\

“’m Tfiafiws it

Q
I a ‘
‘ s .
I

'iE‘Cs‘

we,
15..

This is to certifq that the

thesis entitled
An Approximate Method of Solving

Boundary Value Preblems

presented by

Clyde M. Hyde

has been accepted towards fulfillment
of the requirements for

_M_°S_'__ degree in L

 

v Major Hieflr

Date July 29.9 1955

0-169

 

T R T

The boundary value problem of vibrating mmbrnnes is inves-
tigated. The boundary of the membrane is assumed to be rigidly
fastened, such that the displacement is sero at all boundary
points.

in apprOpriate solution is obtained by assuming a series
solution. consisting of Bessel functions and trigonometric func-
tions. substituting this solution into LaPlnce's equation and
imposing the boundary values at a finite number of points.

, In an effort to determine the accuracy of this method.
several problems are solved for the rectangular membrane bound-
ary value problem. A summary of Hideo Itokaua's work with vi-
brating plates is also presented.

A test of the accuracy of the method when applied to an
irregular boundary is conducted using a rectangular wave guide
as an example.

A superficial treatment of the higher order nodes is pre-

sented in the later sections.

Afl APPROXIMATE HETHGD
0F SOLVIE

BOUNDAPY VALUE PROBLEMS

by

Clyde M. Hyde
a

A THESIS
Submitted to
The School of Graduate Studies
of
Michigan State University
of Agriculture and Applied Science
in partial fulfillment of the requirements
for the degree of

.MASTfiR 0F SCIEHCE

Department of Electrical Engineering
1955

 

The author expresses his deep heartfelt grati-
tude to Dr. J. A. Strelsofi. Protessor¢ Electrical
Engineering at Hichigan State University. for his
guidance and valuable counsels every new and then.

Also. thanks to Dr. G. E. Swenson. Jr. for pre-
senting the problem and rendering valuable assis-
tance throughout the work.

The help rendered by Arthur Learn. a graduate
student at Massachusetts Institute of Technology.
in checking the computations and proof reading the

text cannot be left unaccounted.

v’f‘.._i:fi““ t),
1.! ,sr'Ju. 111“)?» .1

9.
10.
ll.
12.
13.
l“.
15.
16.

mm R 9: 9.9mm

IntrOducbioneeeeeeeeeeeeoee

Solution of Boundary Value Problems

Pomcoordm‘°.eeeeeeeseeee

The Wave Equation . . . . . . . . . . . .

Summary of Boundary Value Problems . . . .

Hideo Itokasa's Method . . .
The Elementary Vibrations of

Its Sides Supported . . . .

a

The Elementary Vibrations of a

FixedSides eeeeeeee

Vibrations of Higher Orders in

The Rectangular Membrane
Direct Solution . . . .
Eight Boundary Points .
Twelve Boundary Points .
Higher Order Modes . . .

The Ridged have Guide .

Conclusions and Discussions

0* I O O O O O O 0

Square Plate with

C Q 0 O O O O O O

Square Plate with

0.0.0....

the

Case of

Fixed

OVER 31°F

10

11
13
1e
15
16
16
17
17
19

The aim of the present work is to investigate further the solution of
vibration problems by using polar coordinants in a grid network. as outlined
by Iideo Itokawa. the vibrating membrane is used as an example: however. it
is important to note that the boundary value oroblem of the vibrating membrane
is identical to the field problem in a metallic waveguide.1

Analytical solutions of vibrating onbrane boundary value ,roblems at
present are possible only when the boundary of the monbrane matches a coor—
dinate system. That is. a rectangular membrane dicta as t? use of rectangu-
lar coordinates. a circular memorane dictates the use of polar coordinates.
etc. Althouah it is possible to obtain the solution of the elliptical membrane
by using elliptical coordinates. the labor involved almost oreiibite extensive
use of this and other of the more complicated coordinate systems.

,For this reason an aphroximote method is usually used when solving prob-
lems with boundaries that are not easily matched vith eithor‘the Cartesian
or the Polar coordinate systems. “hen the boundary is of an irregular shape.
an exact solution is not known.

There are several methods of aoproximating the solution of the irregular
boundary value problem.2 loch method has its a’vnntages and disadvantages.
and each should be considered for each .ew problem.

The present method is not intended to replace any existing method, but
should merely suoolement the existin; method and thus enable the engineer to
have another tool at his command.

the scope of this work does not allow a rigorous proof of th method.

It is hiwhly unlikely that a rigorous proof exists. This paler merely shows

that certain definite problems may be worked in a simple manner and these

examples provide an accurate cabin firing apgroxination to the known solution.

-1-

S’JIAE'TI-"r S 0? BffltfszfiPY VAN}? moon-.45

Hang physical laws and the behavior of many physical systems are dcrcribod
mathematically in terms of partial differential equations. T3939 equations
might involve space variationa in more than one direction can might also in-
clude ti o Variations. The eloctromagnctic countiono are typical and similar
equations arise in tbe ntnay of vibrations. fluii flow. fynamico. claoticity.
etc.

In order to understand the nature of boanfnry valno problems. onppone
that a {iron phyoicn. phenomena in mutiomntically ioccrioci by partial dif-
ferential equation. For a given not on. tho differential eons ions munt be
solved ani the bouniary conditions satisfied. A problem of this type is called
a houndary value problem. Eonentially it in the name no the problem of solv—
ing an orninory Fiffarentinl equation. except that more than one independent
variable in to be considered.

Hany mathematicinno have attempted to valve the boundary value problem
involving odd boundary conditions. to Hate. no oatirfactory exact mothod
has been devised that will fit all boundary conditions. Recover. many Approxi-
mate methods are avnilablc to solve almost any problem with varying do room
of accuracy.

For certain bonni+ry coniitionn. irawrical metrodo ouffice. fitner approxi-
mate methods include the so-cnllad relaxation method. finite differences. oo-
orntion nvthoflo. ani conformal mapping.

Analytically tie boundary vnlu orohlcn in narroached by solving laylnce'a
equation using a technique known on the separation of variaolec. In trio
method. a potential function in aooumnfl to be the product of two or more quan-

tities which are functions of a cinclo inflevcnfent variable.

As a single example of the method of separation of variables. let us
first consider tun-dimensional problems in rectangular coordinantn 1 inc y.

Laplaco'c equation in these coordinants is
fi‘f—Lz-éuo (1)
31* 2:

he wiat to study proéuct solutians of the form Z = ?Y where x denotes t func-
tion of x Alana. and y denotes a function of y algae.

Substituting in (1) m + :2?" a o - (2)
Here x“ canotec the second total derivative of x with respect to x. and y“
denotes the second derivative with respect to y. To eenarate variables in

ttia exnmplo, divide by 1y:

i'.fl (3)
1 y

Kart followc the key argument for this method. tquation (3) is to apply for
:11 value: of the varieblca x and y. Since tke r1 it aida does not contain

1 and 90 cannot vary with x, the left side cannot vary with x eitrer. A func-
tion of x which Foes not vary with x is a constant. Similarly, the left rifle
does not vary with y, so the right cannot; anfi therefore must also be equal

to a constant - the same conctant. Let this convtant be denote& by 12.

Then
31:: ‘2 (L1,)
1
:1. ..2 (5)-
3

Equation (h) is r cognized as the standard form having colutione in ex-
ponentinla or hypcrbolic functions. let us write tram here in hyperbolic
form!

x 3 A each ax +.fi sigh ax (6)

-3-

Equation (5) is recognized as tho form having solutions in cinnnoids:
y a C coo my + D sio ay

90. oubotitutin. we have:
s “E cosh ax + d eioh ma E3 cos a; + .1) sin oil

It remains to motcn the given boundiry con*itioo9 with the potential

function (a).

POLAR CJQFDIRAI7S
Very often it is more convenient to apply boondovy confiitiono to the
solution of Laplaco'o equation in terms of polar cooroiooteo. (r 6). In

this case. Laglaco's equation is givoo by:

”'22 1.2L: a2 .
d22+rar+§1ifa 0 (7)

The solution of this equation is founi in a fashion similar to that used
in the cone of rectangular coorflinates.
To obtain a solution to looleco‘s equation in polor cooriinotes, a profi-
uct solution may to ornamed of tro form
I r f.(r) f,(o) (8)
{5(8) can easily be fieteroioofl to have the form.
a sin m0 + 3 cos m9 (9)

its differential equation for f,(r) will have the form

:2 GEE-33+ r Elf-$53.}- + (:2:er - m3) {Jr} =2 o (lo)

Toe solution of this equation is not so obvious. In fact. it is famoua enough
to dooorvo a special some, Hoosel'c equation.

Aseumin; t?e function f.(r) in given by a power series of tie form

f,(r) . {An r1 + n (11)
r=o

-u-

we evaluate tho constant by taking tto derivatives with reopect to r and
substituting into (10).
?hen the function f,(r) may be writton as a Bessel function of the first

order (1 a m) 3:
m + 2n

Jmh) - go an}; (-5) (12)

Another choico of 1“” -m profluces another Bengal function of the first
kind. order (-m) which is liven by

J_m(r) . (-1)“ Jam (13)

for into ral values of m these two solutions are relate} hy

m1») =- Jm(r) A1 + A2 [I m] (in)
“m

T31: is the aesoel function of the second kind, order m.

f,‘

she general solution to " oosol‘a eqzation now becoznae

{1(r) a A Jmfr) + 3 Ym(r) (15)
and the solution to equation (?) is
fi
2 a (C sin m9 + 3 cos m9) [3 Jm(r) + 3 Ym(rE] (16)

Vffii i“’? V IC"
Vbe differential equation of special intereot in this pa er is tic two

dinanaional wave equation

3:31: J 2
+ 2 V .
Assume Z a f(x,y,t) a VYT, (1?)

substituting into (15) gives
out? + KY"? 9 a2 If?” (18)

Giviaing by 3?? lives

Y' ‘ n

Since

2

T“ e - :2- T. (20)
the solution is reconnized no

'r =- F o‘ 3" (21)
a: above

5.1+ 1:... -62 (2.2)

X Y

3-1:: .b2 (23)

K

I” - -bzx (2b)

@30 solution to this equation is rec04nized as

c

I a A. coo bx + B sin bx (35)

35.23.824'132342 (26)
a

Y s 0' coo dy + D sin dy (27)

- t I 9
Z = F o 5- [3 cos bx + B'ein bé] [é coo iy 4 3.313 dq] (28)
In the oaeo of a rectangular vibrating meoor no with all the edfeo so-

curely fastened

///////// /////

\\ \\\A 3
\\\\\\

 

 

 

\\

//////7///////

l 1“'0 at a 3:0; x s‘w; y a O; and y a h

at all times. We can rewrite 2 combinin. B with the Otto“ constants and have

2 = a- 3' ‘Es" cos by + 23" sin by] E6" (:09 “:x + .3" sin fix] (39)

~6-

since 1 3 O at x a 0 for allot
0 ‘3 o” 2- " CA" coo by + '33" aim bfl C" (30)
.'. C“ ‘ 0

and at y ' 0
o - o 3' [MEI E5” sin :5] (31)
.'. A“ 'I' 0

Thus, 2 a 0‘ 8‘ ‘ Ed" sin by 1‘)" min Fig (32)

Ao.§‘

at y a h. a . 0 for all t ani x

2%!

or sin "0;: «in *1: (33)

“'1 bh ' 0 and bh a n7 :1 =2. o.1.2."'

.11 'e
b h (it)

At 1 a w, z «'0 for all t and 3
III
.‘. oindVHOandd 1135177 2330.1,2,

«is-EI- (35)
V

If we displace a certain poiat on the monorano b; a known amount we can
determine A. T?us all the unknowns are detorminei and we have
5“ r
2 u A 0 sin 55—: oin 3;— z (1-)

where 02 a d2 + b2

172
cnflffl ()3? . (37)

S'Ijeifif‘fY o? Eift‘if‘fléfY 1?”)th :-:mc2.3..‘=r'<:

 

The msin features of tlo process for solving a oi ple bonnlary value prob-
lem can now be summarized. By nonumin_; t.V :Rt p.olutions for t}.0 do .enlent vari-
aolea exist in too form of orofucta of functions of reenactive in’opondent varia—

bles. the original iifferontial equation is broken iown into aeveral ordinory

diffsrontial equations. each of which involves a parameter (c) ranging over
a continuous infinity of values.

When the boundary coalition: of the problem are imposed upon the product
solutions after the component total fiifforential equations have boon solved,
it 1a accessory. in order to nvoii solutions which ore iflentically zero. that
the parameter satiafy a certain equation. This countion is known an the char-
acteriatic equation of the problem. and its rootn. in general infinite in num-
ber. are known an the choracteriotic values or eigenvaluoa of the problem.

Only for these values can solutions be found antiofying toth the partial dif-
ferential equations and the givvn nounfiary conlitions. In a vibration prob-
lem. the characteristic valuoa determine the natural frequencies of the system.
sad the characteristic equation is therefore usually called the frequency equa-
tion.

The solutions which corresoona to the respective characteristic values
are known as tho characterintic functions or the oiganfnnctiono of tho prob-
lem. la a vibration problem they are usually called the normal noéea because
they define the extreme positions betveon which the eyrtcm occillatea when it
is vibrating at a ainglo natural frequency. io.. in a normal manner.

To satisfy tho initial conflitiono of the problom it is necesoary to be ublo
to expreso an arbitrary function no an infinite rarica of the characteristic
tnnctiona o! the problem. T113 can be fione in moot cacao of interest because
under very general conditions the characteriotic functions of a bounflary value
problem form an ortho;onal cot over the particular interval related to the

problem.

HISTO ITnfin-t's H“*TOW

(SET-.2 mistomry mvthofl for solvin, the equations Of v13xratin;},, plates is to

ueo a coordinate system suitable for the given boundary confiitiona. towever,
this does not provide a general method the: can be auplien to the case of any
shape plate with arbitrary boundary conditions. For instance, the problem
of vibration of triangular plates has not been colved because of fiifficnlty
in fin ing a coordinate eyotem suitable to those particular bounfiariee.

In Itokawe'e metbod, a series of functions which are composed of Hansel

functions and trigonometric functione(oboeinefi by solving the equation of vi-
bration in polar coordinstee) in choren an the general aolution. the coef-
ficients of these functions are determined at representative points on the
boundaries. Thus, the boundary conéitione are only approximately fulfilled.
The method was applied to the trianhuler pletc mentioned above with reenltn
accurate enough for practical purpoeee.
The equation of the vibrating plate in given by
nvhuowzzwuo (3&2)
where (.0 a density a), 8111131121! frequency
a 1' dignlacement
P a dieplaciig force
. D a elasticity constant.
Puttint I I ¢(r. 9. ejwt), we can rewrite tte above equation in polar co-
ordineten as

J! . d2
(m‘ifi’im)'-W° ‘3‘”

fin
D

where k a

 

Unler the condition that z romaine finite at r u’fi, the solution of tfle equa-
tion is of thecgprm

I a 2:21, Eanfirr) + LBnInO-tri) [Cncoe n 6 + Dn sin n 9] (ii-C)

-9-

(1) T373: t:Lti:-.::t:*:.ri:ry v1 awn-or»: or .5. "r 91,..ng 1" ms: aims SLni-Pm'rfm

In this case the solu ion is obtained in Cartesian coordinates. and it
in known that k'u'2.22. Therefore, this case was chosen for comparison with
the solution obtainefl by the present method. In Figure 1. the dots and air-
cles corrocoood to point: whore

3 on 0 3nd 52$. 0. ram-)octivelzy’.

Using symmetry prooortios. we assume that
a"! [CoJof-rr) 4' 811°(krfl 4- [0231.(m) + 0311,1er can he (“1)
Chen we tako f3 mints. as from in Piano 1. and determine tho coefficient.

such that the conditions 2 a'o and,€%§5;£ o are satisfied at those pointa.

 

 

 

 

 

 

 

Than k I 2.15. the accuracy being; 3}; (cooporrjd with k a 2.223 which wao obtained

by the rigorous methods). The coefficients are

0. ea 1.0 :1 a 0.0039
C2 =2 14.2063 63 2 0.8113

~10-

(E‘ho center was taken no the origin anc‘i the length of tho ciieo was taken to

ha 2.)

The contour lines are shown in Figure 1 from which it is inferred that the bound-
ary conditions are only aggrozimatcly fulfilloé at any point other than the co-

lected 8 points.

In?“ 'V'P"‘¢“'.“~“'0Vj y "fix-am ray” “1 I (1".93“ 'r'arr". '. rtv- :1 “'1'“ (“"143
(2) :uu -L~n-'.h T ilouzrldhu Or a Snurfi. 0y~l. >1,“ xl J .i.no

EM 1 '- [60.10 (h) 4* (1110(er 4' 02J4(kr} “ Panther} co: 1+9

+ Eamon) 4» cslgtuj con 89 (ha)
Lot the casflitionc a I! 0. and ELI.” 0 (“3)
r

be fulfilled at 12 points shown in Figure 2. 'l‘i-en we get.

I: a 3.00
03 v 0.!»931 Co '= 557.273

C5 . 238.261

 

 

 

The contour lines. which chow the pattern of tPe elementary vibration. or.

-11-

drawn in Figure 2. For comuarinon k'a obtained b Faznwn's and Kate's methods
I. . I

are @1183 in T~b1e 1 totethar with the k Itokawa obtained.

 

 

 

 

562333 (2) k.= 3.06

Vato (3) 3.00

Itokaun (b) 3.60
Table 1

It can be seen that the solution obtained by tke present method gives a cut-

f1cient1y good approximation.

If we throw away paints on lines of oft multiples of ‘fl78 Ind fix the 8

points, wa fiet

k ” 3.0?

fiha nulueu for the contour lines near the four corners are slightly unga-

tive. (?1gures 3 and a.)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

-12-

(3) VI‘WiTIJiJS 0? $513.35.? "I )EL'FS If! 85.": ‘3'." "”711“: TI“???

k

Fut s a [EOJ1(kr) + Cllllkij cos a

* EEJ'BC‘G') 4' C3130")? (:99 36
than I‘m 0 on the line of 9 3'22; an& thin given the Halal line.
2
Em z a 0. 511.20 at points (as o. r-= 1) and calf. r r: 2)
r 4

then k ani C's are determined afi follown:

k a b.07, 01 a 0.0227, C2 a ~0.6670, C3 = 0.0395
R a 5.3. 01 2: 0.00217 92 a 0.6307 03 . 0.0059.

The contour linea 1n the former case (k‘w 4.0?) are clown 1n Fivure 5 and

shore in thc latter (k a 5.3) in Figure 6.

 

 

 

 

 

0‘s I.\
I, “ I, \ Q
I ~ , ,- \
I \ i O
' ' ‘ ' _-' O ‘ l'
. ‘ '0 . .
' ‘0’ . ___
\ .
o I o
' ‘x , ' o
“ \.I I, 'f ‘\ r"‘
C ' I \. I
O- .’ ’ ‘.' ’ I C’ J
\..p" ‘--”
Al

 

 

 

 

 

-13.

THE R‘ICTAITGUIAP. P-TEPJIBFAITE

In the case of the rectanflular membrane. tha eigenvalue is given by
9-. f/% 32'
c n

where Au = fife natural remonant angular frowueflcy of the mambr-ne

k

c fi'elanticity canetant
a E Wifitfi of the EM flrane

b 3 length of tie me brane

*5

 

l»

‘ ‘7' —> x

\ \

 

 

 

float of tha follow1u¢ examples warn galvefi for tke reCfianq¢1ar membrane
where a a 2 and b y a,

In this caed

k =1|/-l-+ J- ae 1.7 mm
b 16 5

-11....

DIRECT SOLUTION

In an attemyt to solve directly for tie eigenvalue. n daterninnnt was
set up after matchin" eiiht boundary uoidtn on the ofiée of the Zzu vibrating
membrane. By takin. advantage of symmetry, the rscultiné 3x3 ieterminant has
9 terms with 2,2 terms heir; zero. inn first 5 tarno of tne power series,
regrenentinb the fivnoel function, were noon on a fu*ther approximation to the
problem.

the operations leadin; to the result were len;tby and ta ions. fho ap-
proximate oi envalne obtainefi was 1.533 as conu;refi with the analytical value
of 1.756, on error of 9.57 yorcent.

Althovjh this result in not within en insurind accuracy it noon provide
a starting pIWCG. 3? cfoooing thin value as an aggroximnte cinenvalue, 901v-
1n; the determinant. fietermin 2g the error, tryinf onot er .alne nni determin-
ing its error, a 500i third .nnon in porniblo by intergolation or extrapola-
tion.

manor fire “guesfiee" rcsnlting

4,
I

Tfig remainjar to the axnmvles workefi in tfifi
from this atto.pt at a direct a; roach to the yroblco.

In these examples a “Queen" is m«&e an: than tie associated error of tie
determinant couguted. ;ext. a noconn ”guess" in mace ané a second orror com-
puted. By interpolation. a third guano io mad. where tte error function agyovrs
to be zero on the gragh of tho error vs oinonvnlne. the error of the thirfi

solution provifios a point noar enough to zero in all comes to allow another

(rather accurate) interpolation to the eigenvalue.

-15-

V} fl":=.-'. ”-‘xfi a’j‘r‘w "“f ! ‘Y'Jf‘
.:¢IU;LA ‘31}\:n'6¢J.5‘5‘-* RO‘ \‘n‘hS

The initial prohlem attempted in this effort won that of matching ei;ht
bouniary points. 33 taking aivantame of the symmetry involved. it is noceeeary
to salvo only a 3x3 fietorminant.

In tho firnt example (Example I). the points were chosen at

(r. a) a 1. o: 1.h14. h5°; 2. 90°.
The results are 5001. fiko error obtained was 0.63;.

Next. in an effort to ietormine if the pooition of tho m;ddle point,

(r H 1.h1&. O‘a Q50), baa an Appreciahla affect on the answer. the "bad" point
at r a 2.236. Gun 65°5fi' was inveatiflmted. The error was raised by only a

slight amount. See Example 11.

T2217: B?U§W&*Y POI T9

Twelve bouniary points were Charon for the naxt example. Again symmetry
was employed to good aflvantage. The two "middle" points were choson two-thirds
of tko way up each side. 369 7xample III.

The results wors somewhat better thaa ttose obtainod with only 8 boundary
points being matched. the error being oniy 0.1?E.

Continuing the invostigation of tho imxortance of the "miédle” points,
the same problem was eolvoi after moving the "mifidle" points to points tri-
socting the 93° finale. 10.. at 33 flogroos an& 50 $6 rees (Tranblo IV).

1

Again tLa error of 0.2}; is loan than that obtained with eight bounfiary

points being matched.

~16-

I‘ll?»t E17? WIFE 1%? {S

In order to investigate the action of this method in the neighborhood
of the higher armor modes. the value of the 3x3 determinant (c) wan computed
for values of k between 1.5 and 1.0. T36 results were ylottcd vs the value
of k as shown in Figure .

The Analytical'valuoa of the eigenvalues were obtained from

The funiamcntal mafia agrees well. Th9 aecanfi mofie is missed entirely.
while the third mode a rees within reason. (See anmnle VII)

If more of the bouncary points had been matched. it seems likely that the
h1;her order mofies would all eggear more accurately. Lo ever. no definite con-

clusions can be firawn about the Liufer orier modes from this example.
THE FIDGED 1AV733133

In an effort to test tke methofi on a nystcm wtera tie boundary con&1tiona
must be inpoeefi upon two points of 936 ray. the rifluui waveguifie problem was
tolvcd.

The confiwuration of the guide was c;osen as

 

 

 

 

 

 

 

 

 

 

-17-

Four boundary pointo wore matched. One at (r. G) ”'(1, 0), another
at (2.2}6. u5°) and (1.118. u50) and the last at (1. ,o°).
éccording to Toma and Ihiunerya tho firnt eigenvalue of a rifiQGd wave-

guide is given by

k - (fix-V

From this formula the natural frequency of the funflamontal mofia of a
ridged wavomuida with tkis configuration is
hivvfiij
The results obtainea in Trample VI. by unin; the polar coordinate approxi-
mation. Ihov an error of 0.9955. zio field present at any point in the guifie
could easily be plotted after eolvinfi for tie constant“ Cl, 02, C3 and Cu.

These regulta are cuitabio for most enginczri.g problems.

- 18.-

Cfifi LWFIégfi A}? 317CWSFIDVF

The analytical solution of vioration problems. at present. is possible
only when tie boundary metcicc a coordinate system. It is highly desirable
that approximate methofia be fieveloped that provifie the necessary accuracy
with a minimum of effort. the present method apneare to be one that can be
quickly applied to eny boundary value problem with a reaeonohlc degree of
accuracy.

The fact that tie equipotontial lines of any phyoicel nyctom tend to
be smooth. continuouo curvce as the bounfiary is left, makes this mothod par-
ticularly suitable. ‘Thot in, in tre cone of rectangular menbreneo, as one
opproachee tio center. th. qnigotontinl linen approach a circle. Thus. it
seems to the writer that the most natural moths? vault use polar coorfiinatee.

The method present o here is very Similflr to the analytical method used
when the bounflary is alrealar. It recon li*ely then that wten thin metlod is
applied, more accuracy can be expectefi for bowngary confiitions tbat are nearly
circular.

$he success of this type of colution relies upon ”messing the prayer
values at the offset of the work. Successful guessing may be fione by real-
izing than tie natural frequency of the syetem will be in the vicinity of
known oyrtems that poroese rectangular or circular eymnctry.

As an examnle consider the following figure. It is 10 ical to cnoyect
that tie nytural frequency of thin sveton would lie between lot of tie cir-
cwlnr nonhrfnee with aéii of r1 and r2. A goofi starting point for the guess

of tho oignevalue wonli be halfway between these two values.

-19-

 

If the membrane in nearer to a enuare in shape. t}o aparorimations could
be obtained hy a 81 11a? scheme unin; square or roctqnqulnr membranes as otart-
135 points.

By using such a system of guessing, the third solution for the error of
the éotorminant should be fairly close. 0f courro. when the desired error
in leaa, another apuroximation may be in order.

232:: error exi-iuitod in the emmplca wrenentod in t; in fiche? are small. com-
paroé to tho error cutained from oimilar approximations when coin; a rectangu-
lar grid apyroximation tth involvon the same amount of work.

finly a small portion of the work cu bentod an a raouit of thin affort is
prenanted in this theois. The writer feels trat more inventigfiticn should be
done with tie direct approach to the problem. It is porrihle that some ot?er

Approach to he flirect oolution wouli yiul? better resulto.

-20-

?he method shows such pOFaibilitie9 f0; tie two diivnsional nyvtcmn that
some work afouli be Cone inveytiiating the ttrwe Gimwuvioual boun.wry value
problems. Instead of a polar system, agh1ricnl cooriin~tep vauld be urea with
the Le endwafuncticns replficind tic fivnnnl functiofiv.

Ar way statefl earlier. no r140rsus proof in .iveu for t?e mctfioa.
ever. it in reasonable to beliavo that tfie amtEoi will war? with any fie ree
of accuracy. fiepamiira ;. of course. umn the n'. saber oi‘ bomwut'y pOiE'Ltfi matcieri

ad tius tia amount Of effort ex enfiaa.

the hiqher orier modes prencat unonher definite point thgt slauld be in-
vestigated. It ac ms lively that imyoninfl b)unuary con itinns, nupyliud by
the fierivatives of tin functions. wanla force tie eccond anfi L1 be? 0758?
modem to hopes: more accurateiy on tie plot of the cigeavalue va error. This
poi t should be investi,eted in an effort to evolve a be tar method of ap rowi-
nating the first "guess”.

It wouli be minleafiin, to beliive tkfit tie m?thoi QT?T83$QJ h‘re 18 best
waited for every vibration problem. Unfinutteély tier? are mzny problcma which
can be ewlved mor- easily $34 quic*i7 by otfimr mstrofls. For example, the method
of aeparwtin; variables 13 very suitfibla for differential eguations whera the
boundary confiitiona are exyrwssvd by circul» or rrctan2113r baunfiariee.

Alno the accuracy with wiich t?e ans er in rv uireé helps to determine
wrich ayatnn nEould he need. An anpraximate rotkofi would not be used where an

exact salution is required.

Fora biou ht niaull b3 plscefi ucnn a riaorous groof of this awtkofi also.

1)
2)

3)

1)

2)

3)-

 

anmplo I

y a COJO(Kr) + CZJ2(Er) coa 29 * CuJu(7r) cas 49
l.

 

 

v- .4
wrror a .63y

6 = 0. £5, 90
K a 1.767
,/// r a 1. 1.u1b. 2

3y analytic method E a 1.756

 

 

I
C°J°(¥r) + CZJ2(?r) con 29 + CgJ4(?r) coa 59 a O

 

Qha set of equations to be satisfied are:
0050(1.h1§3) + 8252(1.b1ufi) con 90 + C434(1.h1&1) cos 130 a O

0050(2r) + 0232(27) cog 150 + chu(zv) cos 0 r o

COJO(E) + 8252(F) + C4&u(Z) = o
COJ°(1.uluv) - cuza(1.u1ak) = o

COJ°(2'¢") - 52-52(17) + CQJL‘,(23:) 3" 0

Solving tha eguationa far K

 

J°(K) 32(3) JQ(K)
Jo(l.41hK) o -Jg(1.h1&x) a o
3°(zx) .J2(2H) 34(2K)

J0 J2 J“
K a 0.3590 .29u0 c.0209
firz = ~0.935? 0.h%25 0.0?13
Era a ~0.3101 o.u§&6 0.2obh

$36

ifl

x a 0.3574 0.2989 0.0218
Krz s -o.0459 o.uuéz 0.0739

Zr} ” ~0.3553 0.“5}u 0.2105

1.75

.3690 .29h0 .0209

-.0359 o -.o713 = -.00232
-.3501 -.$§:6 .zoah

1.77
.35?“ .29i9 .0218

-0 3353 "oufig‘ .2105

1.75
.BQQO I .0713 I .3601 + .0209 I .0359 x .5556 * .0359 x .99QO I .QOQQ
"" 03690 X 0:31.66 ’- 00713 a

.00??8 + .ooojb + .00313 - .01250 B -.00232

1.77
.gggg z .G?39 x .3353 + .8218 I .UQL? I .553“
+ .curg z .2939 x .21c5 - .453“ x .0739 1 .BL?“

.DOCSb * .COOfiU * .CCSOB - .01195 t .00015

K t (1.76) 4* .35322 3,2gguu 2.03135
2K ' (2.LP?) u-.0&?9 g.hh&5 3.0727
2* * (3.5?) =~.3fi??9 u.u5aca =.2u?u7

.36322

-.0529

~.3&¢

79

.296hk

.03135 +.0032u96
+.osc£13u
*.0626385
.2314 _ .:
C7 7 ‘OC 20333

-002‘2? ‘

D 3' -.OG'O 71413

y = 65.100?) + Cszfitr) con 20 + CgJME’Zr) cos #0
l

 

 

t

 

1.

'30
x - 1.75 0.3690
5K ‘ 3.91 —0.b015
RR = 3.5 -0.3501
K s 1.?7 0.35?“
5H ' 3.955 ~o.39¢?
20 . 3.50 -0.3853
K 8 1.?1 0.3922
5:: a 3.532 40.3402?
252 a 3.02 ~0.3678
1.?5
.3690 .20b0 .0209
-.u015 «.2562 -.O?88
~.*'01 -.b586 .200»

-.01940 + .00??? +

Example II

0 = 0. 65’553'. 90°

:- a 1, 2.236, 2

Actual

2’
J2

.29h0
0.335

0.b536

0.2909
0.3?51

0.0534

0.2fi52
O.h052

0.0577

n .OOMES

-0-

K a 1.756

50
0.0209
0.26?6

0.?Ouh

0.0218
0.27hb

0.2105

0.0192
0.2838

0.1922

K " 1.?0 .
Error '3 3...?)

.3534 .2959 .0218
-.399? -.2500 -.0296 ".00609
“03853 -oufiju .3105

—.018?8 + .0030? + .00395 - .00216 - .00079 + .02525

Indicating (by extrapolation) 1a are too high.

m

Lry K - 1.71

.3922 .2942 .0192
-.u337 -_g7oo -.o??h B +.00116
-. 3678 “0156?? o 1922

-.02030 + .00286 + .00361 — .001907 - .00502 + .02200

K a 1.66
Jo(1.66) a +0.uaio 32(1.65) a +0.2?19 Ju(1.66) - +0.01?2
J°(}.715)==-0.h000 32(3.715) = +0.0256 34(3.715)'w +0.3376

00(3.32) a -0.3u06 32(3.32 a +0.b?65 Ju(3.32) a +0.1772

.“210 .2719 .0172

$.3386 “0&766 01772

Yxample III

3 ' ?rr. ‘ .17;
/ 01a tan‘l u/3 .. 53° 00.

 

Oz” 3 = 71° 30'

-,3r01 E a 1.753

can
cos -.U¢308
cos +.742hl

coa ° a «.80003

can +.2501

 

 

 

con +.3518h

1)
2)

0030(0) + 0232(g) + cugu(y) + 06:6(v) a 0
0630(1.é7¥) + C2J2(1.57K)(-.2301) + cu;b(1.57g)(_.;0305) . 9656(1-575)(.752513= 0
3) 0050(2,112) * 0252‘3o113)(-.80003) * 0050(2.11k)(.?301) + 0506(2. 1w)(.351ch)= 0

u) COJO(2V) + 03:2(2?)(-1) + 6054(2?)(1) + CgJ£f2T)(-1) = 0

J. :2 an :5
K = 1.75 0 369 0.290 0.<9.9 .00056
er a 2.0? «0.232 0. 80 0.1216 .00?9
Kr) = 3.605 -0 30- L.’?Q 0.“:u" .0934
Fr“ = 3.5 0.3%0 c.05“ 0.9000 .'?%¢
K - 1.?5
0.369 0.200 0.209 .Ccofié
~0.252 ~0.155? -.1025 .00?05 a +.0002009
-0.309 -o.3u32 .055? .03175
-O.BEO ~O.@59 .zobh —.0?50

-6.

1.

3.

5.

7.

9.

10.

11.

* .000205
- 00
+ .000069

+ Or 9:

J! 'v

* .000589

+ #3.
+ .000617
- q- «n

+ .0005656
- figqpaal
+ .0005h35
- .9001269
+ .ooou175
- 00 C1

+ .000060
+ 00 0

* .000099
+ Q“Qlf2
+ .000288
- 9909]]
+ .000277

+ QQQH?§
K ' 1.75

all a 0.359 3,1“ :1: 0.0015
812 ‘ 0.7957 ‘ 022 * —O.1357«

‘13 t 0.0566

.065? (-.0250) = -.001658
.2000 (.011?5) = -.002002

(~.1357 -.:0h0?0 a (+.0005523)(.369) + .0002038
(-.000070)(-.29h) a + (—.0001197)(+.23?) * + .QQQQZZZZ
(.0209)(.00745) '3 .0001557
+ (.1025)(.00056) * .00005?0

(2)
(-.3h32) + .0002131 . (.0000?31h)(.3flo) a-- 00002?

   

(+.309)(*.u59)(.0009131) . +.oqnqgggg
(.0209)(.011?5) ' .002h5575

-(.00056)(.065?) '

 

+.00241896

(00232)(+.002“19)(-.b59) I - finygfizfiz

  

(.300)(+.1357)(.002019) a +
(.1025>(.0254) 8 .00260350
«(.aohb)(.oo71+5) '3 W
+.0010&072

(2)

(3. 69) (+. 13032 ) ( . 00108) «- +w

(«.309)(.294)(.00100) - 9.0001zfiéa,

(-.1025)(.o1175) - —.00129u375

-(.oo7u5)(.0657) . - , 000000

   

(. 369)(~.1+59)(-.00169M - ._000?F: {fl}:
(.300)<.200)(-.00169u) a —.Qflp}?92fi}z

(.0209)(-.025u) - (.00056)(.200b)

(2)
(.232)(.1432)(-.00065306) a -.000051990205
(2)

(-.399>(.1357)(-.00065306) = +.00003535907?

-8-

J6
+0.0006
+0.0105
+0.0350

+0.0269

“ -.001159

J0 J2 J“

x a 1.7? 0.3570 +0.2939 +0.0218
Krz a 2.955 ~0.zuuu +0.0851 +0.1261
Kr3 a 3.?3 ~L.b007 +'.0229 +0.2390
m. = 3.5“ -o 3953 +0.’*534 «0.2105
1.7?

.35?u .2939 .2513 .0006
..2000 -.1350 -.1050 .0079
-.000? -.}380 .0672 .0123
-.'053 -.b§}u .2105 -.0269

(.2010)(.00?9) + (.1060)(.0006) = .0022000
(-.b007)(-.b5}b)(.00229) = +.0000160b
(.3053)(-.3330)(.00229) v
(+.06?2)(-.0260) - (.2105)(.0123):= ¢.00u396
(.357b)(-.1360)(.-00#396) =
(.244h)(.2939)(—.oou306) =

(-.1060)(.0121) - (.99?9)(.06?2) a -.00153b7
(.3574)(-.b53h}(-.001835)
(.3853)(.203?)(-.001835)

(.2010)(-.0269) - (.2105)(.0006) a -.0o77067
(.zhhh)(.3380)(-.00?70?) a
(.h00?)(.1360)(-.007707) =

.2018)(.0123) - (.0006)(.0572) n *.0034230
(.244h)(-.4534)(.003424)
(.3853)(.1360)(.003h24)
(-.1o£0)(-.0259) - (.01233(.2105) = .0002622
(.357a)(.33€o)(%6) -
(~.&00?)(.2959)(05) =

$41

+.000h160&

+.00021508

+.00029735
-.00021133

~.00063665
-.00001009

-.000379“2
+.00017942

+.00003167b

-ngggfilhgé
“0001169

3 Equal Angle (4 x u)

 

 

 

Txample IV
Error a .23fi
91,“ 0 62 = 30 93 u 60 90.‘ 90
r151 r281.155r3=*2 rufiz
can 292 3 gas “92 a cga 692 n
x (:00 292 a can “62 ’3 cos '69? '3

 

 

1)

3)

h)

0

coJ.(1.155‘~<) + C:gtTg(1o1557')(.5) - c..n.<1.1ssvc>t.s) - c6.rér1.15sz<) . o

C JO * C232(§) * Cb5u(§) + 0656(V).3 0

0050(21) - 02J2(22)(+.5) - 6050(2r)(.5) + 65:6(21) = o

CoJo(2y) - 6232(2?) + 0050(22) - 0505(21) a 0

K = !!72

Kr3 ‘ 3050

K a lag:
Krz . 2-02?
11-3 - 3. 52

Yr“ ‘ 3.52

K=L%
K u'1.76

K 3 1.?66

JO
0.30003

+0.211?9

-O.35013

~0.33013

+0.36323
+0.20830
-0.38279

52
0.20000
+0.35751
*0.#5063

10.115363

+0.2964u
+0.u5508

+0.h$.08

Err a +0.0000123

Fr: a +0.00000U9

Frr 3 -.0000619

~10-

J0
0.02091
*0.03529
+0.20hb0

+0.?th0

+0.02136
+0.03567
+0.20?47

+0.207Q7

J6
0.00056
40.00128
+0.025h3

+0.02543

+0.00058

+0.06130

+0.0?517

K ‘ 1.75

0.3690 0.2900 0.0209 0.00056
-0.3801 -o.3293 -o.1022 0.02543

-0.3801 -0.U586 0.20M -0.025U3

(.1022)(.025h3) - (.2004)(.025h3 = ”1 = -.002999
(.3690)(.1788) M1 1) -o.0001979
-(.2188)(.29uo) M1 2) +0.0001929

(.0176)(.02543) + (.2ouu)(.00128) a 02 = .0007092

(.2293)(.369o) M2 3) +0.00006oo
-(.3801)(.2940) M2 0) -o.oooo793
-(.0209)(.025h3) — (.2ouu)(.00056) = M3 = —.0006460
-(.2118)(.2293) M3 5) +0.0000310
+(.3801)(.1788) u) 6) -o.oooon39
-(.01?6)(.025h3) - (.00128)(.1022) = 0“,: —.0005?8h
-(.369o)(.h586) M“ 7) +0.0000979

(.3801)(.29h0) M4 8) ~0.00006h6

(.0209)(.02503) + (.1022)(.00056) ='M5 = .0005887
+(.2118)(.b586) M5 9) +0.0000572
-(.3801)(.1788) M5 10) -o.oooouoo
-(.0209)(.00123) + {.0176)(.00056) a 05 = -.00001689

(.3801)(.b586) M5 11) -o.oooooz9

(.3801)(.2293) M5 12) 19.292221:

-11-

1.76
.3632 .2960
.2083 .1790
-,3828 -.2280
-.3828 -.h561
(.1037)(,02617) -

(.3632)(.179h) M1
—(.2083)(.296h) M1
(.0178)(.02617) +
(.2280)(.3622) M2
—(.3828)(.296u) M2
-(.021#)(.02617) -
~(.2083)(.2280) M3
(.3523)(.1790) M3
-(.0178)(.02617) -
-(.3632)(.h561) Mu
(.3828)(.296h) Ha
(.ozlu)(.02617) +
(.2083)(.h561) M5

-(.3628)(.179h) M5

.021“ .00058
-60178 -000130
-0 1037 002617

02075 -002617

(.2075)(.02617 = M1 = -.002716

(.2075)(.0013) = M2 = .0007356

(.207b)(.00058) = M3 . -.00068

(.1037)(.00058) = M5 = +.00062

—(.021u)(.0013) + (.0178)(.00058) = M6 = -.0000175

(.3828)(.h561) M6
-(.3828)(.2280) M5

+.ooouz15

-12-

2) +0.000168

3) +0.0000607

u) -0.0000837

5) +0.0000323

6) -o.oooou67

7) +0.0001oo

8) ~0.000068

9) +0.000059o

10) -0.0000h26

11) ~0.0000031

12) +0.0000015

Err = + .0000004

J

K = 1.766
Krz ' 2.0b0
Kr; " 3.532
Kr). " 3.532

.3598 .2979

.2009 .1808
'038u3 ‘02173
’038u3 1 ‘ou5u5

(.10h7)(.02663) - (.2093)(.02663) = M1 =

(.3598)(.1808) M1
-(.2oo9)(.2979) M1

(.0102)(.02663) + (.00135)(.2093) = M2 =

(.3598)(.2173) M2
~(.38h5)(.2979) M2
-(.0216)(.02663) -
-(.2009)(.2173) Hg

(.38u3)(.2979) M3
-(.0182)(.02663) -
-(.3598)(.b5#5) M4

(.3843)(.2979) Mu

(.0216)(.02663) +

(,2009)(,h5h5) N5
’(.3843)(.1808) M5
~(.0216)(.00135) +

(.38H3)(.h5h5) M6
-(.38h3)(.2173) “6

.35975
.20088
“038h31

-.38N31

J2
.29790
.361?0
.05050

.45“50

.0216
'00182
-010“?

.2093

+.0003271

”00003890

Ju
.02163
.036h9
.20932

.20932

_.00059

“000135

.0266}

-.02663

(.2093)(.00059) 3 M3 =

(.1067)(.00135) = Mu==

(.1007)(.00059) ='M5 =

(.0182)(.11159) a “6 3

J6
.00059
.00135
.0266}

.0366)

-0.002786

+0.000767

‘000006987

“000006260

+0.0006370

“0.000018“

1) -o.0001812
2) +0.000166?

3) +0.0000600
h) -o.0000879

' 5) +0.0000305
6) ~0.0000800

7) +0.000102u
8) ~0.0000717

9) +0.0000582
10) ~0.oooouua

11) -0.0000032
12) +0.0000015

Err n -.0000619

Example V

4°(x)=1-fi+gfi,___x5 +_§3 .- 61°

+ 0.0
b, 6a 2306 147956 14,705,600

 

J2(K) ..§_,, K“ + K6 _ x8 + x10
8 96 3072 18102 1.768.824

 

J4“) 35—... ._§..+__K,,8, _, X10
7580 358670 30.965.760

K“. K6 + K3 310
16 216 9216 060.000

2 L“ 5 8 10
J 22 . L ¢ K K + x
2( ) 2“ 304 3072 552960

“ 6 8 10
J 23( :- 5... -K + .3!" K
0( ) 95 950 23000 967680

Jo( 2K)=1-Ei?:.+

J°(2K) =11 - K2 +'L“ x6 g8 _ K10

1* 36 576 W

J2(2x)-1E-K“+6§-.fl+ x10
2 48 720 17280

 

J0(K) = 1 - .2562 + 1.5625110*2K“ - 9.2735x10-“x5 + 6.7816110'6K8 - 6.7816110‘8K10
J2(K) = .12562 - 1.0016x10'2K“ + 3.2552x10‘b36 - 5.0253x10'5K8 + 5.653h110’7K10
J6(K) a 2.6oulx1o-3K” - 1.302x10'475 + 2.7126x10-5K8 - 3.2293110’8K10

Jo( 27) - 1 - .5K2 + 6.2510046+ - 4.6296x10‘3x6 + 1.085:10~“68 - 2.1701110-6110
J2( zx) a .25K2 - 0.1666110"2K“ + 2.6091110-365 - 3.2552:10“‘K8 + 1.808h110“6K10
J4( 21) = 1.0016110*ZK“ - 1.0016x10-BK5 + 0.3002x10'515 - 1.0333110'5K10

J°(2K) a 1 - 12 + .2574 - 2.7777110-236 + 1.7361110-318 — 6.994hx10-5110

J2(2K) a .5K2 ~ 1.6666x10‘16“ + 2.0833x10‘2K6 - 1.3888x10'3K8 + 5.787110’5K10
Ju(2K) a u.1666x10*2x“ - 8.3333x10'3K6 + 6.9uuux10-418 - 3.3068110’5K10

-114v-

(1)

(1:2)
(2")
(15)

m . 6 .. 8 10
11 “1738-: “#33“ K (11%;:’
.. I8(§-;%g) '* K10(--177)

621w

10
+ a (A. $1,713) - ..

12 1
K
( 77 lw-UO)

 

a 16(1.302210*3) K‘Q‘SU ".‘.3$7x10"‘) + 110(1. 96C5110'5) - 3612(1.119z10'7) +...

" K8(13.O')1104) + ’{15(23. {37310 -5) "’ 312(190665310-7) +000
X12(59.67110-7) ‘00.

+ ,10(, T32.‘ , 3x10 ’5)

11 a 15(1.30211o‘3 - W(1 54:10‘3) + 610(.5“36110’3) - Kl?(.117110‘“) +

I . .. 8 . 1!) - 12
’2 " (7311’ (1‘53?) (7172(0) ("L-‘4

01931520
~r F3. 10 1’12 1-
- I; + q in “we“
( 2";10"‘-E ) ( 6060 ) (2211160 )

+ 11°(w A132) .. KIN-47%,“: )
.. KIN—545....)
flair-51:30

a 15(1.3110-3) - K8(6.51x10’5) + 210(1.36110-5) - 613(1.615x10’5)
.. 18(616110-5) + 110(213110'5) - K12( 1752x1043)
+ 110(56.2110‘6) - K12(2?1.3x10‘8)
- 112(361.7x10-8)
{3; I1.. K671. 3,_10"3) - 128(4321110’“) 4- 3110(7.726113'5) - 3(12( 6.?98x10'6)
.. K8(6.50x10“‘) + 21.1% 26.92064) - 11% 38.63160")
+ 210(3.125110*5) - 112(31.125110-5)
.. 1:131 6.0181104)

‘32 E6(1.3110“3) ~ K8(1. 11451110‘3) + K10(.i+(;?5;r,10-3) - 3f12(.825x1=3"b)

-15-

K12(365 16631‘77) +00 0

TS 3' 35 1-1-2- - 39(3)?) + Kiwi?) - VIM—5113‘)“
- Katiéj) «1» K10( 1 1- [1% 1'2, 0)...
*K10( WEZO)- V1?(‘-1K5')
" ”(mm

a {6(5.2053x10'3) - 35(1.73éx1o“3) 4- 31W 2.1?x10‘“) - 312(1.Ms6x10'5)
.. KW 5.20am“) + 310(1.?35x1o-’*) - K12(2.1?x 1035 )
+-310(2.17x10'5) - 312(7. 233x10’6)

-312(5.16£x10’7)

3; . K5(6.2110’3) - xitz. 2 5:10‘3) + 310(n 12119-“) - 312(b.391x10-5)
_. 9:54 1. 3110-3 ) + K10(5.£ 511104") - 312‘ 10.3110'5)

+ amuemm-‘U 3W 3.53:10-5)

- K12(.2257110'5)

T3 = K5(5.2x1o-3) - 38(3.56£10'3) + 310(1.033110'3) - 312(1.8h5210'“)

T; - Mfg?) - 118%?) + mm) - 312(,._._, .. ..
- 3<m> + 33w - FM

+ 10 - 12
V (7&5) gm)
3 (r—3§rr)

134' ' :11:

 

It K6(5.208210‘3) - 33(1,OU1110-3) + K10(8.68110’5) - K12(h.13hx10'5)
.. 13% n.3ux104) + K10(8.68x10’5) .. 1:12( 7.2331935)
‘ + 310(1.36x10‘5) - 31% 2.71:10‘6)
- 313(2.2sx1o“5)
36(5.208x10‘3) - 38(1.u75z10~3) + 310(18.?2110-5) - 313(1é.3311o*5)
- I{B(2.€~Cfl+xl{)'3) + KW 73.8x10'5) - 31% 93.6:10-6)
+ 310(32.55110-5) - K12(92.19110'6)
- 111% 23.1110’5)

I

1“,: 35(5.2eax10-3) - 39(3.039z1o*3) +'310(1.25x1033) ~ 312(3.25310'“)

~16-

T1 u z5(1.30211o—3) - K8(1.5ux10-3) + K19(.583Cx10'3) - K12(.01?x10-3)

T2 ' (1.302x10'3) (1.1uax1o-3)
T3 ' 5.2x10-3 3.55110-3
Tu.‘ 5.2x1o-3 h.0?9x10’3
0 ' 13.00h - K2 10,237

x6 - 6.u7:“ + 20.0732 - 25.5 - o

12y fievton'l fiethod. solving for K2

K6 - 6.h73“ + 20.0?32 - 25,5 3 o

1 —6.h7 +2n.o7 ~25.5
2 -8.9u 22.26
1 -u.u7 +11.13 -3.2#
2 -h.9u
1 42.“? 5. 19
2
1 -O.h7
¢he depressed equation
1 43.117 6. 19 45.2.1;
.5 .015 3.1625
1 +.o3 6.205 -.13?5
.5 +.265
1 .53 6.u7
.5
1 1.03

-17-

(,b075x10'3)
1.05B310'3
1.25x10-3

+ K“ 3.299

2
3.2“
""- 05
6.19

.5

LEZZE .02
6.3?

(.0826x10'3)
.1335110-3

.226x10'3

- E6 .510

1.03 6.“? -.13?5 .02

.02 .021 .1298
1.05 6.b91 .007?

.02 .021u

’ 00077 )012
1.0 6. 123 "'""' "
7 5 e. 512

.02
1.09
1.09 6.5120 .0077 .001

.001 .00109 .3065
1.091 6.51%9 “.0012

0 .{I‘ o J

001 .0109 0012 .0002
1.092 6.51138 6'51””

one root 10 K2-= 2.5212
K a 2.5212 c 1.588
The actual value 18 1.756

Thus the error in this eyctam amounts to

.1a25§———1~523.x 100 8 9.57% error

1.755

A correspondingly larger error would exist in tfe higher nodes.

-13-

Examyle VI

Examining I. rialged wave guide

I

 

 

 

L.

K 3 1.h

 

 

 

 

 

J Error 8 .9953 error

 

 

Choose 3 raflialo such that

91 ‘ 0 r1 3 1
92 " 65°51“ r2 ‘3 5 I” 2.235
03 3 65°51" 23 3' 5 =3 1.118

Oq'90° rufi'l

Using previous equations

J. J20?) .1143.) JW)
JO( 5%) ~J2( ST)(.666) ~Ju( 5?)(.103) J5< 53)(.015)
M 3-3) -J2( Emma's) -Jz.( E00100) .151 3310815)
Jo(3) .J2(3) - Ju(3) -J6(3)

1
~ 7 «.3
by Fields & Waves in Eudora fiadio . Famo & Whinory, p. #10, c R (——-‘F-—-94
2
g'Z since‘KZS-fip C23 .1...
0‘
1.- %

h a u K's ( )313 ( )fi’u' 2
(18%

~19

Eilged wave guide

Assume K a 1,b5

Kr J

O
1.h5 .5195u
3.2“ -.35032
1. 521 .14113141
1.35 .5395“

Assume K‘s 1,b

1.“ .5é636
3.13 -.30092
1.565 .h?5°8
1.h .56686
K t 1.h
.56686 .20736
-.30092 -.323n3
.#?520 -.1655
.Sééas -.29?36
x . 1.05
.5195“ .21969
-.33032 -.32079
.53341 -.1?h79
.5395“ -.21959

J2
.21969
.4816?
.26219

.21969

.20736
,u0510
.2h826

.20735

.00906
-.00109

.00906

.01035
-.01787
-.00159

.01035

.01035
.16549
.015?4

.01035

.00906
.13979
.Oljfio

.00906

.00015
.0116)

00002“

OUOOIB
.0139“

.0002?

“000013

-29-

’6
.00018
.01710
.00036

.00018

.00015
.0102?
.00029

.00015

K’t 1.“5

.00159 1

"05395“
.33032

.0178?

.5595“ x.

.43341
-.01035
.33032
#3331
f.01757
-.S39Su
-.5395h
.01035
-.33032
-.5395“
.01035
-,h&3h1

+.5395#

X

I

I

I

I

2

-53607937

+133.0012

.00018 - .01035 I .0002? "H1 ' 2.7153
.320?9 ”1

.ZIQ59 ”I

N

.00018 - .01035 .01394 - H2 . -1u1.06}4
17339 “2

.21969 M2

.00018 - .01035 x .00018 9 H3 - 4.726
.17479 M)

.320?9 ”3

.0002? * 100159 I .0139“ ' Hg_-'16.9823
.21969 Mb

.21069 34

.0002? + .00159 x .00013 =335 2 30.3012
.21969 3
.32079 35

.0110h + .0173? x .00010 a "6.” 107.0956
.21969 M6

017Q79 M6

Error 9 ~.00039299?5

-21“

1) * 4.6996
2) Q 1.9?0“

3) -13305é96
h) -137ch10h

S) - 201513

6) - 5.2992

7) - 20.1267

8) - 20.1267

10) - 52.hh29

11) “1&306838

12) +139.101é

K-l.h

.00149 x .00015 - .0002» x .00006 = 31 = -1.9530110-5

-.56606 x .32338 ”1 1) +0.000003503
.10032 x .20736 31 2) -0.00000122

.01610 x .00015 a .00906 x .0116) a 32 - -1.0297110-5
.30116 x .16551 02 3) -0.0000096607
,b'4’528 I .20735 Hz 1}) 4.0000101LH31

-.00106 x .00015 - .00906 x .00015 a 33 ='—2.713110'5

.30092 x .16551 33 . 5) -0.0000001350
.07528 x .32303 H) 6) -0.000000&1?9
-.01518 x .oooeh + .00139 x .0116) 3 Wu - 13.000110-6
-.56306 x .20736 ”n 7) -o.0000015107
-.56536 1 .20736 “a 8) -0.0000015£07

.00006 x .00020 + .00100 x .00015 c 05.: 2.3975x10-6
-.30092 x .20736 “5 9) -0.0000001u96
-.5€606 x .32338 15 10) «0.000000u306

.00905 x .01163 + .01618 x .00015 = “6N“ 10?.Ex10‘6

--“7523 x .20736 “5 11) -0.0000105201
.55555 x .15551 ”6 . 12) +0.0000101139

~000017o108

+0000103722 Error 2 -0.0000065586

Very near 1.h

2K

2?

ZK

2K

2K
2K

2.0
2.83

4.0

2.1
3.9?
h.2

2.2

3.11
a.»

2.3
3.25
h.6

2.0
3.“
0.8

2.5
3.53
5.0

J0
0.5110
0.1555

-o.2601

0.2939
-0.1958
‘003971

0.1100
«0.2051

“'0. 3H2 3

0.0555

0.0025
“00 36113
-0.2300
-0.h€b

-0.3832
“0.1776

Example VII

Jo

0.2321

(‘)Oo’+fi61

0.3523

(-40.36111-

0.3746

(-)00 3105

0.3951

.‘

(-)0.2501

0.0139

(’)0018%

0.4310

0.1061

“

-23-

30
0.0110
(-)0.ou10

0.1330

(-)00 1103

0.2811

0.0005
(-)0.1281

0.3100

0.0076
0.3365

0.0553
(-)0.1569

0.359“

0.0603
(-)0.1091

0.3780

0.0738
0.3912

K ' 2.6
2K 3 3.68

ZK“ 5.2

2K ' 3.8.2

23" 5.0

x a 2.8
2K 1' “.1

2?: "3 5.6

K 3 2.9
2K 3 4.1

2K ' 5.8

3*
I

3.0
23 3 “.25

2x - 6.0

K ‘ 3.1
2? 3 “.38

2x ' 6.2

JO
«0.0968
“003981

~0.1103

-o.1uzh
’Oob027

-0.0h12

~0.1&50
"093887

0.0270

-O.2243
“003587

0.0917

~O.2601

0.1506

-o.2¢21
-o.3h63

0.2017

-0.3302

0.2033

32
0.0590

(-)000217

0.0696

(”)000867

0.0777

(*)0.1“5“

0.0832

(—)-0.1990

0.0861

(~)-0.2029

0.0052

0.0835

(-)~o.3000

-2“-

Ju
0.0800
(-)o.2322

0.3935

0.0950

0.3990

0.106?
(“)002953
0.3926

0.1190

0.2958

0.1320
0.3576

0.1h55

(-)0.3339
0.329“

0.159?
(-)00 3507

0.2005

K a 3.3
27.. =- 0.66

23 a 6.6

I ‘ 3.7
2K 3 5.24

2K.= 7.4

K ” 3.9
2'5 '3 5. 52

23 a 7.8

Jo
43.31093
-.2003

0.2700

'00 3&3
”.23?“

0.3931

~0.3801
“'0 19 38

+0.3001

-0 o 3918

0.2951

~0.3902

0.2706

-0.0026
0.0712

0.2516

'001’018

0.2150

J2
0.0700

(-)-0.3119

0.0697

(‘)‘003123

0.0586

0.0008

(-)-0.2000

..

0.0093

(-)-0.2097

0.3???

(-)—0.1630

J0
0.1703
(“>003655

0.253?

0.1892
(‘)003?E38

0.20?7

0.2000
(‘)003fi?5
0.1578

0.2198

0.1051

0.2343

.4-

(-)0.3991

(“)OOBQw

K t 1.5

K t 2.0

K 3 2.2

K ' 2.3

K" 2.u

(.1555) .1320

“.5118

’02951

- . 11“”J

-00555

-ojfiuj

.0018 x .2601

- .2321 I .0097

+ .hfiél x (-.0232)2

I .2311 -0039?1 1 .1103
- 03523 ‘ “.0991
x .1103 * .1958 x .0350

* .350 (-.0130)

x .3355 - .3b23 x .1070
x .1Q?O + .2951 x .0#76

- .2501 x .00218

f: I .1591“ " .2961 x ‘1559

+.h13? x .1690
x .1559 * .3328 I .0553

+ .11h5 x .0091“

x .3?80 - .18?1 x .EQOb
(-.h}10)(-. 833)
x..1891 + .3603 x .0633
+ .1161 (.0230)

D1.5

D2.0
D2.2 '
D2.3 “

.00225

.03h76

.00655

.02341

.05911

.0005;
.05856

.06995

.60169
.0;.6u

‘
C

.0?396

.0096?

.0315}

K - 2.6

K ' 2.7

-. 832

-.3901

+.0953

“54037

.2203

.3912
.4061

.2030

.0065

.3985
.5590
.2332

.0217

.3990

.2000 x .1776

0
pa
C:
"Q
0

C
C1
“\1
KAI
«'33
N

.3832

.1103 x .2322
.1802
.3081 x .OEQO

.0559“

.0012 x .2538

+ on696 3 .1712

+ .1024 x .2530 * .302? x .0950

-.3887 x

+.1850 x

on. 389.7 I

I

.3926
.b???
.2958
.1060

.2953
.0032
.2958
.1990

f

00270 3 .2958
.1h45
.308? x .1067

.0962

.2958 x .091?

. 887 x .1190
.1126

.27.

* .07395

- IOIEQB
+ .0550

+ .02006

K a 3.0

K ‘ 3.1

3 "3.2

K ' 3.3

K I 3.“

-.3533 + .3576 + .315? x .1506

* .2501 x .3169 * .3663 x .1320

-3 316‘?

+.3203

.3443

‘02)?“

.3611}

'3‘

.2329 x .1311

032?“ + .2017 I .333“

.uasz

«0&6?
.3363 I .1h56

.1379

.2433 I .3507
.00”8
03150 X .1597

.1628

.0617

' .1E32 1 .23?b

.1827)

3)3.0

93.2

”3.)

DB.“

.0309}

Igggsua
.0090?

.02268

.ObOG

.0182?

.0037?

.ouaau
m

..Oh502

.01386

.05h49
.06835

.02898

.05”05?

m

.085337

K a 3.5

K a 3.6

K ' 3.7

K B 3.8

K a 3.9

-,1;33 x .1573 + .3883 x .3001
(-,u5a6)(.0a60)
+.3801 x .3335 + .aohh 1 .1938

(-.3Glh}(.13?3)

-.1h7? x .1051 4 .2951 x .3753
- .4434'8 I .1012
+.3918 x .3953 + .137? x .2198

— .2800 x .1801

(-.ca55)<.0510) + (.3991)(.2?56)
(-.u283)(+.1063)
(-,3..2;(-.3991) + (.0965)(.2353)

(-.249o)(.1820)

-.0712 x .0031 + .?516 x .3794
- .1109) x .1003
+.h026 x .399» - .9712 x .2507

- .2007 I .1}*30

(-.1233)(-.0557) + (.3960)(.215h)
(“03379)(00922)
(-.uo18)(-.3sao) + (.2661}(.1233)

(’01638)(+01919)

-29-

D3.5

D3.6

t1

‘3.7

93.9

.039#h

.05902

gogohz
.099u5

.05553

.ouggz
.09085

.OHIOS

.0’°98

.0710)

.03576

..- ,ogihg
a ’ 006719

010.!

”0.0..

mMDJcSZMG—N

 

It
I?

V n M x

\006 +.

2.0+.
Mommm

 

1.

5.

BIB LI OGRAPHY

wy11.. "Advanced Engineering Mathematics“, McGravaill Book Compnny.

1951.

Hildebrand. “Methods of Applied Mathematics“, Prentice-Hall. Ino..
1952.

Rama & Uhinnory. “Fields and Wave: in Modern'Radio“. John Vile: &
son.. 1% O

Timoohonko. 'Thoor: of Plato: and Shells". McGrav-Eill Book Company,
1940.

Churchill. "Fourier Serio- nnd Boundary Value Problems“. MoGrnvaill
Book Company. 19h1.

ROOM USE C3;

fl \1

e”):

55:;

‘ ' " A . 1H
"""\wv~\.

"U-fl

(>14

 

 

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