Page 316 - Electromagnetics

From (4.328) we find the electric field associated with the wave:
˜
˜ H z0 (2)
E φ =−Z TE H 1 (kρ), (4.334)
4
where
k
Z TE =
ω˜ c
is the TE wave impedance.
(2) (2)
It is not readilyapparent that the terms H 0 (kρ) or H 1 (kρ) describe outward prop-
agating waves. We shall see later that the cylindrical wave may be written as a su-
perposition of plane waves, both uniform and evanescent, propagating in all possible
directions. Each of these components does have the expected wave behavior, but it is
still not obvious that the sum of such waves is outward propagating.
We saw in § 2.10.7 that when examined in the time domain, a cylindrical wave of the
form H (2) (kρ) does indeed propagate outward, and that for lossless media the velocityof
0
propagation of its wavefronts is v = 1/(µ ) 1/2 . For time-harmonic fields, the cylindrical
wave takes on a familiar behavior when the observation point is sufficientlyremoved from
the source. We mayspecialize (4.330) to the time-harmonic case bysetting ω = ˇω and
using phasors, giving
j (2)
ˇ
ˇ
E z (ρ) =− E z0 H 0 (kρ).
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If |kρ| 1 we can use the asymptotic representation (E.62) for the Hankel function

2 − j(z−π/4−νπ/2)
(2)
H (z) ∼ e , |z| 1, −2π< arg(z)<π,
ν
πz
to obtain
e − jkρ
ˇ ˇ
E z (ρ) ∼ E z0 √ (4.335)
8 jπkρ
and
1 e − jkρ
ˇ ˇ
H φ (ρ) ∼−E z0 √ (4.336)
Z TM 8 jπkρ
√
for |kρ| 1. Except for the ρ term in the denominator, the wave has verymuch the
same form as the plane waves encountered earlier. For the case of magnetic polarization,
we can approximate (4.333) and (4.334) to obtain
e − jkρ
ˇ ˇ
H z (ρ) ∼ H z0 √ (4.337)
8 jπkρ
and
e − jkρ
ˇ ˇ
E φ (ρ) ∼ Z TE H z0 √ (4.338)
8 jπkρ
for |kρ| 1.
To interpret the wave nature of the field (4.335) let us substitute k = β − jα into
the exponential function, where β is the phase constant (4.224) and α is the attenuation
constant (4.225). Then
1
ˇ ˇ −αρ − jβρ
E z (ρ) ∼ E z0 √ e e .
8 jπkρ

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