Page 526 - Electromagnetics

Separation of the Helmholtz equation
2
2
1 ∂ ∂ψ(ρ, φ, z) 1 ∂ ψ(ρ, φ, z) ∂ ψ(ρ, φ, z) 2
ρ + + + k ψ(ρ, φ, z) = 0 (D.68)
ρ ∂ρ ∂ρ ρ 2 ∂φ 2 ∂z 2
ψ(ρ, φ, z) = P(ρ) (φ)Z(z) (D.69)

2
2
k = k − k z 2 (D.70)
c

2
2
d P(ρ) 1 dP(ρ) 2 k φ
+ + k − P(ρ) = 0 (D.71)
c
dρ 2 ρ dρ ρ 2
2
∂ (φ) 2
+ k (φ) = 0 (D.72)
φ
∂φ 2
2
d Z(z) 2
+ k Z(z) = 0 (D.73)
z
dz 2

A z F 1 (k z z) + B z F 2 (k z z), k z
= 0,
Z(z) = (D.74)
a z z + b z , k z = 0.

A φ F 1 (k φ φ) + B φ F 2 (k φ φ), k φ
= 0,
(φ) = (D.75)
a φ φ + b φ , k φ = 0.

a ρ ln ρ + b ρ , k c = k φ = 0,

k φ
P(ρ) = a ρ ρ −k φ + b ρ ρ , k c = 0 and k φ
= 0, (D.76)

A ρ G 1 (k c ρ) + B ρ G 2 (k c ρ), otherwise.

 jξ
e

 − jξ

e
F 1 (ξ), F 2 (ξ) = (D.77)
sin(ξ)



cos(ξ)

(ξ)
 J k φ


(ξ)

N k φ
G 1 (ξ), G 2 (ξ) = (1) (D.78)
H (ξ)
 k φ

 (2)
H (ξ)
k φ
Spherical coordinate system
Coordinate variables
u = r, 0 ≤ r < ∞ (D.79)
v = θ, 0 ≤ θ ≤ π (D.80)
w = φ, −π ≤ φ ≤ π (D.81)
© 2001 by CRC Press LLC