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(48)

1

1

1

*

34

1131 1411 1111

10

1111

0

31

11

0

51

0

4

(1) 1

2

1,for therma lly insulated

2

,for isothermal

1

12

sk

k

Bk

nn

sk

k

n

k

sk

k

n

k

n

cc

EnncQs cskAsiDs

skD sP

E

Ds P

E

k

P

1

1

111

1

0

7111 1

01

,

5,6,7, 8,and7, 8

1

12

sk

Bk

n

k

il il

sk

k

Bk

nn

skQs APP

EEfori l

EskQsAPP

(49)

Here the elements

0

11 2,3,4,5,6

0

k

ll

EE l

for k

,1

of determinant Equations (47) and (49) can be obtained

by just replacing l

s

l

1,3,5,7

il

Ei

2,4,6, 8

il

Ei

1

in with

l, while are ob-

tained by replacing

, 2,3,4,5sl,6

in with 2

1, 3, 5, 7

il

Ei

.

The element il and in (48) can be

obtained by replacing Bessel’s function of first kind

,Ei5,6,7,88l

J

with that of second kind Y

and the elements

and l can be obtained by replacing 6, 8

il

Ei7, 81

in 5,7

il

Ei

and 7,8l

with 2

respectively.

The Equation (44) holds iff each term vanishes sepa-

rately. This implies that

0

00for =0, 0

il Xkn

E

(50)

0 for0,0

k

k

il Xkn

E

(51)

The Equations (50) and (51) have a non-trivial solu-

tion iff

00, ,1,2,3,4,5,6,7,8

il

Eil for k = 0, (52)

0,,1,2,3,4,5,6,7,8

k

il

Eil for (53) 0,0kn

After lengthy but straightforward simplifications and

reductions, the determinant Equations (52) and (53) lead

to the following secular equations.

77 8878870EE EEforn 1 (54)

2d

10

d

n

n

PmPfor n

1

where

(55)

0

det0,,1,2 ,3,4, 5, 60,0

ij

Eil kn (56)

det0,,1,2, 3, 4, 5, 60,0

k

ij

Eilkn (57)

1

1

1

1

034

113410 11

1011

0

31

011

0

51110 11

4

(1) 2

1,for thermally insulated

2

,for isothermal

1

12

s

s

s

s

B

cc

Ennc csAs

sDs

E

Ds

EskQsAs

(58)

1

1

1

771111

*

34

1131 14 11 1111

1111

31

11

51

3

2

4

(1) 1

2

1,for thermally insulated

2

,for isothermal

s

k

k

kk

sk

k

k

sk

k

k

B

EJJ

cc

EnncBs cskAsiDs

skD s

E

Ds

EQ

k

1

11 111

1

2

sk

k

sskAs