Page 321 - Electromagnetics

Nonuniform cylindrical waves. When we solve two-dimensional boundaryvalue
problems we encounter cylindrical waves that are z-independent but φ-dependent. Al-
though such waves propagate outward, theyhave a more complicated structure than
those considered above.
For the case of TM polarization we have, by(4.212),
˜
j 1 ∂E z
˜
H ρ = , (4.349)
Z TM k ρ ∂φ
˜
j ∂E z
˜
H φ =− , (4.350)
Z TM k ∂ρ
where Z TM = ω ˜µ/k. For the TE case we have, by(4.213),
˜
˜ jZ TE 1 ∂ H z
E ρ =− , (4.351)
k ρ ∂φ
˜
˜ jZ TE ∂ H z
E φ = , (4.352)
k ∂ρ
c
where Z TE = k/ω˜ . By(4.208) the wave equations are
˜
2 2
∂ 1 ∂ 1 ∂ 2 E z
+ + + k = 0.
˜
2
∂ρ 2 ρ ∂ρ ρ ∂φ 2 H z
Because this has the form of A.177 with ∂/∂z → 0,wehave
˜

E z (ρ, φ, ω) = P(ρ, ω)!(φ, ω) (4.353)
˜
H z (ρ, φ, ω)
where
!(φ, ω) = A φ (ω) sin k φ φ + B φ (ω) cos k φ φ, (4.354)
(1) (2)
P(ρ) = A ρ (ω)B (kρ) + B ρ (ω)B (kρ), (4.355)
k φ k φ
(2)
(1)
and where B (z) and B (z) are anytwo independent Bessel functions chosen from the
ν ν
set
J ν (z), N ν (z), H (1) (z), H (2) (z).
ν ν
In bounded regions we generallyuse the oscillatoryfunctions J ν (z) and N ν (z) to represent
standing waves. In unbounded regions we generallyuse H (2) (z) and H (1) (z) to represent
ν ν
outward and inward propagating waves, respectively.

Boundary value problems in cylindrical coordinates: scattering by a material
cylinder. A varietyof problems can be solved using nonuniform cylindrical waves.
We shall examine two interesting cases in which an external field is impressed on a
two-dimensional object. The impressed field creates secondarysources within or on the
object, and these in turn create a secondaryfield. Our goal is to determine the secondary
field byapplying appropriate boundaryconditions.
c
As a first example, consider a material cylinder of radius a, complex permittivity ˜ ,
and permeability ˜µ, aligned along the z-axis in free space (Figure 4.25). An incident
plane wave propagating in the x-direction is impressed on the cylinder, inducing sec-
ondarypolarization and conduction currents within the cylinder. These in turn produce

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