Cavity with one side permeable

-------Zhonghai Liu

Hertz Potential

For a rectangular cavity, it’s convient to single out one particular direction, say ,and choose the Hertz potentials as:

Then: Correspondingly:

3e

),(),( meteA

), 33 ee me

3021123 , eeeA

322

12

2013210231 )()()( eeeAE t

322

12

2013213102 )()()( eeeAB

Cavity with one side permeable

For conducting surface: For permeable surface: Without loss of generality, assume the permeable

side is on z=c

0;0 BE

0;0 BE

czyx

zz

byy

axx

BB

B

B

B

0

0

0

0

0

,0

,0

czz

zyx

byzx

axzy

E

EE

EE

EE

0

0

0

0

0

,0

,0

Finally we get the boundary conditions for

So

tit

z

y

x

lmne

zc

n

yb

m

xa

l

)2/1(cos

sin

sin

czzzz

byy

byy

axx

axx

00

00

00

0

'

,0

''

,0

,0

''

,0

,

czz

zz

byy

axx

00

0

0

'

0

,0

'

,0

'

tit

z

y

x

lmne

zc

n

yb

m

xa

l

)2/1(sin

cos

cos

)2/()()2/1(

cossinsin

)(

,,

*

,,

lmnlmnti

lmnti

lmnnml

lmnlmnlmnlmnnml

keaeazc

ny

b

mx

a

l

aa

lmnlmn

)2/()()2/1(

sincoscos

)(

,,

*

,,

lmnlmnti

lmnti

lmnnml

lmnlmnlmnlmnnml

kebebzc

ny

b

mx

a

l

aa

lmnlmn

tilmnlmn

lmnzec

ny

b

mx

a

lN

)2/1(

sincoscos

tilmnlmn

lmnzec

ny

b

mx

a

lD

)2/1(

cossinsin

lmn lmn

lmn lmn

zkzkykykxkxkkk

zkzkykykxkxkkkBB

]coscossinsincoscos)[(2

1

]coscoscoscossinsin)[(2

1,

'33

'22

'11

23

21

'33

'22

'11

23

22

lmn lmn

zkzkykykxkxkkkBB ]sinsincoscoscoscos)[(2

1, '

33'

22'

1122

2133

lmn lmn

zkzkykykxkxkkkEE ]coscossinsinsinsin)[(2

1, '

33'

22'

1122

2133

lmn lmn

lmn lmn

zkzkykykxkxkkk

zkzkykykxkxkkkEE

]sinsincoscossinsin)[(2

1

]sinsinsinsincoscos)[(2

1,

'33

'22

'11

23

21

'33

'22

'11

23

22

)(8

1 2200 BET

)12

1)4,,,(

2)4,2,,(

2(

4

))1(1

)1,1,1,1()1,

2

1,1,1((

4

)1

)1,1,1,1()1,

2

1,1,1((

4

)22

2])()()[(])2()()[((

4

))2/1(

]))2/1(

()()[((4

1)1(

4

1),(

3232

33

133

11

2

12222

1222

2

1222

0000

000

ccbaZ

abccbaZ

abcabcccba

Zcba

Zabc

nccba

Zcba

Zabc

c

n

c

n

c

n

b

m

a

l

c

n

b

m

a

labc

c

n

c

n

b

m

a

lxxTdzdydxE

R

n

nnlmnlmn

nlmnlmnmllmn

cba

]2

cos2

cos))2/1(

())12(

cos2

cos)()12(

cos2

cos)([1

4

1),(

)(cos)(cos)(cos

)(1

4

1),(

222200

'33

'22

'11

32

22

122'00

321

zb

my

a

l

c

nz

c

ny

a

l

b

mz

c

ny

b

m

a

lxxT

zzkyykxxkVwhere

VkVkVkVkxxT

lmnlmn

lmn

lmn lmn

Another approach (Carlos Villarreal’s paper)

2'2'2

2'2

2'1

2 )())12(()2()2(

1

4

1321

ttzzcnyymbxxlaF

lmn

)(2),( 32

22

12

02'00 FFFFxxT

ccbaZcbaZ

abc

nncncmblancmbla

abc

nccnmbla

abc

cn

c

cnmbla

abcE

nnlmnlmn

nlmn

nlmn

32)]4,,,()4,2/,,([(

16

))2(

11(

4)

])()()[(

1

])2/()()[(

1(

16

)12(

1

4])2/)12(()()[(

1

16

])12[(

1

4]))12(()2()2[(

1

3

332

22222222222

222222

222222