(67)I

(68)1

(69)I

(70)I

(71)I

(74) Equation in ¢.

Boundary oonditions at vertical interrace; I lat hor~~ntal(54) (46)=(47) (56) (50)=(51) (55) (48)-(49) (76)

I ,I I I(60) ~(61) (82) (80)

I (62) J I I(63)~ (83)' (81)

\. I I--"""l (86) (8$»)

Add & subtract. I ./.. (87) ¥

I (89)(~8) /

(90)

Figure 7. Diagram or procedure used in1mathematical derivation.

From (~9), for Rn H1,o wave in guide 1 we have

't (7f")( -~~~I "1- 'f3 X) co I (J, 7T ) '6~"'X-14. H~I e- tUj

:=~ ¢o ~I t2- +R Q.. (. I +f;.2 CJ.t ~ 'J; t l a-.

b H - L'~t1/ ~

(' (J

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Nro

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Now, writing everything in terms of x,y ooordinates (lee fig.6), we have in guide 1

<\ Hx-, e- zwt:;::. ~[i("1-""':"'B+¥-~8)][£~H")('~9-,f~ 8)

+R <l.+L ~ (X-4ra8- ~~ 9)J00 .

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- (3 +war e ~ () 'to 0 ~J e.

-R l2. +i p(?!"-r1-9-~~9~

;-, '(... ( n. 71") . lh7T ( ~\1:~ c.., '(...~-tw't.rC 1-: ~ '10 ~~e"'~~~J.

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The oomponent ot H in guide 2 along the x-axis when x D 0 is:

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Sinoe expressions (46) and (47) m~lt be the Bame for all values of y~13)we haTe trom(46) minus (47)

(n) 0 ={e--[;' 'J-4-c-el[;i(3~9+ (JZ-T) £i ~r:- 8]

/+t~ (C~ -C~)c.-e-[~ f4-<-~a..-y",("""B} ~e

{~ f3

+ ,.+w 'I.t' f::-~

(r~).M-[¥. ~e-..9][~+ 'P,-o.:-~ (-R+rye.-'-('1~ 9]

OQ ~ • 9}I ". 1fJ\. )"7T • ,,:77"" - '"'I~ ~~-L c,,-C~\ • .. (~)~[- 1~81e ~8~::; 2. / ~.,. + t.u ft = ~o J

(13} Bee aPEend1x A.~o

Sinoe expressions (48) and (49) must be the same tor all values of 1,(14) we have trom(48) minus (49)

{

) +~~~~e ) _L~'t~eJ(Ss> 0:::: Cfr"-' [-fa 't-~e]r<Z.. +(-1<:-1 e

~ ( I -a.) [~~~el---(~f~e}. e+L- '\ C/o,. - C .... ~ ~." JIZ.- ~

h::: 2-

{i ~ f!!..). [11" ~[ +-L'f~~8 . ~ e J

+ -~"'+w't<E\t. ~ ¥.'fCr<-~ e.. +(-R-T)£'P:;' J

co_~ (c I ~C ~) CI"" (ill\ · [!i!! &1 _"?I.,#-~ B)f:;:;. h I,. Y,,'- +- w ~f< (; ~.J- :t-. ¥-e-- J <Z- J~ e

(14) Ibid.

1:

Sinoe the expree.lonl (50) and (51) must be the sarne tor all values ot y(16) I we havetrom (50) minus (51)

(SIJ).

O (.. tV A. (Tr) f" 1= _ ". Tr +L ~ 9 ..-(3~+-"'I'-E ~. ~[~ '1-e--'1, (l- f->¥ + (R-T) £ I..f';~ 9

04 •

-L-. IC~ - c:\ t.I..Wf< (!2JI..)~l~ ~e7 - '6;-.. 'f~ eh::.2.. V / Y.,. -f-J.-u Lr- f: 1-6 ~D ~ Je.

iNow let U8 simpl1fy our expressions (54), (56), and (56) by making the tollowing sub-stitution.1 .

(s~ d ~ = C' - C 2-~ k

). (>17r

(.8 Rk == n. .A..<-,.,.. ~ ~~ e)?f':+ w1.r-t:

<Z. - ~~ 'f~s

(59) 13k

==~(~ 1-e-... e) CL- 'f'.. 'f~ e

(15) Ibid. -t:-­I\.)

(bo)

Then rewriting equ.ation (M),and dividing through by ("/'10> sin', we baTel

~ [ J[ +L(3~9 ) -~ 13~ 8)o =TO CNt 9 ~ ~ 'f~9J (1. 1- (R.-T 12.

¥ ~ {.. (b [rr c,:. Bl[ + ~ ~lf ~ e+~~BL- :B~ d~+ '& I. 6 ~ ~ t J Cl..

1i ..., :2. -~ +'w t<

-(-R-T)a..-':(3¥M.... 9ti.. R d 'z{J J,.::: 2. >-t ~ h-

And rewr11ilng equatlcm- (6G), we haTe, it we dlTlde tbrol.lgh by (p Jo)/(w J.L ",) andtranspaee taral

(61)06

- i g L R),. d.~r h::2-

- ~~ [- -f"+/-v>.r-l:~ [f; ~~8J :L (31"':""19

+(-R-T) <2.- ~ r~ ......:.... 8]-t­\A>

From equat10JT' (61):

(112)[

+lp'~8 ) -c:.~~1 -p~ -t- tA../I.f<~e. +( R-T (l, == - -----

~ [;0 t;.~8J

QCJ

L AhtLhh::1.

8ub8titutlng the expression from equation (&2) into equation (60) and rearrangingterru, we have I

(l,3)~ [;0 ~~ 9] (4- L' ~ M- e ( ) - Lf~ el

i [3 e. - R-T a.. J_~"'+w",....e

0() ~

= L Rk. d.. ~.. - f e,,:C sI- 13" d ..h~L h=~

1. t. oa

[71 ~1 - (3 -to W f< & ~

+crt e cd "¥. if. '---8J f; f;-~ Rl.. d ....

t

Adding eql1at1ons (61) and (83), we have:

( ~ I. f ,...:..... [f. '1 c.,... 9] e.+- t. (' If~ e = r R d (¥...- .: (3) +c.t e~[If; ~ Cr<.~•t.t 2. .. ~ .. " '- "I,. 'f~ .. h~~

ClO

-fS"+w'"f<E;" A d -~c.:tB£ "B~dl,L- h h.,,- ;,=2-.2!:. ~ :: 1-

-#0

8ubtraot1ng equation (61) tram equat10n (63), we have.

(b5')~ • B ~

I ~[¥;~~ej(R_T)<Z..-Lrs1"""'" =L R...d.... (1I,,-f-lf)_~f.~ ~ '&. C h'=2.

-~ + w f'

[71 ~1 - f!'t..+w ~.k,f 01 ':/- r

+c..;t f) 4ft ~ f~ 9J ;r- £. A~ d",- - ---!...~8 L-"B~ d>...~o J1 =2.. T ~:"3.

~\J"I

iOWriting the oomplex oonJugate or equat10n (64), we have:

(b6 ) - 2- i ....:.-. [f, 'f~9) e.- LI'~"':- ~i.... Rhd: (1f'h+i.r) + c..te c.etC;' 'f~9}~ -f"+w"-f<-E: 11=2.

00 '*-f3 ~+w1.;<-E:;- A d* _!!. 4t 9L 'Bft d.J,.

-rr '-- k. k. 1T h c."1-- .. ),::2;-

Mult1ply1ng equat10n (66) by (R-T) and subtracting the result trom equation (65):

(b ,)00

o =L~ -::: '2.

2. ]7T -~+w~6

R>.. [r Y... -t-': (3) +4t& wr['f6 ~"--~ t -~~ e 13" •

{ d" -(J(, - T) 4"'1<" ]

1;

A solution of equation (67) 1s:(16)

(b8) (R-T) = d k

~d.."

~i'/"= e ~

Now we shall proceed to substitute this value of (R-T) from equation (68) into

equation (62) and at the Bame time remove a factor e i ¢ from the lett aide of

the equation (62):

(69)~~

e.i.(-~+~'f~e) i(ep-~~~e)

<Z.. +<2..

"2-

:=. _ -(3 +UJ7..f4-i::. 00

~[TT LR d.~o '4~ eJ Jo\:'2- J., ~

(16) See appendix B.~~

Writing the expsnentials 1n tr1gonometrio form:

e ~ 4 [Co-<>- (- ~ +-(3tM.... 1'1)+1.~ (- (jl +~¢ ....:...... e)

+- i ~(I}- (3~~ 1'1)+ e.."..o...( cp - ~ ~.-:.... e~== _ -;3"l..+'-'O &.Ii b ~

~[:. '1-c.-...~ k~ A/,. d;,

(70) ~f:L <Z ~(~-rJ~e)

<>0

~"- - .... 2..f<-6 L Rh

dh.::: - ),=2-

Writing out the value ot An and dn trom equations (58) and (57) in the equation (70):

('11) ,-e..i4>~(tP-(J~~e)~[f.~e-<-&I] ==

~ (n~ ~(p"--W"-r-~L. (C~ -c:) n~ ¥: ~~~ a.-¥,,-~~G

11=2 « + L-v1.r<-c.1;;

equation (40) and the suooeeding discussion

-.L

[ w~j<e - (;:r] 2-

-L

== [( ~:r-w ~~~] ~

~ =

~h(72)

(72)

We shall proceed to simplify equation (71) by 8ubstituting for r n ~nd ~. From

we see that

Henoe

(7+) a. i 4>e-- (4) -(3~ ~ e) M- [f. ~ M- e] =00 c'2.._ I

~ h. Ck • (nTr ~ -1f~'J~8t-- -~ -;;- ~~e e.h ::::: ;z. ~ M. ".0

~

Now let us consider equations (52) and (53). If we replaoe x in equation (63) by

-x then the range of values tor x over whioh equation (53) must hold true will

be the same/range ot values of x over whioh equation (52) must be true. Repl~o­

ing x by -x in equation (53), we have:

(75) E a.,-iWt']r:z.

~;;d.

-- Tl.'wft

2-- ~ +w '1-f<. C

(~)~[i; rX ~B+d~ e}].iPf~~"-J~9]

e

;- ?- ito?- (Yo.1T) [rt.TT"f ,1]-~ c... ?f...2.+UJ7'-t ';f- • .-k..... F x.....:...etd e-. IlJ •

~ fx~e-d~8}~

\Jlo

Expressions (52) a.nd (75) a.re each equal to zero over the same range of values of x.

Hence, if we add we have:

(76).

(WI" ()0=- Ir. · 1/-(3" +w·"I"·6 ~. ~[-v.fx.~e+d.~e}J.

[-i~f-x.~6-J~9) . ""iP{X~e-J~e}J

a.. +(R+T)~

00 .

-L(C~ +C~) I. l..W;.t (~)~[!!I!:[X~8+d.~e}].h=2. ~h +-1AJ2..f<f: ,"0 ~6

1S'~ {"it~ e - J~ e }e

\.J1I--'

In a similar manner if we subtraot equation (75) from equation (52), we have:

('1 '1) o =- _~:::~~~(1f.)Ai-[f.fX~B+d~e}].

[-i~ f-x~e -J~ e} +if {-X~8 -d. M.-B}]

(L +(R-T)~

_t (C~ -C~) ~ iw

,: (~)~ [?i![~~ e + JCfrL. e}] •~:::2- '(~ +w;.<e ~(J ~o

'l!~ {):~8 -J~ e}fL.

V'tN

-DK~CDrl..--

,~

Q)

53

"4<tl

Q)

~r;--a

<l)

-~-~

j-u+

--...)1

.-

Q)

~*

)0-<

~

I

-~"-J

-"'~

I"

"""---'~

..--.J

<!)

~~\

0

r

jL~

*"r-

..."

\

IJ

en\

tt:'-{

~""""--

~(1)+

I

-t~

---....-c

7

...........~

\a

+'--v---J

.i~

~\

0

~~

L~

ea-'-.J

'0

..J

-~-+.

:-.~

~

)~

~

--...."-J

+-

~\

I)

t!~

~)c)

~;-

'

:t.. 1-~

:1

u

~

I

•.J+..

-~

C!L

~

IaW

r.,\I

'.r

C)

In e~u8t1on (78), we 8ubetitute the right member ot equation (62) for the quantity

inside the braokets and write out the value of An and dn :

o =•

L. w)-{ (11:.).' [71-r'"+W"f<.{i ~. ~ v.{(J.-r)t.-e~9+-~~9)].~ ~ .

[-(-~"+4J'f'-t:)r.(C~-C::) '" nOb - i" "~_SJJ1::: 2. y -J,.. lJ.) ;-<i:

OG iUi~

-£ (~~-C:) Yo .. + Ll'e (¥)~[~f(d-J:)tAt.....e~s~ =. 2.. . it.. I,.() If D t/. D L'

+J.~8}J ~_~k.'4~9

(19) 0 == 0\.n-t--

Now we shall prooeed to eliminate (R+T) from equat10ns (55) and (76). Solving

equation (76) for (R+T), we have:

(BO)\ 1'0 ) +i.B[.x~9-d.~9} . f"I In +T e. r == - a.-<~ JL':'-9-d.-:""S}

CIO ~ [>11Tf' }]~ (' '2. ) -p. +w ...~~ ...k- V; ~ AUt.1 e -hr1~ s }(" f;;t 41l.9 -J.~9- L- Ch + Ck" 7..)z - e~... 1f... +w"t<-E: ~[.f. {)i.~9+.I~8}]

In equation (80), it we let x. (d - y) tan e, thene

, -iB#~8 +if3#M.-9 ~ _~I.+wt.~!:(81) (1i'+T) tZ \ = - (£ -L (C~ +C~) 1. ~ 11·

J, =z. ~" + w ~ E:.

~[-i: f(cl-~~9~e +d.~9]] Y~~~9-----------t.L~[~{(J-~)~ 6~8+d4_c.9]]

\Jl\Jl

Multiplying out equation (55) in anticipAtion of solving for (R+T), ~m he.va:

(8 ') '('IT' ~1 +L~}~8 ( J -, _'e"_~9L4. O:=. _~ ~tc.,c..~e- ~S-~~1~~(R+7je I..

rr ~8

+ ip ('][.), [1T . ~91 + ~~;. ~8-f- +w~f<.e '11)~ Vb 'f J tL C-p-tz... 9

,I~

_(+'<JL,w.t (;.) .......... [f. lf""'",(R+T) (/.- i.f~ ~9~ 8

t:JJ. ~ (, 2.) [ hrr ] - ~k 'f~ 8-L- CJ.t+C~ ~ ~lif..~fJ tZ . ~()h=~ ~

~ (' 2.) -6 (HV) [- ~l -y q-~ I)- b1. C" +Ch. 't'..... +-;;'r f -v: ~ "i. ~~eJ ~ ... Cv<L.6

\J10'

57

~~

(I)~

<I)<b

~S

~~

~rq;-,

rq;-"'rq;-,

~~

~g

~~

~~

t::'~Q

.t~

~l';;"t::\:'

t:::\~~

'----J

·f~

'----'

·t.~

-{-~

~--...

~\~~,;

~~

~\;t,---'

"'--'~~

\IJ-.u

•'U

..u•

;-~

en}

~

"-~

..~

en...~~

~)G~

1<n..

~

-~."

+-..,

{.

+•.,J

+~

to!~

tl.i

..CIL

~C!1-

,\

pot,.

t~

++

+~

-+,

.......,

CI)

en~

<n<

I)CD

.{j.

+.~

"7"'

-{.~

~J

~Q

;"l'Q

)1'C

i;I~

n~

~-\-

~~

-.tE-c

Q)

i!~

~~

~r,..

-~0

-J=:1~

k:':'I::\Q

~ht&

...:..-4

8W~

",,::»-M

!'---J

t......:..-I~

0

U\-to

~

!-

en....I

til-..,

Ico

'~'--w

---J-s::

-\-~

0......pas+

:::I0-

....e;..,C

)

bDs::~rl

W0tn

~~

Let

(84-), 2-

S =Ch+C h~

SUbstitut1ng equations (B4), (72), and (73) into equation (81) and transposing:

(85")• .' 00

( )-if;'~9 +L(3~~6 _ \"

1?+Trz. +<L --LJJ::z.

.s~ - ?f~ '1-~e~~ .

[>17r{ cl . :a. e }]

~ ;fo e"c.. fJ - ~ ;;: 9

· [Tr{ d ~'Z.e1J~ ¥D ~8 -+. ~9

\.rI00.

59

~~

<b<b

Q)

Cb

~5

~~

,.....,rq

;t~

'Q)'

1il»

~~

g£

-*~

)t;.~

~;

klQ

~):tM.

..tn.

-l-.-"

'----.A•~

'----J

--~

-{-{

-ttt:>

~co........

-~r::

---:"'..

l'~

--;--...

0~,~

;'l~oM

~~

.lb"

.p.:t\>

l&l~

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

............:.,

::sc:!L..

c:rL'P~

~~

~0

'.~

•...J

Cl

-,J

•+

+I

++

0.pC

I)<0

<n~

r::Q

;).~

....-f

-~-{

.~..--.

~-~

-(Y

o)V

)~........

t'f,.~~

(i)''Q

)1"d-

'lL

~~

gW~

r;.

...,

~~

co+

..~

110-~

I-

Cl

.3'\.~

C\l

IIt::1~

J:::'~~\

~t'-

l...,,;-J~~

~\~

........

~"'-'

..CD

~L

.--J~

-of

~~

.qt

co,

........ttl

~~

~s::

lXL.

0'-..J

....-f,

.p~~

:;j0

-

~CDb£js::

4-o

M

..s:s,.p:;j.poM.p•.0

\Q'

:;jttl

e

Multiplying both aides ot equation (86) by the denominator on the right:

(87) [e--[f. fe-..8].M-. 8 + i.~(~)~[t.~~~C-r'- !/}(R +T) tZ. -. n"":- 8

={_ ~lf.~""""~9 +ip(f)~[f. ~cr--~~8} ~""N";'" Bco\ -Yh1-~9{ --L- S~<2 ~[!!!. ~~81~8

11: 2- :to 'T :J

+ '0. (~)~ [y: if.~ ~~9]

0"­o

•

61

(J)

Q)

~"{

<I)

"4'C

i)""~

<J:)~

~~

~

)oC

1"1'~

J::lO~\;

'~x

*~

~-~

L.......J

~~

~l~

'b-~

••c.

-J~

A,

~{1~13

i-:l\.}

:n

cn....-.....;.....-

r~

•..J

)o~

\!L+

+r

-.."

+~

<b.

-j~

~<I)'

-{•

I~

s::~

I

Q)+-~

...~\1

I"'(f)-tCD

.~

..Ii-i

~,.,

.f.

~

CD

~

--~

to~+

'--'

~-..I

0~\O~

ea-,

~\,O

0

"."

~..t'~

-1~~

~I()

I

L--...

!'oa

1='~

'---'

~

~

-~

"~'-

--t

0

~-+-

~'{L

-~0::~

.........~

~

s:I~

..~<J

.j-

-~

L..I

\.tl~}"

I:"-~~

'1f)1

1='

CD~

U1

-~

0+

t::

<[)~-<

0~

0-~

'~

............8W

tl

4'i

n.~~

p

<I)

.1

ea~\~

V}~

p.t

:I~

--{

0"

--..,1

0

e~

-{8W~

CD

-s::

~

....)'\,

I~

s::~

l'0

•t:::\•

....

a~

~t::

II~

Sot"-

~

CI

!'lL

:sJ::I~

.p

0-

-...,•

tlO+

'---'

t::

if~

....s::

4'i

4'i

~

~

.....Pc

0~.

.....

0

..p

Q)

rot

...........~~

~

~

SUbtra~t1ng equation (89) trom equation (as), and then dividing by

i~ (10111)

(90)

sin (11/10> Y 008 9] 008 e , we have:• CO

( )_(J~v.~e +i(3~~,-B_\

1?+T e r -fZ, -L-~='-

{ ~[f; ~e-9J~8 •

, ['>1"" f d. ~2.6]]~~ l~9 -'l-~8

. [.".. { d . 2..~]]~ ~ ~e -; ;::9

- ~[~~~91~6

}

5" ~"'''!f M.- (;;

_?f.(~)~[~~~91~8 . (!I-~) . fIr. '4-~Bl4-c-8~ hlT '1- 0 J '(3 -:;r~ ~o J

~N

..~~o()')

-\0co-

,Q

:)

~~j*t::\~I......:......J

-4~--­ en......•-JI

~~.~

rq;-,

~~~,;.

a-......

L8W~

\1~

Q)

oJ

6.3

64

•(()

Q)~

Q')

~~

-{r"(b"\

~Q

)s

~4

~-{

J+k:\~

~k\~

L---J

.{.)\:.

'---J

~-.:-

-{~~\~

k:\Q

,-t"

t\>~

......-:--.~

~\~(!L

L.--...J

"--J

~

~\I)

-...'

,en--.J

<l)

~

"~+

(b~..CD

'q)I

,....----,,.

.,~

,---....f~

~

~\1~

~~

~~~j

..l::\~

-L

.-.J~

.0

~*

~CO

1::\0-

,\

""~

s::

~\1~\1

'----'0

I-{

-rt+

3'---v----J

CIS~U'

8W~

'-v--J

\.-v--J

11'G

)

~'4J::l~

a~~

~A

0

"-....,..."

JotL

-.--I~

~0

.{.~

~.J{

-.{

+0m-s::

~0

tn.........

•.,J

+3

-4-as

~::$tr

ctCDbDs::-rt.p0cdM.p

~,p::s

mrtl

"'"-"

~•

~~~

Q)

.~,---,..,

Qi'

en~.S

<br-(

I)'

./Sf>

l\>-

~0

~1;'

\::,ltrL

...:-..l~

L-.-...I

.~*

'P~-{

kt0

~~

,~

~~

~~Ik

~L

...JttL

~*

.t.-

'-...,I

~\J)

'!LQ

).-J

..-4

~CD..

,.,

,.

~~~

~~.

~\~~\1~

s~4

~r<

b...,

-~

t\2~

enQ

-t::1~

31>~

.N.

s::L

-...J,

~)

0

0I

....~

Q)~\1

~*

.p

(

Q1

~~

.....,:::s

-{0

''---v

---'C

)

\-ofxW7.

'--'--'0

'-v-'

~~l

QQ

C)

IIr

~~

\:::\~~~

p

'---'"as

L.-..-..I

'----J

til<1J

)o~

::s

.~-{

or-,

.

.~S

::.

,00M

~G

)en-.

r-f.....

~I

0~

0G)

~.s:::.pbes::....~

...,....~<1'\~

65

Let us multiply equation (93) by (R+T) and then subtraot this from equation (91).

We then have:

(~~o ==t {~rf. 'J-~1~ ()- (p( fF-)~[f. :r:~e) ~ B

•

. [h7l{ J -!t~:e]] [n7r1-~87~9~ ,#-0 1"-<rl-9 e.-- - ~!fo J' .... e}]71 do ~.

,.......;.,.[~O{~8 -1-r_,<; _If,,¥~8[5 -S::(R+T)]

( ~(1) . [~~~el~ s}e (¥-9 . [E."" 'i~9}e-..9v -.............. "0 J . B _ ............. ~o-~ h~ ~ ~r ~

Q'\

0'-

(9S)

A 801utlon of equation (94) 18:(17)

*Sh.-S~ (1?+T) = 0

(R+-I) = Ski5: - :1.. i- Ife.

SUbstituting this value ot (R+T) trom equatlon (95) into equation (85), and separating

a factor elf trom the lett member ot the result, we have:

(36) <2. i. Y' [ a,.L ('1'- ~1.....:...e~ 12. it- '1'+ r;:~ 8) ] ==

rJJ

-L-.),=~

(17) See appendix B.

51-\.-"k[

),7r f d ~2- f) } ]~~~9-t~8

, r7r[ J. ~2.{;)}J

~ VD ~(!) -~ ~!)

_'rS~ ~~ 8tl-

0'.-...J

Writing the left member of equAtion (96) in trigonometrio form, and simplifying,

we have:

(97)

• ." oct IM1r f J. ' 2- If; }]

2.(J.~ e",v(Y'-~'1-~9)=£S ... .....:... ~1~9-;t=8h=:z. >t ~ [2r. {_J. _ ~'3.S }]

~o ~8 }~9

e.-"?fh '-f~S

Rewriting equation (97), we have:

• a. G]J't 7J d. ~ _(98) e: e--(r-~1A.h..9)...u:..,[~{C-r'9 -~ ~G -

_; . C~ +C;:~[~{ d. _ -#-~"&.~}J _¥~~~9~ :z.. n.. 1-0 eo-- 9 ~ 8 tZ..

Q'\();).

69

CONCLUSION

Equation (74) expresses a relationship between ¢

(the argument of R - T ) and the physical dimensions

of the guide in terms of an infinite series of constants.

EquC'.tion (98) expresses a relC'.tionship between 'I' (the

argument of R + T ) and the physical dimensions of the

guide in terms of an infinite series of constants. It

is hoped the.t i t ~'ill be possible, vlhen given the num­

eric8l values of the physical dimensions of the guide J

to obtain reasonably C'.ccur8.te approxim2.tions to the

values of ¢ and 'f by satisfying the boundp.ry conditions

at a finite number of points in the region of the cor-

ner, thus neglecting all but a finite number of the

constants in the infinite series. If reasonably ac-

curate values of ¢ and " can thus be obtained then we

have the values of R - T and R + T from equations

( 68) and (95). By taking half the sum ana. ha.lf the

difference of R - IT' and R+ T '....e IDPY obtain tr-:e... ,

values of Rand T.

APPENDIX A

We here show th8t setting the comnonents of the

fields in guide 1 at the boundary x = 0 eqQal to the

c·orrespond.ing components of' the fields in guide 2 at

the boundary x = 0 is 8ufficient. That is, it follows

from Maxwell's eqQations thet the derivatives with re-

spect to x of the components in 8uide 1 will also be

equal to the derivatives "(.nth respect to x of the cor-

responding components in guide 2 at the boundary x = o.

From the fa.ct that Hy in gQide 1 at x = 0 alw8·Ys

equals Hy in guide 2 at x = 0 it follows that ~HY/~t

in guide 1 at x = 0 equals aHy/at in guide 2 at x =o.Remembering that the time derivatives of Hy on the two

sides of the boundary x = 0 are equal and that Ex = 0,

we see at once from equation (lIb) that aE lax in gUidez

1 at x = 0 eQuals ~Ez/ax in guide 2 at x = o.

70

From the fact that E in guide 1 at x = 0 ~lw2Y8

equals E in guide 2 at x = 0 it follows th2.t oE/at in

guide 1 at x = 0 equals ~E/at in guide 2 at x = o.

From the fact that H in guide 1 at x = 0 eoue-ls H in

guide 2 at x = 0 for all values of y 2nd Z" it f'ollovlS

that dHloy and oH/oZ" in guide 1 et x = 0 eauD.I, respec­

tively, ~H/ay and 'OR/oz in guide 2 at x = O. Renewber-

71

ing that the time derivatives of Ey on the two sides of

the boundary x = 0 are equal and thpt the derivatives

of Hx ~Ti th respect to z on the two sides of t!le boundary

x = 0 are equal, we see at once from eduation (lOb) that

aRz/~x in guide 1 at x = 0 equals ~Hz/ax in gUide 2 at

x = O. In a similar manner, ~ve see at once from equat­

ion (lOc) that dBy/()X in guide 1 at x = 0 equals dRy/ax

in guide 2 at x = O.

72

APPENDIX B

vJe here sho'Y that tl:e ill2.gnitudes of the quanti ties

(R + T) and (R - T) are equel to uni t~r. Since the re-

flected energy plus the tr3nsmitted e!J.ergy !!lust eaual

the incident energy, 'V-rhich 1'le have 8.88U!!led to be unity,

we h8ve

(99)

(102)

Let us break Rand T into reed ana. imagin2.ry parts as

follo~rs:

R = Rl + iRZ (100)

T = Tl + iTZ (101)

Then if we substitute from equations (100) 2nd (101)

into equation (99) we h2.ve

(R1+iRZ)(Rl -iRZ) + (T1+iT2) (T1-iT2) = 1

R12 +'R2

2 + T1Z + Tz2 = 1

MUltiplying by 2 and. rearranging ter!!lS eives

[2R12 + 2T1

2 J + [2R22 + 2T2

2J = 2

[(R1

+T1 )2 + (Rl -41 )2]

+ [(R2+TZ)2 + (R2-T2)2] = 2

[(R1+T1 )Z + (R2+TZ)Z] Z 2+ [(!l-Tl ) + (R2-TZ) .,] = 2

I(RI +Tl ) + i(R2+TZ) (Z 2+ I(R1-TZ) + i(R2-Tz)/ = 2

\R+TI 2 + IR-TI 2 = 2

Since Rand T differ in phose by TT/2(18) we know that

(18) Rice, S. 0., op. cit., p. Ill.

the me.gni tud.e of (R+T) eau21s the mpgni'cucLe of (R-T).

Thus we hl?ve

73

IR+TI = JR-TI = 1. (103)

BIBLIOGRAPHY

Rice, S. O. Reflection from corners in Rectengu12r

W~ve ~uides - - Conform21 Transforu8tion. Bell System

Technicel Journal. Vol. 28, pp. 104-135 (1949)

Sarb8chcr, R. I. flIla. 86.80£1, r:!. A. Hyper 1"1':0. Ult;ra.­

high Fre ouency Ane.lY8is. i·j. Y., John \'J'iley & Sons,

1943. pp. 1-227.

Skilling, H. H. Fundame.utals of Electric 1'l8ves, Sec­

ond Edition. N. Y., John Wiley & Sons, 1948. pp.

193-204.

Slater, J. C. MicrowHve Electronics. Reviei'!s of' 1-100.­

ern Physics. Vol. 18, PP. 441-512 (1946)

Terman, F. E. Radio Engineers Eandbook. N. Y., Mc­

Graw-Hill, 1943. pp. 2.51-256.

74

The .riY'i ter l·res ·corn 8 pr 923 i

~·chig8n. He is the son of Cloyd H. Ba cro t. d

Elorence oo_ey BF.'.rcr01't. He [Y'pdupted fro. "'ps111 gton

Gardner High Scnool, Albion, ~icnig~n in 1940. He el­

tered Al bio1 College in this se.me . ecr and receive the

de~ree of aehelor of rts in 1944.

e served i the U. S. Nav from 6 March 19 __ to

21 Aue;us t 1946. ':!hila i. the } 8.VY • e co pIe te

-idsh:p ents School a~ Col mb 2. Uni-ers' y, ~e-rp .1'

School at H rvard U_ive~sity, Radpr School et lIT a!~

spent most of the remaining time as _a 21' 'ainte ance

off~cer aboa:d the U.S.S. Casabla~ca, CVE-55.

In 1946 the w~iter became Instructor i Mathena­

tics at his alma mater, Albion College. In 1947 he did

gre.duate 't'l!orl{ in physics at the Universi ty of Hchi[an.

In 1948 he spent t TO months with Westinghouse Electric

Corporation 10rkin[ on a project for developing an al­

loy wi~h certain desired m gnetic properties.

Since 1948 the author ~_2S been I struc~o in Elec­

trical Engineering at t ~ Universitj of 1issouri School

of ines and etallurgy. He has Iso been working to­

ward the degree Master of Science in P_ysics.