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A x
formulation -
2 A 1x / x 2 +
2 A 1x / y 2 +
2 A 1x / z 2 + (
2 - i
(8.3.1a)
) A 1x = -
1 J x ,s
1 1
1 h 1
2 A 2x / x 2 +
2 A 2x / y 2 +
2 A 2x / z 2 + (
2 - i
(8.3.1b)
) A 2x = -
2 J x ,s
2 2
2 h 2
(8.3.1c)
A 1x = A 2x
1 -1
2 -1
1 -1
2 -1
A 2z / x
(8.3.1d)
A 1x / z -
A 2x / z =
A 1z / x -
0 as x 2 + y 2 + z 2
(8.3.1e)
A 1x and A 2x
A y
formulation -
2 A 1y / x 2 +
2 A 1y / y 2 +
2 A 1y / z 2 + (
2 - i
(8.3.2a)
) A 1y = -
1 J y ,s
1 1
1 h 1
2 A 2y / x 2 +
2 A 2y / y 2 +
2 A 2y / z 2 + (
2 - i
(8.3.2b)
) A 2y = -
2 J y ,s
2 2
2 h 2
(8.3.2c)
A 1y = A 2y
2 -1
1 -1
2 -1
1 -1
A 1z / y
(8.3.2d)
A 2y / z -
A 1y / z =
A 2z / y -
0 as x 2 + y 2 + z 2
(8.3.2e)
A 1y and A 2y
A z
formulation -
2 A 1z / x 2 +
2 A 1z / y 2 +
2 A 1z / z 2 + (
2 - i
(8.3.3a)
) A 1z
1 1
1 v 1
2 A 1z / z 2 +
2 A 1x / z x +
2 A 1y / z y}
= -
1 J z ,s - {(
v 1 -
h 1 )/(
h 1 + i
)}{
1
2 A 2z / x 2 +
2 A 2z / y 2 +
2 A 2z / z 2 + (
2 - i
(8.3.3b)
) A 2z
2 2
2 v 2
2 A 2z / z 2 +
2 A 2x / z x +
2 A 2y / z y}
= -
2 J z ,s - {(
v 2 -
h 2 )/(
h 2 + i
)}{
2
(8.3.3c)
(
1v + i
) A 1z
(
2v + i
) A 2z
1
2
(8.3.3d)
A 1z / z
A 2z / z
0 as x 2 + y 2 + z 2
(8.3.3e)
A 1z and A 2z
V formulation -
2 V 1 +
2 - i
(8.3.4a)
1 (
) V 1 = - i
(
+ A z+ -
- A z- )
(z - z bed )
1
1 1
1
1 h
2 V 2 +
2 - i
(z - z bed )
(8.3.4b)
2 (
) V 2 = - i
(
+ A z+ -
- A z- )
2
2 2
2
2 h
(8.3.4c)
V 1 = V 2
(8.3.4d)
V / z ] nonzero, consistently with an integral of Equations 8.34a,b
0 as x 2 + y 2 + z 2
(8.3.4e)
V 1 and V 2
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