7.2 Scattering on a Parallel Cylinders
133
and satisfy the wave equations:
ΔU + n
2
o k
2 U = 0,
ΔV + n
2
o k
2 V = 0.
We write the relations for longitudinal, azimuthal and radial component of the electric and magnetic fields which expressed through U , V in cylindrical coordinates
(ρ, ϕ, z):
E ρ =
∂
2 U
∂ρ∂z
, E ϕ =
1
ρ
∂
2 U
∂ϕ∂z
, E z =
∂
2 U
∂z 2 + k
2 n
2
o U,
(7.3)
H ρ = −
n o
ρ
∂U
∂ϕ
, H ϕ = n o
∂U
∂ρ
, H z = 0
( 7 . 4 )
H ρ =
∂
2 V
∂ρ∂z
, H ϕ =
1
ρ
∂
2 V
∂ϕ∂z
, H z =
∂
2 V
∂z 2 + k
2 n
2
o V,
(7.5)
E ρ = −
n o
ρ
∂ V
∂ϕ
, E ϕ = n o
∂ V
∂ρ
, E z = 0,
(7.6)
where U , V is
1
ρ
∂
∂ρ
ρ
∂U
∂ρ
+
1
ρ
∂
2 U
∂ϕ 2
+
∂
2 U
∂z 2 + k
2 n
2
o U = 0,
1
ρ
∂
∂ρ
ρ
∂ V
∂ρ
+
1
ρ
∂
2 V
∂ϕ 2
+
∂
2 V
∂z 2 + k
2 n
2
o V = 0.
The field incident on the surface relative to jth cylinder consists of several parts:
the initial incident waves, the primary wave scattered by the jth cylinder and wave
scattered on all other cylinders. Then we can write general expression for the scalar
field potential functions U , V :
U
j
= U
j
inc (R j p ) + U
j
scat (R j p ) +
N
k = j
U
k
scat (R kp ),
(7.7)
V
j
= V
j
inc (R j p ) + V
j
scat (R j p ) +
N
k = j
V
k
scat (R kp ),
(7.8)
where U
j
inc (R j p ),V
j
inc (R j p ) are potential incident field on the surface of the jth
cylinder, U
j
scat (R j p ), V
j
scat (R j p ) are potential of primary scattered wave at j-th cylinder, U
k
scat (R kp ), V
k
scat (R kp ) are potential of the scattered field at all other cylinders.