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THE LOWEST EIGENVALUE OF THE NEGATIVE
LAPLACIAN IN TWO DIMENSIONS:
A MODIFIED PERTURBATION METHOD

presented by

Ling-Huang Yu

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6/01 cleIRC/DateDuepes-pJS

THE LOWEST EIGENVALUE OF THE NEGATIVE
LAPLACIAN IN TWO DIMENSIONS:
A MODIFIED PERTURBATION METHOD

By

Ling-Huang Yu

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Mathematics

2000

ABSTRACT

THE LOWEST EIGENVALUE OF THE NEGATIVE
LAPLACIAN IN TWO DIMENSIONS:
A MODIFIED PERTURBATION METHOD

By

Ling-Huang Yu

The eigenvalue problem for the negative Laplace operator in two dimensions is
classical in mathematics and physics. Nevertheless, analytical methods for estimating
the eigenvalues are still of much current interest. In this work, a modified perturba-
tion method is formulated by applying perturbation method, reflection method, and
the Fredholm alternative theorem. The method provides the asymptotic expansion
formulas of the lowest eigenvalue to bounded doubly connected regions having the
inner boundary which encloses a region with the maximum dimension of 2c, 0 << 1.
The first three order terms of the asymptotic expansion formulas are found explicitly
by correcting the inner and outer boundary conditions alternatively and by applying
the generalized Green’s functions. The relations between the first three order terms
of the asymptotic expansion formulas and geometric properties of the regions are also

investigated.

To my Mother

iii

ACKNOWLEDGMENTS

I would like to express my deep gratitude to my advisor Professor 0. Y. Wang for
introducing me to the world of research applied mathematics in the best possible way,
for sharing his knowledge and insight, and for his belief in me. I also would like to
express my thanks to each of the committee members, Prof. Gang Bao, Dr. Chichia
Chin, Prof. Charles R. MacC’luer, and Prof. Jerry D. Schuur for their assistance and
time.

I thank my family and friends for their encouragement and support.

iv

TABLE OF CONTENTS

LIST OF FIGURES
INTRODUCTION

1 Perturbation Formulation

2 SB is a Circle of Radius c Centered at (20,310)

2.1 Membrane With a Circular Core of Radius c Centered at (mmyo) . . . .
2.2 Circular Membrane With 3 Circular Core of Radius c Centered at (20,310)
2.3 Annular Circular Membrane With Outer Radius 1 and Inner Radius c . .

3 5;; is a Strip of Length 2c Centered at ($0,310)
3.1 Membrane With a Strip of Length 2c Centered at (2:0, yo) .........
3.2 Circular Membrane With a Centered Strip of Length 2c ..........

4 Conclusions

APPENDIX A The Pin-point Phenomenon of Simply Connected
Membranes

APPENDIX B The Generalized Green’s Ifimctions

APPENDIX C The Egg; Series Expansion of K
0.1 Annular Circular Membrane With Outer Radius 1 and Inner Radius c .
C.2 Elliptic Membrane of Area 1r With an Internal Confocal Strip of Length 2c

BIBLIOGRAPHY

vi

10
12
20
24

32
36
53

54

60

62

8322

69

LIST OF FIGURES

2.1 SB is a circle of radius c centered at ($0,310) ................ 10
3.1 SB is a strip of length 2c centered at (2:0, yo) ................ 32

4.1 The comparison of the asymptotic approximation and the exact solution 57

vi

INTRODUCTION

The eigenvalue problem for the negative Laplace operator in two dimensions is

¢=OonC‘, (2)

where R is a bounded region with boundary C in two dimensional space. It arises
from separating the time variable out of the wave equation, so it occurs in many
applications; particularly in applications to vibrations of membranes and to acoustic
and electromagnetic waveguides. For instance [12], we can consider the case of a
fixed, uniform, flexible membrane R, of mass p per unit area, stretched under uniform

tension T per unit length. The equation of motion is the wave equation

- 162T! T
A‘P—pw-‘ZO; 02='p—', (3)

where I1 is the vertical displacement of the membrane from its equilibrium position.

The boundary condition is
\IIzOonC'x[0,oo). (4)

By requiring simple harmonic dependence on time, we can separate out the time

factor:

\II = 1/16”” , w is the vibrational frequency , (5)

where

2

, — w
-A¢=WmR;u=;2-, (6)

2,!) = 0 on C’ . (7)
Eqs(1),(2) has a spectrum of infinitely many positive eigenvalues

0<#1<#2£#3£-~, (8)

with no finite accumulation point [8]. The closed form solutions for the eigenvalues
p of —A exist only in few geometric regions. In two-dimensional space, they exist
only in regions [12] which can be described by rectangular, parabolic, polar, or elliptic
coordinates, regions such as rectangles, circles, ellipses, annular circles, and confocal
ellipses.

Numerical techniques such as the finite difference method, finite element method,
point matching method, and eigenfunction matching method are often used to solve
this problem. However, for doubly connected regions with the region bounded by
inner boundary which has maximum dimension of 2c, c << 1, the disadvantages
of numerical techniques, such as repetition of the evaluation for each different c and
serious scaling problems due to the small size, encourage us to develop the asymptotic
expansion formula for ,u.

The dissertation consists of three parts:

In Chapter 2, the formulation of the modified perturbation method and the re-
sulting asymptotic expansion formulas are presented; it is performed by applying
the perturbation method [1], the reflection method [7], and the Fredholm alternative
theorem [4].

In Chapter 3 and 4, the applications of the modified perturbation method to

special cases are executed, such as regions with an inner circular boundary, regions

2

with an inner linear boundary, annular circular regions, and circular regions with a
centered strip ; it is achieved by correcting the inner and outer boundary conditions
alternatively and by applying the generalized Green’s functions [4, 12]. The first three
order terms of the asymptotic expansion formulas are found explicitly.

In Chapter 5, the accuracy of the first three order terms of the asymptotic ex-
pansion formulas resulting from the modified perturbation method is compared. The
first three order terms of the asymptotic expansion formulas to general regions are
exhibited explicitly. Relations between the first three order terms of the asymptotic

expansion formulas and geometric properties of the regions are also investigated.

CHAPTER 1

Perturbation Formulation

Consider a membrane having region R0 enclosed by a boundary So, with an internal

core which has a boundary S B. The governing Helmholtz equation is

AW(:1:,y) + K2W(:z:,y) = O, (1.1)

where W is the normalized vertical displacement and K is the normalized vibrational

frequency, K = w - L - l/E’l—O’T' The symbol L is a characteristic length defined by

 

\/ (area of R0) / 7r. Let region bounded by So and S 3 be the region R. Consider

AU(x,y) + K2U(:r,y) = 0 in R, (1.2)
U(a:,y)=OonSoUSB, (1.3)

and
AU(:1:,y)+ K20($,y) = O in R0, (1.4)
I7(:1:,y) = O on So. (1.5)

 

Let K0 be the fundamental frequency to eqs(1.4),(1.5) and U0 be the corresponding
eigenfunction. K0 is simple [8]. According to eq(A.1), we assume that [1, 7] the

fundamental frequency K to eqs(1.2),(1.3) and its corresponding eigenfunction are

many) = Uo(rc,y) + iA'UAxw) , (1-6)

(:1

K = K0+ZA‘F,, (1.7)

l=1
where the parameter /\ is introduced as a formal way of separating out approximate
solutions of various orders in eq(1.7) and as a sequencing tool in eq(1.6).

Substituting eqs(1.6),(1.7) into eq(l.2) yields

(AU0(:L~, y) + K3U0(a:, y)) + A (and, y) + K3U1(x, y) + 2KoF1Uo(a:, 21))

+ Z x" AUm(x, y) + K3U...(x, y) + 2K0 2 Fij_.-(x. y)

m=2 j=1

+3: :im was y) =0. (1.8)

8:1

For similar orders of A, this leads to

AUo(:c,y) + K3Uo(rc,y) = 0 , (1-9)

AU1(:c.y) + K§U1(w,y) = —2KoF1Uo(:v,y) , (1-10)
and

AU", (:13, y)+K2Um (3,31):

m— lm—a
"2KOZFj-(Um -j 13 y) _2 Z FsFtljm —(s+t)(x y)!
jzl 8:1 i=1

mzzsApu. 01D

Let

Un(:c,y) = Vn(x,y) + Wn(x,y) , n = 0,1,2,- -- ,

where V0(x, y) = O, Vn’s are defined in R, Wn’s are defined in R0,

and [7]

W0($,y) : 0 on SO,
: W0(:13,y)+ Al[1(33iy) : O 012 SB,
Wo(-’Bay) + A (V1($,3/) + W103)?!» = 0 0’” 30, ’

W003)?» + All/1013a?!) + W1($ay)) + ’ "
+/\m—1(Vm_1(x,y)+ Wm—l(xay)) + )‘me(3ay) : O on 53 i

m=2,3,4,...,

W0(931y)+ /\ (V100,?!) + W1(:c,y)) + . --
Hm (Vm(x.y) + Wm(:c,y)) = 0 on So ,

m=2,3,4,....

Set A]: 1. Then eqs(1.6),(1.7) become

where

U(1L‘,y)= W0($,y) + (V1(1L’,y) + W1(x,y)) + . . .

+((Vm(11y) + Wm($ay)) 'l' ' ' ° ,

K2K0+F1+...+Fm+...,

AI/V0($1:y) + K02W0(31y) : 01:17, R0 1

6

(1.12)

(1.13)

(1.14)

(1.15)

(1.16)

(1.17)

(1.18)

(1.19)

(1.20)

W0(:c,y) = 0 0’” So,
‘ AV1(x,y) + K§V1(a=,y)= 0 in R ,
(i.
[I l[1(may): —W0($3y) on SB,
AW1($,y) + K3W1($,y) = —2KOF1W0($,y) 271 R0 ,
W1(xiy) = -—V1(a:,y) on SD:
AV2(x,y) + K02V2(2:,y) = —2K0F1l/1(x,y) in R ,

V2033) = -W1(x,y) on 53 ,

AW2(x,y) + K§W2(:L‘,y) =

—2K0F1W1(:1:,y) — 2K0F2W0($ay) - FEW/M504!) in Ro ,
W2(a:,y) = —V2(:z:,y) on So ,

AVm (:6 y) +K3Vm(x,y) =

m-——1ms

_2KOZF'1_Vm—j(xy) ZZFsl/mFt —()8+t(3y)inRi

j=l s=:=lt1

m=3,4,5,...,

Vm(xay) : —Wm_1($,y) on SB 3 m = 3,4)5‘i"' i

AW". (It 31) +K§Wm 23,3) =

m— lm-s

—2KOZFj—ij(xa_y) 2:17:17th —(s+t)(m y)anOi

j=1 8::1tl

m=3,4,5,...,

Wm(a:,y) = —Vm(:z:,y) on. So, m = 3,4,5,....

(1.21)

(1.22)

(1.23)

(1.24)

(1.25)

(1.26)

(1.27)

(1.28)

(1.29)

(1.30)

(1.31)

(1.32)

(1.33)

By the Fredholm alternative theorem, the existence conditions [4] of W“, n =
1,2,3, . . ., give

6U , -
/ 2K0F1Wo<x.y)vo(x,y)dA=—f MVdayMs, (1.34)
R0 50 5n

[R0 (2K0F1W1(:c, y) + 2K0F2w0(:c, y) + FEW0(x, y)) (10mg) dA

_ 6U0(x,y)
— £0 an V2(xiy) d3 3 (135)
m- 1m— 5
/ (ZKOZF Wm_ _,- (a: ,y) + F,F m_,(,+t)(:1: 31)) Uo(:c, y) dA
R0 j=l 3:1 t=1
6110(1))!»
2 _ —_ = 4 1.
£0 an Vm(x,y) ds, m 3, ,5, ( 36)

Thus, the corrections to the fundamental frequency are found

f Midas!) d3
50 an

 

 

 

 

 

F1 : 2 , (1.37)
—2K U ac, dA
0/30 0( y)
1. EU 2:,
/ (2K0F1W1 + FfWo(a:,y)) U0(:c,y) dA +f #1601331) ds
F2: R0 2 SO n i
—2K / U , dA
0 R0 0(33 y)
(1.33)
m—l
/ (2K0 Z Fij_j($,y)) U0($,y) dA 6U0($iy)vm(x y) d8
F = R0 j=l __ So an ,
m —2K / U2 , dA 2K / U2 x, dA
011000531) 0R00( y)
m— 1m— 1
[R0 ( F sFtW —(ms+t )(23 ’30) U0(xiy) dA
_ ”1‘21 2 ,m=3,4,5,....
2K U( , dA
0/120 0 (1’3 31)

(1.39)

Remark:
Fm, m: 1,2,3,...

are unique up to a constant multiplier to U0 .

CHAPTER 2

S B is a Circle of Radius (3 Centered

at (330, 310)

 

(301 3’0)

 

W

 

Figure 2.1: $3 is a circle of radius c centered at (2:0, yo)

The coordinates (m’,y’), (23,31), (r, 9), and (r',9’), as in Figure 2.1, are related by

z’ = r—zo , (2.1)
y’ il-yo

10

x = r' cos 6” , y = r' sin 0' , (2.2)
and
:1." = rcosB , y' = rsint9 . (2.3)

Due to the invariability of the governing Helmholtz equation under the translation of

coordinates, the governing Helmholtz equation can be written as

62U(r,6) + 16U(r,6) + _1_62U(r,6)

or? T 8? T2 802 +K2U(r,6’)-—=O. (2.4)

Eq(2.4) can be separated by U (r, 6) = \Il(r)<I>(6) resulting in Bessel equations of order

 

 

 

b [12]
zde‘NZ) + Zd\II(z) + (2:2 - b2)\I’(z) = 0 where z = Kr (2 5)
dz2 dz ’ '
and
d2<I>(6) 2 __
£102 + b <I>(6) —— O , (2.6)

where b is a separation constant. The periodic solutions to eq(2.6) are

) sin(n0) , cos(n0) , n =I‘IQ,’1,2,... . (2.7)
. .\

The corresponding solutions [12] to eq(2.5) are J", the 71“ order Bessel function, and
Yn, the nth order Neumann function, where

(£)(n+2l)
2
“(Tl-fl”, "' 31"")

 

 

Ym(Z)=-72:(ln£)Jm( z)_;1r,m llm—jH—l) (2)(m-22')

1:0 z

1 co _ 1)t(§)‘"‘+2"(¢(m+l+1)+¢(z+1))
;Z(u(m+z)1 ’

=0

(2.10)

 

~

m=1,2,3,...,

where 7 z 0.5772,1,/)(m + l + 1) = (1+ % + + m)— ’y, and 1/1(1) = —7.

2.1 Membrane With a Circular Core of Radius 0

Centered at (2:0, yo)
U0(r, 6) is finite in R0, we assume that

U00", 0) = BoJ0(KOT) + i Jm(KoT) (Am 81110716) + Bm cos(m0)) (2.11)

m=l
with appropriate constant coeflicients Bo, Bm, and Am determined by the boundary
condition, eq(1.5).

{1, cos(m0), sin(m6)°° _1 is a complete orthogonal set of functions and

21r
/ sin2(m6) d6 = 7r , (2.12)
0

21r
/ cosz(m6) d0 = 7r . (2.13)
0

Thus, eqs(1.22),(1.23) give

 

 

V1039) = 80(6) (J0(Ko7‘) — £E;::;%(K0T)) — 302E;::;%(K0T)
+ i [1,..(3) (Jm(Kor) _ ——;:[’I:::]Ym(xor)) -— AmSJ/fE—fi—ggmwor) sin<m6>

12

 

+ §3[B,n( c)Jm(( (K0r—) ){ZEfiZZgYmmOrO —B,,, Y—E—ggg—gy m(Kor) cos(m6),

mzl
(2.14)

To correct the boundary condition on So to 0(E173) for (U0+V1)1 Bo(c),
Am(c), and Bm(c) must be at most 0(E1—C). Green’s 2Ml identity [4, 11] and

eqs(1.20),(1.21),(1.22),(1.23) give

£0 Vl(x,y)aU(:a(:1y) d3 2 £8 U0($,y) (LKQZ—ly). + W) d3, (215)

then eq( 1.37) becomes

 

. (91/103111) 0Uo(x,y)
flea o(:c,y)( 8n + 6n )ds

 

—2K0 / U02(:c,y) dA
R0
[10]'gives
2
W(Jm(z),Ym(z)) : —, (2.17)
7rz
where W (Jm(z),Ym(z)) is the Wronskian of Jm(z) and Ym(z).
J0(KOC) __—_1r 1 +£(ln2—7—ano) 1 +---. (2.18)

 

 

 

Y0(Koc) — 2 (—1nc) (lnc)2

Eqs(2.11),(2.12),(2.13),(2.14),(2.16),(2.17),(2.18) yield

2” 81/1 (7‘, 0) 6U0(7‘, 0)
f0 U0(c,6’)( 8r |,:c+ 6r (m cd0

2KO Cflof U30, 6) dA

 

 

F1:

2Bo(c )Boi—ZE——;Z ]+ 2(4... )4..+B (1B...) jig]

0C)-—K0/RO(C U02( r, a) dA

 

 

2JOOC)(K 2 2 Jm(K0C)
230nm: c...)+ Z (A ”3 )Ymmoc)

—K0 / Ug( r, 0) dA
R0

 

 

13

J0(KOC) 0° ~ ~ Jm(KOC)

0m + 2.3 (1-014,. + 3433'”) m

—K0 / U020, 19) dA
R0

2Bo(C)B

 

«BS 1 + 7r(ano+7—ln2)Bg 1 +
K0 [R0 Ug(r,9)dA llncl K0 fRO Ug(r,0)dA llllcl2

 

 

+
(2.19)

B0(c), Am(c), and Bm(c) are at most 0(lfi), the simplest choice is to set Bo(c),
Am(c), and Bm(c) equal to zero. Other choices would only lead to a higher order

correction to the first order result F1, eq(2.19). Thus,

”BOJ0(KOC) 0°

 

 

 

V1(r,6) : Y0(Koc) Y0( Kor) —mZ=I'—}],—:—— Y(Kor) (Am sin(m6) + Bm cos(m6))
(2.20)
and
F1: «83 1 + 7r(ano+'7—ln2)B§ 1 2+...,
Ko/ U§(r,0)dA llncl Ko/ Ug(r,9)dA |1n Cl
Ro R0
(2.21)

where ’7 x 0.5772.

The Green’s 2'” identity [4, 11] and the generalized Green’s function
G(r,6;F,R),eq(B.5), yield

600,19; 120')
an

 

W1(r,6) = EU0(r,9) + £3 V1(F,é) ds (2.22)

 

°° "N? U r, a) 6U -(r a)
-E ”3( f 41—bv ,6 d , 2.23
U0( (,1‘ 9) )+ szlJZTUQV —K0)HUN,j“2 so 611. 1(1‘ ) S ( l

where E is a constant and

UNJ-(r, 0) = B0(N,j)J0(KNr) + i Jm(KNr) (Am(N,j) sin(m9) + Bm(N,j) cos(m0))

mzl

14

(2.24)

with appropriate constant coefficients BO(N, j), Bm(N , j ), and Am(N, 3') determined
by the boundary condition, eq(1.5).
Eqs(2.18),(2.20) yield

BUNJ(T',6) _ 7TBo 6UN,J-(r,0) 1
£90 an V1(r,6)ds—( 2 £0 6n Y0(Kor) ds |lnc|+nu

(2.25)

To correct the boundary condition on SB to 0(1'117) for (U0 + Vl + W1), the constant

E must be zero. Thus,

(90(1‘, 6; 1", fl)

 

 

W1(r,9) = lie. an V1023) ds (2.26)
°° ’(N) U - 0 6U . 0
N1](r? ) f NJ(T’ )V(T 0) d
= —— , s . (2.27)
~21; (Ki; - 11(02)||UN.1||2 So an 1
Let
15(3)?!) = V2i(3313/)+ V2h(xiy) 1 (2-28)
where
AW($iy) + Kill/{(93131) : —2K0F11/1($,y) in R1 (229)
AV2h(a:,y) + K02V2h(:r:,y) = 0 in R , (2.30)
‘6”(23) = —W1(x,y) - V3003) on 58 . (231)
For the non-homogeneous equation
n 1 I 2 m2
R +;R + K°_—r?_ R=Ym(Kor), m=0,1,2,3,... (2.32)

15

the particular solution is

 

 

 

 

 

 

 

_ TY7;(K07') _
R— _2K0 , m—0,1,2,3,... (2.33)
Thus,
1' _ ‘BoJo(KoC) 7
V30", 0) — F1 Y0(K0c) TYO (Ko’r')
— Jm (K0 c) rY'(Kor)(Amsin(m6)+Bmcos(m0))
m—1/7nc(K0 )
(2.34)
Eqs(2.12),(2.13),(2.30),(2.31),(2.34) yield
~ J (K c)
h _ _ o 0
V2 (7‘19) - 170(0) (J0(K07‘) Y0(K0c)YO(KOT))
00 ~ ~ Jm(KOC)
+2; (C (c)sin(m0)+ +Dm(c)cos(m6)) (Jm(K0r) Ym(Koc)Ym(Kor))
+D0Y0(K0r) + Z Ym(K0r) (Cm sin(m0) + Dm cos(m6’)) , (2.35)
m=1
where
_ —1 2" BoFch0(K0C)I/O,(K0C)
D0 _ 27rYo(Koc) f0 W1(c,6) d6+ Y02(Koc) , (2.36)
_ —1 2" , AmFICJm(k0C)Y,;(KoC)
c... _ ”YMKOc) f0 sm(m0)W1(c,0) as Y..2.(Koc) , (2.37)
_ -1 2” BmF1CJm(koc)Yr;,(KOC)
Dm _ Koo) f0 cos(m9)W1(c,0) 39+ 1m K06) (2.33)
Eqs(2.24),(2.25),(2.27) yield
2 °° "”0 «23 B (N “)J (K c) 6U -(r 0) 1
W()6d6= °°(”°” ——N-’3——’-—YKd—
f0 10 LE; (Kt— KrilllUNgl|2 fs. an °( ”’"l 3 llncl

+...,

16

(2.39)

_ °° “Ni 231114.171) (KNc) 01/173039)
lo S‘“(m9)WI(C’9)d9—[:Z 2133— K3>IIU~,.-Ir KTWKOTW

N21 j=1
1

llncl

 

 

+...,

(2.40)

 

2n on 1N) 7r2 C ‘7‘
f0 comm/1W 9=[2212(KN 308 mm j)Jm (KN ) £61131 ,0)

K—3>IIU~.-12 6n WK”) d3]

1
|lnc|

N—l J

 

+...

(2.41)

To correct the boundary condition on So to 0(I'1n—5 CI) for (U0 + V1 + W1 + V2), D0(c),

Dm(c), and Cm(c) must be at most 0(l—ln—él7). Green’s 2’“ identity [4, 11] gives

0U0(r,6) _ 6U0(r,6) z.
350 an V2(r,0)ds—f;o an V2(r,6)ds

6Uo(r, 9) h 8V2"(r, 9)
__ £3 (Ti/2 (736) —— TUo(r,0)) ds. (2.42)

Thus, eq(1.38) becomes
8U0(1‘,0) '-
F2: F12 _fSO—an V2(r,6)ds
“Mo 2K U2 6 dA
Ol/‘R0 0(7‘, )
6Uo(r,0) h 01/2"(r, 0)
£8 (Tl/2 (7‘19) TUOO‘ 91d
2K U2 ,0 CIA
O/Ro 0(7" )

 

 

(2.43)

Eqs(2.18),(2.21),(2.34) yield

6U0(7‘,0) i __ BoJo(KoC) BUO(7‘,6) ,
£90 an V2(r,0)ds— F1{ ham) )2 an rY0(Kor)ds

17

+ i}, m( (KOZ) £0 {—92% (Am sin(m0) + Bm cos(m6))rY,§,(K0r) ds}

m=1Ym(

 

 

 

71.233
f erfl,’(Kor) ds 1 2 +--- . (2.44)
2K0 [R0 U36, 6) dA So 5" llncl
Eqs(2. 11),((2. 17), (2.35), (2.36), (2.39) yield

7‘ h 7'
1.. (Lauren/.1...) .. ___6%agf>3.(.,.)) d.

21r h
_/ (L5— *6U0( (,7' 6) I =c %h(c,6) + W lr=c U0(C,9)) C d0

 

 

 

8r
- J0(K C 0° ~ ~ Jm(KOC)
_ 4BoDo(C)YO(K C 2;1(Amcm(c)+ B...D,,.(c)) Ym(Koc)

+4BODO + 2 Z (AmCm + BmDm)

m=l
~ J0(K0€) 0° Jm(KOC)
:—4BD C_— AmCmC +BmDm
0 0( )Y0(K0C) 7712;1( ( ) ( )) Ym(Koc)
+i 7%)(K2 :2B2BO(N, 3') f My (K r) ds _1_ +
N=1j= 1 ’KolllUlelz So an 0 0 [Inc]2

(2.45)

D0(c), Dm(c), and Cm(c) are at most 0(W), the simplest choice is to set D0(c),
Dm(c), and Cm(c) equal to zero . Other choices would only lead to a higher order

correction to the 2"" order result F2, eq(2.43). Thus,

 

V2h(r, 0) = DOYE)(K0r) + Z Ym(K0r) (Cm sin(m6) + Dm cos(m6)) , (2.46)
m=1
where
“‘1 2" BoF16J0(K0C)Y0’(Koc)
D=——/ W ,6d6 , 2.47
° Browne) 0 “C ’ + 13660 ( ’

18

AmFICJm(k0C)YTL (KOC)
Yr3;(KoC) ,

—1

= 1er(K0c) (2'48)

 

21r
Cm / sin(m6)W1(c, 6)d0 +
o

-1

_ BmFICJm(k0C)Yr;(KOC)
— ’ITYm(KOC) .

Yn2,(Koc) (2.49)

 

21r
Dm / cos(m0)W1(c, 0)d8 +
o

and, by eqs(2.21),(2.43),(2.44),(2.45),

EU 1', 0 ,
W233 «233 f3 —%(1—1—)rY0(K07-) ds

—2K3 ([120 U§(r,0) dA)2 + —4Kg (fRo Ug(r, 0) dA)2

"M «2323(N' 0U . 0
0 0 1.7) f N,J(r? )Y KT d3
(Kg—K511115512 so an °( °) 1

2K0 [R0 030,9) dA Ilncl2

 

F2 2

 

 

oo
2 .
+N=13

+ - ~ . (2.50)

Eqs(1.19),(2.21),(2.50) give

«33 1 + 1r(ln K0 + 'y — 1n 2)Bg
K0 / (130,0) dA |1n C| K0 f U020, 62) dA
30 Ho

 

 

K=Ko+

«23ng a—U£(:’—@-1"Y0'(Kor) ds

7‘2 Bd _ 0n

2K3 (fa. Ugo, o) dA)2 4K3 (LG U029, 9) M)2

 

q

6n Y0(Kor) ds 1

 

N=1 j=1 (K12v " KdHlUNJHz So

2K0 [R0 Ug(r,9) dA 11M2

 

+

 

+ - -- , (2.51)

where 7 x 0.5772 .

19

2.2 Circular Membrane With a Circular Core of

Radius 0 Centered at (2:0, yo)

The geometry of the concerned region is with the outer boundary where So is r' = 1
and the inner boundary where S B is r = c.
For a circular membrane with the boundary where So is r’ = 1, the frequencies I?

and the corresponding eigenfunctions [8, 12] [7 to eqs(1.4),(1.5) are

Km, m =1,2,3,... , (2.52)
Km , p,m = 1,2,3,... , (2.53)

and
J0(K0,m’f") , m = 1,2,3, . . . , (2.54)
Jp(Kp,mr')sin(p6') , Jp(Kp,mr')cos(p6') , p,m = 1,2,3,... , (2.55)
respectively, where Km," is the mth zero of Jn , n = 0,1,2,3,... , m = 1,2,3, . .. .

K0 = KO,1 x 2.4048 and U0 = J0(KOT’).

Translational addition theorems for circular cylindrical wave functions [5, 6, 12]

give
J0(Kr') = i J,(Kr0)J,(Kr) cos ([6 — l(60 + 1r)) , (2.56)
l=—oo
Jp(Kr') sin(p6') = i J;_,,(K1'0)J1(Kr) sin (10 — (l — p)(00 + 7r)) , (2.57)
p: 1,2,3,...,
Jp(Kr') cos(p0') = i J1_p(Kro)J;(Kr)cos (l6 — (l — p)(00 + 7r)) , (2.58)
lz—oo
p: 1,2,3,...,

20

where To is the distance between 0 = (0,0) and 01 = (2:0, yo) and 00 is the angle from

the :1: axis to 001. [10] gives
J__.-(z) = (—1)"J,-(z) , 2': 1,2,3, . .. . (2.59)
Thus,

J0(Ko,m7") = J0(Ko,mT0)Jo(Ko,m7‘)

+ :0: [(—1)‘2J,(Ko,mro) sin(z'90)] J,(K0,mr) sin(z'0)
+ :([(-1Y121(K0,m7‘0) (305090)] J5(Ko,m7‘) 008(i0) , (2.60)

m=1,2,3,... ,

J.<Kp.mr'>sin<p6')= Jp(Kme0)Sin(P¢90)Jo( Kpmr)

+§i[(-1><‘P>J.(K.,mro)cos<<i—p)oo>

+(—1)(‘+1)J,+p(Kp,mro) cos ((2' + p)00)] J,(Kp,mr) sin(z'6)

00

+ ; [(—1)(“”+1)J,_,,(Kp,mro) sin ((z' — moo)
+(—1)‘J,+p(Kp,mro) sin ((i + p)60)] J,(Kp,mr) cos(z’6) , (2.61)

p,m= 1,2,3,... ,

Jp(Kp,m7J) (308090,) : Jp(Kp,mr0) COS(p60)J0(KP.mT)

21

1"]8

ll
H

+ [<—1)<‘-P>J._.(Kp.mro) sin W — moo)

1

+(—1)i i+p(Kp,mr0) sin ((2 + p)00)] J,(Kp,mr) sin(z'0)
+ i [(_1)(inp)Ji-p(Kp,mT0) C05 ((2 _ FWD)

+(—1)‘ i+p(Kp,mro) cos ((i +p)00)] J,(Kp,mr) cos(2'0) , (2.62)

p,m=1,2,3,... .

Integrals of products of Bessel functions [10] give

/ tJ§(t) 5;[Jg(z )+ Jf(z)] , (2.63)
foth3_1(t =(2Z(n + 2l)J J,2,+2,(z (z) , (2.64)
(=0
72: 2,3,4,....

Thus,

(a. J3<Ko,mr'> em = 7r [J3<Ko.m) + Jf<Ko,m>]

: WJ12(KO,m) 1 (265)

m=1,2,3,...,

IIJP<Kp,mr') Known“? = (R. <J.(Kp,mr') simmer»? dA

27r°°(

        

)‘Ip2+1+21(KP,m) 1 (2'66)
m(lzo

p,m — —1,2,3,...,

22

IIJp(Kp.mr')cos(p0’)||2 = f... (J.(K..mr'>cos(p9'))2 dA

00
27r 2

 

= K2 2(1) + 1 + 21)Jp+1+2I(Kp.m) a (2-67)
p.171 (=0
p,m = 1,2,3,....
Law of Cosine :
r2 = r'2 + r3 -— 27w" cos(6' -— 60) . (2.68)
Ja<z> = —J.(z) , Yam = -Yl(z> . (269)

Then,

8J0(K07") ,
£0 77%(K07‘) d3

 

 

21r
: J1(K0)K0/ \/1+ r8 — 27'0 cos 6”Y1(K0\/1 + r3 — 27'0 cos 0’) (16’.
o

(2.70)
Translational addition theorems for circular cylindrical wave functions [5, 6, 12] give

Y0(Kr) = f: J1(Kro)Yl(Kr') cos (l0' — ZOO) , (2.71)

l=-oo

where To is the distance between 0 = (0,0) and 01 2 (11:0, yo), 60 is the angle from
the :1: axis to 001, and the formula holds for points lying ouside the circle with the

diameter 001 = r0. Then,

aJ K ,, '
f; 0(673, 7‘ )Y0(K07') d8 = —27rKo,nJo(K0T0)Y0(K0)J1(Kom) , (2.72)

n=2,3,4,...,

 

0.] K mr’ sin ’ I .
{9 P( P. an) (p6 )}’0(K07‘) d8 : 271.1(1).me(K0T0)},p(KO)Jp(Kp,m)Sln(p60) ,
p,m=1,2,3,... ,

(2.73)

23

 

(9.1,,(K mr cos(p6’) I
£0 1» 8n) Y0(K0r) ds = 2nKp,me(Koro)1/p(K0)Jp(Kp,m) cos(p60) ,
p,m = 1,2,3,...

(2.74)

Eqs(2.51),(2.60),(2.61),(2.62),(2.65),(2.66),(2.67), (2.70),(2.72),(2.73),(2.74) give

 

 

K : K0 + (J3(KOT0) ) 1 [(11] K0 + ’7 — 1n 2)J02(K0T0) _ J3(K07‘0)

K0J12(Ko) K0J12(Ko) ZKdJflKo)

|lnc|

 

 

21r
J8(Koro)/ \/1+ 7'3 — 270 cos 0’Y1(Ko\/1 + 1% —— 27‘0 cos 9’) d6'
0

 

 

4K0Jf(Ko)
+7TJ0( K070) )i 0° K3 mJ p(Kp,m7'0)J;(Kp’m)Jp(KoTo)Y;,(Ko)
—2—_ oo
+12K0J HKOM:1:1(K2,m — Kg) 2(1) + 1 + 21)J3+1+21(Kp.m)
(:0

_7TJ0 (K070) }/o( Ko) 00 K0,nJ0(K0,nT0) 1
2 )Z 2
K0J1(KO) 11:2 (Kan — K3) J1(K0,n) |lnc|

 

 

+ . .. , (2.75)

where To is the distance between 0 = (0,0) and 01 = (2:0,yo), K0 = K0,] 5:: 2.4048,

7 x 0.5772, and K0,", is the mth zero of Jo, Jnm is the mth zero of JP, m,p = 1,2, 3, . . ..

2.3 Annular Circular Membrane With Outer Ra-
dius 1 and Inner Radius 0

The geometry of the concerned region is with the outer boundary where So is r’ =

7' = 1 and the inner boundary where S 3 is r' = r = c. 70 = 0.

J0(0)=1 , Jp(0)=0, p=l,2,3,... . (2.76)

24

Eq(2.75) gives

 

 

K=Ko+( 1 ) 1 [(ano+7—ln2) 1

K0J12(K0) KoJf(Ko) _ 2K3Jf<Ko)

|lnc|

_ 71’Y1(K0) _7_T______Yo( K0) 0); K0,n 1
2K0J13(K0) K0J12(K ,2: (Kgn—K§)J1(Ko,,) |lnc|2

 

+ - -- , (2.77)

where K0 2 KOJ 2 2.4048, '7 x 0.5772, and K0,, is the 1)“ zero of Jo, p : 1, 2,3, . . ..
Alternatively, eqs(B.7),(B.8) [4] become

 

 

 

82@(r,6; 7, 6) 1 666,6, 7, 6') 1 62666 7, 6) -
072 + 7: (97‘ + 7—2 802 + K

 

~ 6
= EU”)? ’6) in R0 , (2.78)
G(1,6;f,(§)=0,036,63271,0<f 31. (2.79)

{1, sin(m0), cos(m.6)}:‘::1 is an orthogonal complete set of functions, we assume that

~

G(r,0; 7,6) = Ho(6)Po(r,F) + i: Hm(6)Pm(r,f) cos(m6)

00

+ Z Lm(é)Qm(r,f)sin(m6) . (2.80)

m=l

Eq(2.78) yields

~ 32P0(r, F) 8P0(r,7") ~ 2 ,_
H0(6) (r—éfi— + T + rK P0(r,r)

+ i Hm(é) 0%.]. +0_Pni,7(rr_r_)+ +(7R2 — ”172) Pm(r,F)) cos(m9)

°° - 2 7 7 - 2
+2; Lm(6) (7.99%: + Q'ggfi + (7K2 — I?) Qm(r,f)) sin(m0)

25

= _6(7,7)6(6,6) . (2.81)

Multiplying by 1, cos(m6), sin(m6) and integrating from 0 to 277 on eq(2.81), m =

1,2,3,... yields
H(6>-—i 1761—1 (2') L16) 176) (282)
0 _277’ m _7rcosm, m —7rSlnm, -
m: 1,2,3,...,
£n(P,,(r,r)) : —(5(r,f) , n = O,1,2,... , (2.83)
£m(Qm(r,F)) = —6(r,7") , m = 1,2,3,... , (2.84)

where [3,, is the Sturm-Liouville operator of order n [12] ;
~ n2
£n(W)=rW"+W'+r(K2——2)W, n=0,1,2,.... (2.85)
r

{1, sin(m6), cos(m0)}::’:l is an orthogonal set of functions, eqs(2.79),(2.82) yield

 

 

 

Pn(1,f) = 0, Pn(0,f‘) is finite, n = 0,1,2,... , (2.86)
Qm(1,i‘) = 0, Qm(0,f) is finite, m =1,2,3,... (2.87)
Thus,
3J,(f{7) (”(1914177) — mum) , 7 g 7
2 MK)
Pn(r,F) = ] , (2.88)
3.741%?) (Yn(I-() Jn(1~{r) — Yn(f(r)) , r 2 7"
1 2 Jn(K)
Tl : 011121 1
Qm(r,f) = Pm('r,f) , m = 1,2,3, . .. . (2.89)

Eqs(2.80),(2.82),(2.88),(2.89) give

G(7‘,9;F,é) = —1—P0(r,f) +

2,, i COS (m(9 - 5)) Pm(r, 7) . (2.90)

m=1

=ll1-|

Thus, by eq(B.6),

f

Y0(K0)J0(K07)J0(K07)
KOJ1(KO)

 

—‘£li [J0(K0T)Y0(Kof) +

_ TY0(K0)J0(K0F)J1(KO7‘)

J1 K0

 

 

_ (J0(Kor)Jo(KOF)Y1(K0) + FJ0(K07‘)Y0(K0)J1(KOF))]

J1 K0

 

—E i COS (771(6 — é)) [Jm(KoT)Ym(KOf) — Jm(K0T)YmE§:)Jm(KO'F)]

Jm )

,TSF

interchange 7‘ and F in the above result of r S F

 

 

1 ,r 2 F .
(2.91)
Eq(2.54) yields
Uo('l‘,6) = J0(K07‘) . (2.92)
Then ,by eq(2.20),
J0(K0C)
6 = -— . 2. 3
V1(7', ) m(KOC)YO(KOT) ( 9 )
Recurrence relations of bessel functions [10] give
, 1 1

27

Thus,

00 ,0;",9~ _ ~
w1<r,o>=f50 (ran’ )vloamds

 

 

 

 

 

 

 

2" 60(1‘, 0; 'F, 5) J0(KoC) ~
= [0 a; |-=1 (—YO(Koc)n<Ko)) d0
J0(KoC) K0J0(K07‘)Y1(Ko) K0Y0(K0)7'J1(K07‘)
: —7rY0(Ko))/O(KOC) [ 2 + J0(K07‘)Y0(K0) - 2
K0
(K0K(K0)J1(KO)J0(K01')+ —J2(K0)Y0(K0)J0(Kor))
_ 2J1(K0)
(2.95)

Eqs(2.21),(2.65),(2.92) give

1 1 1n K0 + 7 — ln2 1
= +--- . 2.96
F1 (KoJf(Ko)) |lnc| + ( K0J12(K0) ) |lnc|? ( )

Eqs(2.46),(2.47),(2.48),(2.49),(2.92),(2.95),(2.96) give

 

 

 

 

V2h(r)6) : D0%(K07‘) i (2'97)
where
2" Fcho(Koc)Y0’(K0c)
—— W( 6 d6+
D0: 21rY0( Koc) /o 1 (’C ) ngoc)
K0
W3 K0 (K0J1<KO)Y1(K0) + 7n(Ko)J2(Ko)) 1
= gYono) Yo(Ko) + 7Y1(Ko)-2J1(Ko) Imclz
(2.98)

28

Thus,

8U0(r,6) h _8Vz"(r,0)
fix an V2030) ——8n U0(r,9) d3

 

21r _
= / M1 : D0Y0(K0c) + 019mm“) | : J0(K0c) cd6
0 6r 1' 6 Br ' c

= 27TCKOD0 (—J6(KOC)Y0(KOC) + Y0’(K06)J0(K0C))

 

 

 

=4D0
K
3 K0 (KOJ1(K°)Y1(K°) + 32Y0(K0)J2(K0)) 1
= 7r Y0(K0) Y0(Ko) + 7Y1(K0) — 2.110(0) |lnc|2
+..
(2.99)
Eqs(2.34),(2.92) yield
' —JO(KOC) ,
t = —— - 2.1 0
V2039) F1 ( Y0(Koc) 7‘Y0(K07‘)) ( 0 )
Thus,
6U0(r,6) z. _ 21r 0J0(K0r) i
fee an V2939) ‘13—], 6,. |r=1V2(1,0) d6
J0(KoC)
= —2 K J K Y K F
7r 0 1( o)1( o) 1Yo(Koc)
— 2Y1(Ko) 1 + " - (2.101)

 

‘" J1(Ko)|1n0|2 '
Eqs(2.43),(2.65),(2.92),(2.96),(2.99),(2.101) give

F2 ____{ -1 7TY1(K0)

2K81i(Ko) ‘ ZKoJi’(Ko)+

29

(KoJi<Ko>Yi(Ko) + -’§’-Y0<Ko>Ji(Ko))
7T2Y0(K0) 2Y0(K0) + K0Y1(K0) —

 

 

 

 

 

 

 

J1(K0)
_. l
4K0J12(K0) |ln Cl2
+ . .
(2.102)
Eqs(1.19),(2.96),(2.102) give
K=K0+F1+F2+---
1 1 (1n K0 + 7 — 1n 2) 1 1rY1(K0)
2K“ KflK 1 KflK _2K3J4K "2KJ3K
01(0) Incl 01(0) 01( 0) 01(0)
K0
2 (KoJi(Ko)Yi(Ko) + —2-Y0(Ko)J2(Ko))
71' “(1(0) 2%(K0) + K0Y1(Ko) - J1(K0) l
4K0J12(K0) * |1nc|2
+ . . ,
(2.103)
Recurrence relations for cross-products of bessel functions [10] give
4
MoTO — N000 = m , (2.104)
where
Mo = J0(K0)Y0(Ko) —‘ Jo(Ko)Y0(K0) = 0 i (2-105)

30

N0 = J0(K0)Y0’(Ko) — J0(K0)Y0(K0) = J1(K0)Y0(K0) ,

Go = J6(K0)Y0(Ko) — J0(K0)Y0'(Ko) = -J1(K0)Y0(Ko) ,

Then,

To = J0(K0)Y0(K0) — «10(K0)Y0’(K0) = 0 -

7r 1
—Y K = —— .
2 0( 0) KOJ1(K0)

Recurrence relations of bessel functions [10] give

Thus,

K=Ko+<

1

KlOJl(K0) : J0(Ko) + J2(K0) = J2(K0) -

 

K013

1
(1(0)) |lnc|
(ln K0 + '7 - 1n 2) 1 7rY1(Ko) 1

 

 

+l

KOJHKO) +2K3Ji(K0)—2K0J13(K0) |lnc|?

where K0 2 Km 2 2.4048 and 'y z 0.5772.

31

+...

(2.106)

(2.107)

(2.108)

(2.109)

(2.110)

(2.111)

CHAPTER 3

S B is a Strip of Length 2c Centered

at ($07 3J0)

 

 

 

Figure 3.1: SE is a strip of length 2c centered at (1:0, yo)

The cartesian coordinates (m’, y’) and (as, y), as in Figure 3.1, are related by

2’ cos sin '2: -— a:
= ‘P0 900 0 , (3.1)
y’ - sin ‘Po cos 900 y - yo

32

where 900 = 90 + 61, 00 is the angle from :1: axis to 6—61, 01 is the angle from 0‘63 to
x’ axis, and O = (O, 0), 01 = ($0,310). Let (£’,n’) be elliptic coordinates related to the

cartesian coordinates (:r,y) by

:1: = ccosh 5' cos 0' , y = csinh §' sin 17' , (3.2)

and (5,77) be elliptic coordinates related to the cartesian coordinates (1," ,y’ ) by

:1." = ccoshgcosn , y' = csinhfisinn , (3.3)

where 2c is the distance between the foci. Due to the invariability of the governing
Helmholtz equation under the translation and rotation of coordinates, the governing

Helmholtz equation can be written as

(cosh(2§) — cos(2n)) U(£,17) = O . (3.4)

 

02U(€,0) + 32U(€,n) + K28
0&2 6172 2

Eq(3.4) can be separated by U(§,n) = ql(§)<1>(n) resulting in Mathieu equations [12]

%—+ [hzcoshzé— ]\I’=O, (3.5)
$+ [b—hzcos217]<l>=0, (3-6)

where h = Kc and b is a separation constant. The periodic solutions [12] to eq(3.6)
are Sop, the pth order odd angular Mathieu function, and Seq, the q‘h order even

angular Mathieu function, where

00

So..(h,cosn) = 2331(h,p)sin(2177); (21)B§z(h,p) = 1
=1

(:1 l

33

if p is even , (3.7)

50,.(h, 0081?) = Z B§i+i(h,p) sin ((2l + 1)77); 2321+ 1)B§i+i(h,p) = 1

1:0 (:0

if p is odd , (3.8)
Se,(h, cos 0) = Z B§z(h, q) C08(2177); Z 35101, q) = 1
l=0 l=0
if q is even , (3.9)
s..(h, cosn) = Z B§l+1(h:Q)((2l + 1)n) ; 2 82.104) = 1

1:0 1:0

if q is odd , (3.10)
p: 1,2,3,... , q=0,1,2,....

The corresponding solutions [13] to eq(3.5) are J09, the 12‘“ order odd radial Mathieu
function of the first kind, Nap, the p‘h order odd radial Mathieu function of the
second kind, Jeq,the q‘h order even radial Mathieu function of the first kind, and

Neg, the qth order even radial Mathieu function of the second kind, where

Jop(h, coshg) = gtanhéi {(—1)1'¥(2l)B§,(h,p)J21(h cosh 5)}
1:1

 

(E i {(-1)"5 ° (hip)

830%?) 1:1 21
1 _£ 1 ,5 1 _£ 1 e _ ,
[J¢_1(§he )J,+1(§he )— J1+1(§he )Jz_1(§he )J} if p is even ,

(3.11)

Jop(h, cosh g) = gtanhéi {(—1)"£;‘1(2l + 1) §,+1(h,p)J21+1(h cosh 5)}

(:0

34

23$.)

 

B.,, i{(— 1)‘ ”T‘Biiiiwp)

:J[ <

(plzzo
he ‘Meéh e)— Ji+i(— adhefiM 9130]} ifpz'sodd,

(3.12)

M

[OH-I;

He, (h coshfi) _—_ EZH—l B§,( (,h q)J21(hcosh£)}
l=0

 

(12? 00 i iseven
= 33(h,q)§,i( 1) B2102 q)Ji( ”M01311 an} fq ,

(3.13)

m
M8

H
0

Je (h, cosh 6) =

q

{(—1)l—1;—IB§t+1(ha Q)J21+1(h @3110}
z

3%

 

= 3.0,,an 1)’ B3302 q)
1

1 . .
[Jz(-h€-£)Jz+1(§he£) + Jt+1(§he—£)Jz(§h€£)]} 1f q 13 Odd ,

NH

(3.14)

Nop(h, cosh g) = g tanhgl: {(—1)‘-‘SZ (21)B§,(h, p)N2,(h cosh .9}

 

fl i{(‘1)l—;B§z(hil’)

_ B§(h1p)z 1
1 _ 1 E 1 _£ l 6 . .
[J,_1(—2-he £)Nz+1(§h‘9 )— J,+1(§he )N,_1(§he )]} 2f P 13 even i

(3.15)

Nop(h, cosh g) = gtanhfi :2 {(—1)’"P%l(2l + 1)B§,+1(h,p)N2z+1(h cosh §)}
:0

35

 

Wi{_1) — ;IB§I+1(h1p)

[JA-hep H)Ni+i(—he )— Ji+i(%he-‘)Ni(%he‘)]} if p is odd.

(3.16)

Neq( (h ,)cosh£ 2 £2“ — )1 §B§,( h, q)N2i(hcosh{)}
(=0

 

(E °° . .
= §B(‘ (h,qN l{he J 1‘he }i is even,
38“,, 200—112, )1 )i(- ) fq

(3.17)

Neq(h,cosh{) = \flgi {(— 1)’ LB§,+1(h, q)N21+1(hcosh£)}

 

= ‘5 -§:{<—1)'LB3.1<hq)

1 1 l . .
_ )N)+1(-2—he£) + J1+1(§he_£)Nl(—2-hei)]} if q is odd,

:-
m
m

(3.18)

p2112333°~- , q:0,1,2,... .

3.1 Membrane With a Strip of Length 2c Centered

at ($01y0)

U0(E, 77) is finite in R0, we assume that

M8

(10(5) Tl) : Agm(ca c[30)502"; (ho, COS Tl)‘]02m(h01COSh§)

m 1

36

CXD

+ Z [A2n+1(ci c700)15'02,.+1(h401COS n)J02n+1(h0’ COShé)

n20

+A§n(c, 900)Se,,, (ho, cos 77)Je2,. (ho, cosh 6)

+A:....<c,vo)s.....<ho,cosn)J.....<ho,coshs)] (3.19)

with coefficients Agm(c, 1,00), Agn+1(c, (p0), A3,,(c, (pa), and A§n+1(c, cpo) determined by
the boundary condition, eq(1.5), where ho = Koc.
{S

02(n+1)

(h,cos 17), So,n+,(h,c0517), Segn(h,cos 17), Se,n+,(h,cos 77)}:10 isacomplete

orthogonal set of functions and [13]

211' 00
j; 332(n+1)(h, c0577 2712] ((83, (,h 2( (n + 1)))22 2(n+1)(h), (3.20)
l=l
0 211532" “(h,cos 77) 5177 = “2(B§1+1(h12n + 1))2 = Mgn+l(h) a (3-21)
1:0
21r °°
/0 S; (h, cosn) di7— - 21r Z(-€1—l)( c,(h ,2n))2——M2‘n(h) , (3.22)
1:0
0 53. .11», cosn) d0 = 7r 83,002,212 + 1))? = Mam) , (323)
1:0
where
1 i l = 0
e, = f (3.24)
2 if I 75 0 .
[13] gives
Jop(h0, 1) : 0, p : 1,2,3, . .. . (3.25)

Thus, eqs(1.22),(1.23) give

V1(€,77) : i Agm(ca 900)502m(h01C057l) (J02m(h01COSh é) — 1%O—m_((};19:1T))N02m(hO)COShE))

171:]

37

 

co ,0 J .. (120,1)
+2 Ame,vo)so....(ho.cosn) (10....(ho,cosh£) — N:n+‘l(ho,,)N3...(ho,coshe))

11:0

J62" (ho, 1)

———Nc h, h
Ne3n(h011) 2n( 0 COS 6))

+1123, 10053.00, 11) (13.010, cosm—

~e JC 71 ho, 1
+£12.33,vo)se....<ho.co:))( 1.....<ho,coshs) — N2 “((110 1))N.....(ho.coshi))]
9271+] ’

 

02

m

 

°° J , h ,1
—Z [Aomzl 211+] (C? (100 )Nozn+ ((hZ,1):N02n+1(h07COSh€)SO2n+l(hO’COS")

n20 0271+]

#2n(h0’1)chn(h0,COSh£)Se2n(h0’COSn)

+148 c, ——
2n( $00) ezn(h0, 1)

Je .. h ,1
+Ai...i(c (00),, ”(01° 1))chn+i(hoiCOSh€)5e2n+i(hoiCOSTl)
N32n+1 0’

 

=21}; m(C $00)Soam(h01C0377)J02m(h01COSh€)
+ Z [Agn+l(c1 wolsozn+1(h01 COS n)J02n+1(h01COSh€)
1120

~ Jean h , 1
+A;n(C, 900)Sezn(h01cos ll) (chn(h0) COSh g) _ flNnn(h01COSh€))
82'; 7

~ J32n+1 h 3 1
+A§n+1(ca $00)Se2n+1(h0?cos 77) (J 8211+! (ho, COSh g) _ N ((h: 1)) Ne2n+1 (ho, COSh €))]
. 3271+] ’

 

_ Ac J-—,——Ne h , h 362 h ,
“ZIO[ 211(6) $00)Ne3n(h071) 2n( 0 COS f) n( 0 COS Tl)

38

Jain-H (ho, 1)
Ne2n+1 (ho, 1) Nun“ (ho, cosh {)Sc2n+1 (ho, cos 17)

 

+A§n+l(ca‘100)

(3.26)
[13] gives
w (Jeq(h,cosh§),Neq(h,cosh£)) = 1 , q = 0,1,2,... , (3.27)

where W(J¢q(h,cosh§),Neq(h,cosh{)) is the Wronskian of Jeq(h,cosh§) and
Neq(h, cosh 5). Thus, by eqs(2.16),(3.19),(3.22),(3.23),(3.25),(3.26),(3.27),

6V1(:c,y) 6Uo(:v,y)
£3 Uo($,y) (T + —6n—) d3

—2Ko [R0 U§(a:,y) dA

2" 6V
A U0(O,17) (M [gr-0+ 6110(6)") lgzo) d7)

F1:

 

0.5 65
—2Ko [R0 U3(€,n) dA

 

 

1 w ~e e .1327: h )1 c
= Z [A2n(cu900)A2n(0,¢0)-N—-(hi1—)M2n(h0)
-2Ko / U3(€,n) dA=o e2..( 0, >
R0

-c e Jean 10?“ ’1)
+A2n+1(C,‘P0)A2n+1(C’(’00)N + (h: 1)
¢2n+1 ,

 

Mama]

 

+ 1 i [(Aan)” (c,soo)1{;“"(h°’ 1) M§n(ho)

—2K0 [R0 U02(§,17) dA ”=0 m(ho, 1)

e 2 chn 1(h0’1) c
+ (AW) (C’¢O)Nc,;l(ho,1)M2"+1(h0)] .

 

(3.28)

figm(c, (p0) and Agn+1(c, cpo) do not affect F1, the simplest choice is to set them equal

to zero. [13] gives
Balm) = 0<h'2‘—P') , (3.29)

39

B§z+1(h,P) = 0(’l'”““") , (3-30)

 

 

33102.4) = 0(h'2”"') , (3.31)
B§z+1(h,9) = 0(h'2'H—q') ~ (332)
J62n+1(h71) __ 4n+2

Ne2n+1( , ) — 0(h ) , (3.33)

J (h 1) r C(53)?) n 2 0
film: 1) = J (3'34)

\ 0(h4") n # 0 .

#540,900) = 0(1) : A§n+1(c,900) = 0(1) - (3-35)

ho coshé is 0(1) on So. Thus, to correct the boundary condition on So to 0(fi) for
(U0 + V1), 113,,(c, 900) and 113,, +1(c, 900) must be at most 0(fi), the simplest choice
is to set them equal to zero. The other choices would only lead to a higher order

correction to the first order result F1, eq(3.28). Thus,

00 J32n(h0’ 1)

 

 

”(6: 77) = _ "2::0 A2n(ci WO)WNezn (ho, COSh €)Se2n(h0a COS 77)
J62“, h ,1
+A§n+1(C:<Po)Nezn+l((h:,1))Ne2n+1(hoaCOSh€)Se2n+1(h0aCOSTI)] . (3.36)
and
00 e 2 J82n(h0? 1) c e 2 Jean 1010, 1) c
F "Er—E) (A271) (Ca WO)WM2n(hO) + (A21H-1) (C,¢0)N62n:1(h011)M2n+1(h0)
l :

 

—2Ko [R0 U3“, 77) (114

40

w3n

The expansion formulas connecting the circular cylindrical wave functions with the

concentric elliptical ones [5] give

 

\/ °° B0 h
Jp(Kr)sin(p (6— (p0)) 2—87: 2 -p——( m) Som(h,cosn).]om(h,cosh§) ,
2 m=l M°( (h)
(3.38)
p: 1,2,3,...,
where p and m are both even or odd.
and
Jq(Kr)cos(q (6— (00)) [1:13:14 Se,.( (h cosn)Jen(h,cosh§) ,

(3.39)

q=0,l,2,...,

where q and n are both even or odd. Then,

0(eh00) _p_h__§EO;O) .
A3( =\/8 87r—li———B ’ v2 A 2 B 2
(3.40)
[13] gives
h2 7h4
Be h 0 1 —- —-— ~- 3.41
0( 2 )1: + 8 + 512 + i ( )
1 2 13 4 )
e = _ _ , .42
M0(h) «(2+ 2h + 12812 + (3 )
(B5(h,0))2 1 2
= —— h+ 3.43
mqw) 22+” ( )
J6 (h,1) 7r 1 7r 1
° =—— —l4— —1K —— 3.44
Neo(h,1) 21nc+2(n 7 n 0)(lnc)2+ ( )

 

m C J32n(h’ 1) e e 2 Jen 1(h11 e
Z [(14232 (C,<Po)1—\f——O—M2n(ho) + (A2,...) 3.100),,“ + (h:,1))M2n+1(ho)

 

 

 

 

 

F _ nzo e2n(h011) 62n+1
1 — 2
—2Ko / wen) dA
R0
e 2 J€0(h0’1) e
(we) (c. w) NM, 1,Mowo)
: + . . .
—2Ko / U§(€,n) dA
R0
3(h B (h 0) 2
{V8 7rBoB —('—)—A(:,(630,:))+ +\/—Z[M 2p( «2:» (A2p3in(2P‘PO)+B2PCOS(2P‘»00)):|}
M0
— —2K U2 , dA
of]?0 0(5 77)
Je(h01)
o i e h
Neo(h0)1)M0( 0)+
«BS 1 7rB§(ano+'y—ln4) 1
2 Ko/ (”(5 n)dA'1nC' + Ko/ Ugo: n)dA |lnc|? "
R0 0 ’ R0 0 ’
where 7 z 0.5772. (3.45)

Eq(2.23) yields

°° '(N) UN,j(£177) f Mil/1(5)”) d3
30

W , 2 EU a
1(E 77) 0(g 77) + 1;”; (KIZV _ K3)llUN,jl|2 an

(3.43)

where E is a constant and

co

UN,j( £1 77) :21‘42 (C) (1003 N?j)SO2m(hN)COSn)JO2m(hN1COShé)

+ Z [Agn+1(629002 N1j)SO2n+1(hN’COS n)J02n+l (hN’ C081] 6)

n=0

+A;n(C, ‘pO: Naj)Sczn(hN1COS 77)J¢2n(hN1COSh€)

42

+Agn+1(ci9901ij)Sezn+1(hNiCOS 77)J82n+1(hN’ 008116)]

(3.47)

with coefficients Agm(c, cpo, N,j), A§n+1(c, goo, N,j), A§n(c,<po, N,j), and
A§n+1(c, cpo, N, j) determined by the boundary condition, eq(1.5), where hN = K Nc.
The expansion formulas connecting the circular cylindrical wave functions with the

concentric elliptical ones [5] give

Seq(h, cos n)Neq(h, cosh 6) = g i Bf,(h, q)Yn(Kr) cos (n(6 — (00)) , (3.48)
n=0

q=O,1,2,...,

Where q and n are both even or odd.

Thus, by eqs(3.36),(3.40),(3.41),(3.42),(3.44),(3.48),

aUNJ<€a 77)
{So ———an V1(€,n) ds

°° e___,,( (1)110,
: _\[§Z{A; MC Manolo: 3: B2 "(h0:ZQ)

q=0

BUNJ-(r, 6)

d
an 3

£0 Y2n(K0r) cos (2n(0 — apo))

CD
Jezq+1 (110,1)

(h 2 1
chq+1(h0’1)nZoB 2n+1( 0, (1+ )

 

+A2q+l (C) 900)

go Y2n+1(Ko7‘)COS ((2n +1>(6 — (00)) W ds)

BUNJ-(r, 0)

8n d3 +

£0 Y2n(K07‘) (2n(6 — We»

43

= [(9% (EWWW is. 2355914545191 d8] 3— + ° '-

|lnc|

 

_ 7rBo 0UN,J~(1‘,6) 1
_ (T £0 Y0(KOT)—an— d3 [111C] + . (3.49)

To correct the boundary condition on SE to 0(lnl cl) for (U0 + V1 + W1), the constant

E must be zero. Thus,

00 ((N) 2U 6U ‘
‘2 2(K N,j (5 0) £0 Mmen) d3 . (3.50)

 

1H j_1_K0)l|UN.jH2 0n
Let
V2(:c,y) : V2i($,y) + V2h($,y) 2 (3'51)
where
AV§(x,y) + KgV,‘(a:,y) = —2K0F1V1(2:,y) in R , (3.52)
AV2h(a:,y) + K02V2h(:c,y) = 0 in R , (3.53)
Vz"(1v,y) = —W1(1',y) - Vz‘($,y) 0n 53 - (3-54)

The method of variation of parameters [3] gives

V‘ , = 2K F Aen c, A—Se n h ,cos
2(5 77) o 1§{ 2 ( $00)N¢,n(ho,1) 2 ( o 77)

lF;n(€)Nezn(hOa C031] 6) _ ;n(§)Jezn(hOv COSh (H

J

e2n+l

N

e2n+l

ho l
((h: 1))S¢,n+,(h0,cos17)

 

+A§n+1(c, s00)

[F;n+l(€)N€2n+1(h0!COShé) — ng+l(€)J32n+l (ho, COSh £)]} 1

(3.55)

44

where

E
Fm) = / J..(ho.cosh u)Ne,(ho,Cosh/1) cm, (3.56)

6 £ 2
0.15): f Neq<ho,cosh 1») d1 , (3.57)

q=0,1,2,....

Alternatively, eqs(2.33),(3.36),(3.48) give

V21( (g ,7) __\/—:F1rZ{A 2,, (,c (p0)N ———((——:ho 11)): B; ”(120,2q)Y2',,(K01‘)cos(2n(6 — (p0))

J32q+ (ho, 1) m e
N 1 (ho 1) Z B2n+l(h’012q + 1)}f2’n+l(K0T)

€32tH-l n=0

 

+qu+1(C,CP 0)

COS ((2n + 1)(9 - Won} ~

(3.58)

Eqs(3.20),(3.21),(3.22),(3.23),(3.53),(3.54),(3.55) yield

V2"(§, 17) = :12 )So,m(h0,cos17)Jo,m(h0,cosh§)

8

+ Z [C2011+1(C1$00)Sozn+1(h0$cos 77)J°2fl+1(h0’ COShg)

n=0

Je,,. h ,
+CZn(c’ $00) SCQn(h0)COS 77) (Je2n(h01COSh£)-_ WNe2n(h01COSh€))

n

 

" I]: n l h ,1
+C§n+1(ca$00)Sezn+1(h0100577)(Jezn+1(ho:<303h§)— N” + (0,: 1))Nezn+1(ho:COSh€))]
3273+] 1

+ "12:10; 1n()c 900)( )(h01COSh6) Sozm(h01COS77)

45

00

+ Z [C2011+1(C’(p0)N02n+1(h01COSh€)Sozn+1(h01 COS 7))

n20

+C§n(cv 4p0)1\'f€2n(h01COSh£)S'e:2n(h01COS 77)

+C§n+1(cv 900)N62n+1(h01COSh€)Sezn+1(h01COS 77)] 1

 

 

 

 

 

(3.59)
Where
211'
— f0 W1(0,n)502m(h01605") d" (3 60)
0° , = ’ .
2m(c $00) M2°m(h0)N02m(h03 1)
21r
—/(; Wl(01n)303n+1(h0’cosn) d?) (3 61)
0° , = ’ .
2n+1(c (p0) M§n+1(h0)N02n+l(ho’ 1)
21r
.. f0 W1(0,n)Se,.(ho,cosn) d"
C" , =
2n(c 900) M§n(ho)Ne,n(ho, 1)
J3 h ,1 c e
2K0F1A§n(c1<p0) 2n( 0 ) [F2n(O)N¢2n(h0’ 1) — 2n(O)J¢2n(h0’ 1)]
Ne2n(h0’ 1)
_ Ne2n(h0’ 1) ,
(3.62)
21r
_-/(; Wl(0177)Sezn+1(h0’COS 77) d"
Ce , =
2n+1(C 900) M§n+1(ho)Ne2n+1(h011)

 

Je h ,1 e e
2K0F1A§n(ca‘p0) 2n+1( 0 ) [F2n+1(0)Nezn+1(h011) _ 2n+1(0)']32n+1(h’0’1)]

Nezn+1(h011)

Nezn+1(h011)
(3.63)

46

Eqs(3.7),(3.8),(3.9),(3.10),(3.47),(3.49),(3.50) give

21r
/(; W1(0»77)Se2n(h0:C0577) d7)

 

1(N) (76 WW“ 17)ds)

0° 50 an ’ 0° '

: A3 C, ,N, Jezq h ,1
NEW (K3 — K3>HUN.,-H2 [2 24‘ We 1) ( ~ )

(21ng(hN, 2q)B5(h0, 2n) + 71' Z B§,(hN, 2q)B§,(ho, 210)]
1:1

2 6n
(KN - Ko)||UN,,-||2

7rB 3U -r,6
0° 1(N)(— if; Y0(Kor )JJé—J

 

d
) [214240 C1‘p01N 1j)Jezq(hN11)

(2633mm 2q)BS(ho, 2n) + 7‘ 2 353W“ 2033““ 212))”

(=1

|lnc|

(3.65)

21r
A W1“): n)S€2n+1(h’01 COS 77) d7?

1(N) (£0 WWW) d8) [co

= A8 c, ,N,jJe,qlh,1
Z: (K3—K3>HUN.,-II2 Z3 23+“ “0" ) +(N ’

 

(771:3 21+1(hN129 + 1) §I+1(h012" + 1))]

1:0

 

00 ((N) (T 5' Y0(K07’)_ é'n 1 - d3) 00
= 0 Ac (C190 aNaj)J€2q+l(hN’ 1)
Z Z (K?v — K8)I|U~.jl|2 [2 °

«1:0

47

(”2851+“th 2‘1 +1)B§l+1(h012n +1))]}

(:0

 

 

1
|ln cl ’
(3.66)
21r
/(; W1(0177)Sozm(h01 C0577) d0 : 0 1 (367)
21r
f0 W1(0a’7)Sozn+1(h0,COS77) dn = 0 . (3.68)
Thus, by eqs(3.25),(3.27),
6UO(€277) h 8V2h(§,77)
£3 (Tl/2 (£177)_ 7110(5)”) d5
__ 21' 8110“,") h 6V2h(£,77)
— [0 (‘T 6:016 (0,11)+ T Igzo Uo(o,n) dn
0° _1 e ”e e
- "2:0 [Em/127(0) 900)Czn(c, W0)M2n(h0)
—1 e ”e e
+ Ne2n+1(h01 1)A2n+l(c) c100)C'2n-+-1(C7(100)1‘4211-1-1(ho)]
+ Z [A546 ammo, (comma) + 113,316, (PO)C§n+1(C, ¢O>M5n+1<ho)]
1120
(3.69)

é§n(c,<po) and é§n+1(c,<po) do not affect the second order result F2, eq(2.43), the

simplest choice is to set them equal to zero.

No (h,1) =0(h“'°) , p: 1,2,3,...,

p

48

N

e2q+l

(h, 1) = 0(1)-2H), q = 0,1,2,... ,

Ne,q(h, 1) = . (3.70)

 

 

_ 4n 2
Ne2n+1(h,1) _ 0(h’ + ) 1 (3'71)
r 0(fi) n = 0
Je,n(h, 1) _
N.2.(h,1) ’ i (3'72)
001‘“) n # 0 .
F;(0) = o + - -. , (3.73)
03(0) 2 0 + « -- , (3.74)
q=0,1,2,....

ho coshfi is 0(1) on So. Thus, to correct the boundary condition on So to 0(7—15167—2)
for (U0 + V1 + W1 + V2), 03,,(0, (p0) and C~'§n+1(c,<po) must be at most 0(lTn1?|7)1 the
simplest choice is to set them equal to zero. The other choices would only lead to a

higher order correction to the second order result F2, eq(2.43). Thus,

00

 

%h(€1n) : Z l 2811(61 cp0)1V82n(h01 COSh E)Sezn (h01 COS 77)
71:0
+C§n+1(c1$00)N62n+1(h01COSh€)Sezn+1(h01COS77)] 1 (375)
where
211'
_/ Wl(0177)5e2n(h0100577) d"
511(61 900) = 0 e
M2n(h0)Nean(h01 1)

49

Je2n(h011)

2K0F1Ac2n(c1900)m
can ,

lF;n(O)Nezn (h01 1) — ¢2311(0)J¢3n(h01 1)]
Nezn(h011) ,

(3.76)

 

211'
_A W1(0177)Sezn+1(h01cos 77) d7?

 

 

 

Ce" 0, =
2 +1‘ “0“) Mgn+l<ho)N.,.+.(ho, 1)
c Je2n 1 h ’1 e e
2K0F1A2n(C1900)N + ((h" 1)) [F2,.+1(O)Ne,...(ho,1) — azalmwehswo, 1)]
_ 3271+] 0’
N€2n+1(h0’ 1)
(3.77)
And
6U0(€177) h 8%h(5177)
7; ( an v2 (6,17) an wan) ds
: Z [A;n(c1900)C§n(c1900)M26n(h0) + A§n+l(c1 $00)C§n+1(c1 900)M2en+1(h0)] '
n=0
(3.78)
[13] gives
8 h2 7h4
Bo(h,0)—1+-§-+'5E+”', (3.79)
1 13
e : 2 _ 2 __ 4 _ . _ ) .
M0(h) 7r( +211 + 123" + , (3 80)
7r
J.,(h, 1) = (E + . -- , (3.31)
and
2
Ne0(h, 1) = (/:lnh + ' -- . (3.82)
11

50

[$291010)

A3(c,(p0)= flBoo BO(h0’0) +\/‘ZB (ho)

M0(h0)

(2
: —B0+...
7T

(A212 3511919900) + 32p COS(2P¢0))

 

 

 

(3.83)
Thus, by eqs(3.65),(3.76),(3.78),(3.79),(3.81),(3.82),(3.83),
h
f (——6U§€’U)W(€m)— ————6V28(€’77)Uo(€,n)) d8
53 n n
= (13(6) 900)Cé(c, (06)M8(ho) +
2 °° “1‘” BO(N, j) BUM-(136) 1
:{B" 2:2 (KN— K311IU~1112£Y°(K°T) 6n d3} |lnc|? +
(3.34)

Eqs(3.19),(3.45),(3.58),(3.79),(3.81),(3.82),(3.83) give

an(€,T]) 1'
£0 TVMEW) d3

71' 00 e Je2q(h0, 1)
_\/——2:F1 (120 {A2q(C1 (100) NegquO, 1)
3U ,0
[2( B; "(hm 2g) )fo rYz’n(Kor) cos (211(6 — (00)) —%%——) d3]

Jezq+1(h011)
N (ho, 1)

¢2q+l

 

+A§q+l (C1 ‘100)

[f3 351.1010, 2g + 1) 7550 rYz'nsmor) ((2n + 11(6 — (001199;? dsl}

7120

51

 

:_\/: 1rB0 If“ fryo' (K0? )3U0(r,0) ds+~~
ZOROKf Ug(gn)dA|1nC| 035—1116 80 8n

8U0(7‘, 0)

 

2 3 1
7r B0 jammy) an ds 1
z 2 +-.. . (3.85)
2K6 [R0 (13(66) dA |lnc|

Thus, by eqs(2.43),(3.45),(3.84),(3.85),

F12
—2K0

f ——BU0(T’0)V;(1',0) ds f (-———6UO(T’6) V2h(r,9) — ——6V2 (130) Uo(’r,9)) d3
50 SB

 

F2 2

 

 

an an an
2K0 [R0 Ug(r, 6) dA 2K0 [R0 U30; 0) dA
, 3U (r,0)
— W233 + 71'ng £0 TYO(K0r)——(:971— d8
— 2

 

 

—4K3 ([120 [162(5177) (M)?

~2K3 ([30 (13(66) dA)

 

 

 

 

°° '(N’ B N ' BU . 0
7r2 0( 1.7) f' Y Kr N,J(r1 )ds
+ Bg7r1vz=1321m1v “Ko)||UNj||2 So 0( 0) 6n 1 +...
2K0 / Ug(r,6) dA |lnc|2
R0
(3.86)
Thus, eqs(1.19),(3.45),(3.86) give
K 2 K0 + F1 + F2 +
K + 1182 1+1r(ano+7—ln4)Bg
= 0
K0 / (133(1) dA |lnc| Ko/Ro (13(66) dA
0U (r,0)
77283 _ “233 £0 7'Y0,(K07’)—051‘1—‘ d

 

 

— 2K3 ([30 113(5)”) cw)? 4K3 (fRo (13(66) dA)2

52

 

 

°° “1‘” WV 1') 6U -(r 0) ‘
827‘. 2 f Y K 7‘ ———N"7 , (15
+ IEIEKN BOKO)”UN,jll2 So 0( 0 ) 8n 1
2K0] U02(1~, 0) dA |lnc|2
R0 1

 

where '7 x 0.5772 .

- , (3.87)

3.2 Circular Membrane With a Centered Strip of

Length 2c

The geometry of the concerned region is with the outer boundary where 50 is r =

1" = 1 and the inner boundary where SB is 5 = 8 = O.

Eqs(2.51),(2.111),(3.87) give

1 1
K = K
° + (mew) |lnc|

 

 

 

(ano+’y—ln4) + 1 _ 7rY1(Ko) 1
K0J12(Ko) 2K3Ji(Ko) 2K0Ji3(Ko) [11142

where K0 2 K0,] % 2.4048 and 7 x 0.5772.

53

+...

, (3.88)

CHAPTER 4

Conclusions

The main contribution of the dissertation is two-fold. First, from the computational
point of view, a general formula to the asymptotic approximations of the fundamental
frequencies K of membranes with an internal core of maximum dimension 2c, 6 << 1,
is derived. It is convergent in asymptotic sense. Moreover, the first three order terms
of the asymptotic approximations are carried out explicitly. Second, from the point
of view of the inverse problem, relations between the first three order terms of the
asymptotic approximations and geometric properties of the regions are investigated
from the explicit formula. These are summarized in Theorem 4.1 and Corollary 4.2

respectively.

Theorem 4.1 A general formula to the asymptotic approximations of the funda—
mental frequencies K of membranes with an internal core of maximum dimension 20,
c << 1, is formed as in eqs(1.19),(1.37),(1.38),(1.39). Moreover, the asymptotic ex-
pansion of K for membrane with an internal core of maximum dimension 2c, derived

from eq(2.51), eq(3.87) and the minimax principle [8], is

K K
K=K0+——1+——2—+---, (4.1)
|lnc| |lnc|2

54

where K0 is the fundamental frequency of the membrane without an internal core,

2
7rB0

_ KO/ Ug(r,6) dA
R0

 

K1

1

and

77(ln K0 + '7 — 1n 4)B8
K0 / Ugo, 6) dA
R0

 

 

733ng er’aKor) d3

 

 

 

 

 

W233 071
_ 2 _ 2
R0 Ro
0° KN) "2323(N' a . 6 1
o o 1.7) f UN,J(r, )Y K
_ 1‘ ds
+"’Z=‘JZ=:1(K12V—K3)IIU~6||2 s. 6.. 0< 0)
”Of 113030) dA
R0
S K2 S
7r(111 Ko + 7 —1n2)Bg
K/ U2 ,6 dA
0 R0 0(7‘ )
5U r,0 1
/1r2Bg 71-233 £0 —(()9—(Ti—)TYO(KOT) d3

 

 

2K3 (fa. U36; 6) 6A)2 4K3 ([110 U30; 6) am)2

i "N’ «2838mm 7 6116,16, 6)
N=1j=1 (K12)! " K8)||U~3||2 So an

2K] 02136 dA
0 R0 o( )

Y0(Kor) ds

 

+

 

 

(4.2)

(4.3)

Corollary 4.2 Eqs(4.1),(4.2),(4.3) show that the geometry of internal core starts to

affect K at K2 while the position of internal core starting to affect K at K1 and the

geometry of membrane without an internal core starting to affect K at Ko.

55

The asymptotic expansion formula of K obtained by the modified perturbation

method is highly accurate and is valid for small c. This is summarized in Lemma 4.3.

Lemma 4.3 (1a) Eqs(2.111),(C.13) show that the first three order terms in the
asymptotic expansion of K, the fundamental frequency of the annular circular mem-
brane, obtained by the modified perturbation method agree with those in the exact
series.

(1b) Eqs(3.88),(C.18) show that the first three order terms in the asymptotic
expansion of K, the fundamental frequency of the circular membrane with a centered
strip, obtained by the modified perturbation method agree with those in the exact
series of K,, the fundamental frequency of the elliptic membrane with an internal
confocal strip, i.e. K0 % 2.4048, K1 % 1.5429, K2 % 0.1208.

(1c) The asymptotic expansion [9] of Km, the fundamental frequency of the circu-
lar membrane with a centered strip, computed by the eigenfunction matching method

is of the form

K1 K2

Knu:K _ —— ”'1
0+|lnc| |lnc|2

(4.4)

where K0 :2 2.4048, K, = 1.55, and K2 2 —0.012 are computed by a least squares fit
on our numerical results for the range c = 10‘2 to c 2 10’6. (Numerical instability
occurs for c < 10-6).

(2) The comparison between the first three order terms in the asymptotic expan-
sion of K (eq(2.111)), the fundamental frequency of the annular circular membrane,
obtained by the modified perturbation method and the exact solution is shown in
Figure 4.1. The error is less than 1 % as c is less than 0.25 and less than 5 % as c is

less than 0.4.

56

Figure 4.1: The comparison of the asymptotic approximation and the exact solution

 

8 , 1
-: the exact solution of K solved from Eq(C.1)

7 c «z the asymptotic approximation of K, Eq(2.111), ,’ -

obtained by the modified perturbation method ,/

 

 

 

 

1/||n cl

57

Propositon 4.4 Eqs(2.77),(2.111) show that

 

= 116%) , (4.5)

A

00 K0,n
112:; (K3,. — K02) J1(K0,n)

where K0 2 KOJ % 2.4048 and K04, is the 1)“ zero of Jo, p = 1, 2, 3, . . ..

remark: The series in eq(4.5) converge slowly.

The future research related to the dissertation would he studies in the convergence
of the modified perturbation method, extensions of the method to higher frequencies,

and extensions of the method to membranes with arbitrary many internal cores.

58

APPENDICES

59

APPENDIX A

The Pin-point Phenomenon of
Simply Connected

Membranes

Proposition A.1
‘1ng 2 K0 . (A.1)

Proof:
First, consider the case where SE is the circle of radius c centered at (110,310) .

Green’s 2"" identity [4, 11] gives

fR(U(z.y) A Wm. 11 — Uo($:y) A U(m.y)) em

= 5.65. (may/15311934!) - U,(z,y)29;_:112) d, , (1.2)

where the normal derivative on So is with respect to the outer normal direction

and the normal derivative on SE is with respect to the inner normal direction.

Eqs(1.2),(1.3),(1.4),(1.5),(A.2) give
(K’ - K3) f3 U(z.y)U.(z.y) em = -— fs. new)??? ds . (1.3)

60

Let zero“ order approximation of U (x, y) be U(x, 3;). Consider U(x, y) on the annular
circle with the inner boundary, S B, and the outer boundary, 1‘ = 0(c), then U(x, y)

is independent on 6’, so

= 0 . (A.4)

 

 

Thus,
Ma=a+amm mm

where Cl and 02 are constants.

where 01 and 02 are constants.

The boundary condition , 0(0) 2 0, yields

—C'1

 

 

 

Cg = , C1 = 0(1) . (A.6)
lnc
Thus,
8061.6) _ 21* 606,6) _ c, [21:
‘fimmm &Ldy_fl;mmmé% W”“_mco%@mw’
(A.7)
so eq(A.3) becomes
(KK—Kh/(flaynM=ffhfhma6yw+nmn (an
0 no 0 ’ Inc 0 ’
The proposition is proved by the minimax principle [8] and eq(A.8). C]

61

APPENDIX B

The Generalized Green’s Functions

A generalized Green’s function [4, 12] is needed for problems of the type

AWOW) + K3W(x.y) = -f($,y) in R0 . (B-l)

W636) = h($.y) on So . (13-2)

where K3 is an eigenvalue of Eqs(1.4),(1.5).
For the case where the eigenvalue K 3 is simple, the generalized Green’s function

G(x,y;fé,g) is a solution [4, 12] of

Uo($,y)Uo(5,‘g) i

n R0 .
lonll2

(13.3)

 

A(:1:,y)G(:r1yi£5137)+ KgG($,y,5I,g) : —6($1y;i',37)+

G(x,y;x,fi) = O on So , (B.4)

where U0 is an eigenfunction corresponding to the eigenvalue Kg .
The generalized Green’s function G (x,y; 51,37) can be expressed as an expansion in

eigenfunctions with terms associated with the eigenvalue K3 omitted; [4, 12]

 

0°“MU x )U (x, )
C(x, ;x, N’j( y N‘j y , B.5
y y): Efém- K0111! .12 ( )

62

where K1211 , l(N), UM,- are the eigenvalues to eqs(1.4),(1.5), the multiplicities of
K12,“ and the corresponding orthogonal eigenfunctions to K3,, respectively, N =
1,2,3,4,....

An alternative [4, 12] is to derive more compact series expressions directly from the

differential equation. The solution G(x,y;x,;i]) to eqs(B.3),(B.4) can be obtained

from the usual Green’s function C(x, y; 5:, g) by using

,..-_. 6 -2 2-
06.6.4.6) — 1116.613?) [(K — K.)G(x.y.x,y>] , (8.6)
A(.,y)@(w, wit?) + Ifzéhwflfl) = -5(w,y;:fi.3}) in R0 , (13-7)
C(x,y;x,g) = 0 on SO , (38)

where K2 is not an eigenvalue to eqs(1.4),(1.5).

63

APPENDIX C

The Exact Series Expansion of K

C.1 Annular Circular Membrane With Outer Ra-

dius 1 and Inner Radius c
The characteristic equation [14] is
Yo(K)Jo(KC) - Jo(K)Yo(KC) = 0 -
We assume that [2]
K = K0 + Kl-lfi + K26(c) + 0(6(c)) ,

where

K0 = KOJ x 2.4048 : the first zero of Jo ,

 

_ [m(Ko) . 6(a) _
1 _ 2 J1(Ko) ’ and c1310 1 — 0'
|lnc|

Addition theorems for bessel functions [10] give

Yo(K> = Y0(K0) — K1Y1(Ko)7fi;l — Y1(K0)K26(c) + . ..

64

(0.1)

(0.2)

(0.3)

K2 2
J0(KC) = 1 "' OC

 

+~~ , (0.4)

1 KfJ2(K0) 1

=—K K
MK) 1J1(0)]lnc|+ 4 |lnc|2

 

— K2J1(K0)6(C) + ‘ ’ ' , (0.5)

 

2 2
Y0(Kc)=—lnc+-(ano+7—ln2)+--- , (C6)

71' 71'

where 7 x 0.5772 . Eq(C.1) becomes

[Yuma — Kit/l(Koilfil — Y1(K61K26(c) + - . ]

1 K§J2(Ko) 1
|ln c] 4 |lnc|2

 

 

— [—K1J1(K0) — K2J1(Ko)6(C) + ' ' {l

[glnc+§(ano+7—ln2)+'~]:0. (C-7)

Balancing the leading orders, one finds

 

 

 

1
(5(0) : W (C.8)
and
K2] K
—£K1Y1(Ko) + M +(1nKo + '7 — 1n 2) K1J1(Ko)
K2 2 2 4 . (C.9)
J1(Ko)
Eqs(C.2),(C.9) give
7rYo(Ko) 1
= K —
K 0 + 2 J1(Ko) |lnc|
K2J K
—%K1Y1(Ko) + —1—24-(—‘12 +(1nKo + 7 — 1n 2) K1J1(Ko) 1
+ 7.07.) (1.62 +"' '
(0.10)

65

 

 

 

7r 1
_y K : —__. CH
2 0( 0) KOJ1(K0) ( )
2
E‘J1(K0) = J2(K0) . (0.12)
0
Thus,
1 1
K = K
0 + (KOJ3(K0)) |lnc|
(1n K0 + 7 — ln 2) + 1 1rY1(K0) 1 +
KoJ12(Ko) “(3510(0) 2Ko-Ji3(Ko) llncl2 ,

(013)

where K0 2 Km z 2.4048 and '7 3 0.5772.

C.2 Elliptic Membrane of Area 7r With an Internal

Confocal Strip of Length 20

The characteristic equation [13] is
Neo(h, 1)Jeo(h, cosh c) — Jeo(h,1)Neo(h,coshc) = 0 , (C.14)

where h = Kc. The asymptotic expansion [9] of K is
l 1

 

K : K K —_ —— .15
0+ 1|lnc|+K2|lnc|2+ ’ (C )
where
K0 2 K0,] 2 2.4048 : the first zero of Jo ,
WY0(K0)
K = —— R1 1.5429
1 2 11m) ’
K2J K
Jim/1m) + ——(—°’ + J1(Ko)K1(1n K0 — 1n4 + 7)
K2 = 2 4 x 0.1208 .

 

J1(Ko)

66

Eqs(C.11),(C.12) give

 

 

 

 

 

 

7TY0(K0) 1
K=— z—z1.5429,
1 2 J1(K0) K0J12(K0)
K2J K
—%K1Y1(K0) + —1——:—(—°) + J1(K0)K1(1nKo —- ln4 + 7)
K :
2 J1(K0)
_ (ano+'y—ln4) + 1 _ 7rY1(K0)
KoJflKo) 2K3Ji1(Ko) ZKOJf(Ko)
% 0.1208
Thus,
1 1
= K
K 0 + (KoJf(Ko)) |lnc|
(1mm, +7 - 1114) + 1 _ m(Ko) 1
K0J12(Ko) 2K3Jf(K0) 2K0J?(Ko) IIDCP

where K0 2 KOJ x 2.4048 and 7 x 0.5772.

67

(0.16)

(0.17)

(0.18)

,

BIBLIOGRAPHY

68

BIBLIOGRAPHY

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Sound and Vibration, 2_1§:195—199, 1998.

[3] E.A.CODDINGTON. An Introduction to Ordinary Differential Equations.
Prentice-Hall, N.J., 1961.

~[4] G.F.ROACH. Green’s Functions. Cambridge University, New York, 1982.

[5] J .A.ROUMELIOTIS and S.P.SAVAIDIS. Cutoff Frequencies of Eccentric
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[6] J .A.STRATTON. Electromagnetic Theory. McGraw-Hill, New York, 1941.

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({8} J .R.KUTTLER and V.G.SIGILLITO. Eigenvalues of the Laplacian in Two Di-
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[9] L.H.YU and C.Y.WANG. Fundamental Frequency of a Circular Membrane with
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[10] M.ABRAMOWITZ and I.A.STEGUN. Handbook of Mathematical Functions
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[11] O.D.KELLOGG. Foundations of Potential Theory. Dover, New York, 1953.

[12] P.M.MORSE and H.FESHBACH. Methods of Theoretical Physics, volume 1.
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69

[14] LORD RAYLEIGH. The Theory of Sound, volume 1. Dover, New York, 1945.

70