Page 265 - Electromagnetics

4.11 The wave nature of the time-harmonic EM field
Time-harmonic electromagnetic waves have been studied in great detail. Narrowband
waves are widely used for signal transmission, heating, power transfer, and radar. They
share many of the properties of more general transient waves, and the discussions of
§ 2.10.1 are applicable. Here we shall investigate some of the unique properties of time-
harmonic waves and introduce such fundamental quantities as wavelength, phase and
group velocity, and polarization.

4.11.1 The frequency-domain wave equation
We begin by deriving the frequency-domain wave equation for dispersive bianisotropic
materials. A solution to this equation may be viewed as the transform of a general
time-dependent field. If one specific frequency is considered the time-harmonic solution
is produced.
In § 2.10.2 we derived the time-domain wave equation for bianisotropic materials.
There it was necessary to consider only time-independent constitutive parameters. We
can overcome this requirement, and thus deal with dispersive materials, by using a Fourier
transform approach. We solve a frequency-domain wave equation that includes the fre-
quency dependence of the constitutive parameters, and then use an inverse transform to
return to the time domain.
The derivation of the equation parallels that of § 2.10.2. We substitute the frequency-
domain constitutive relationships

˜
˜ ¯
˜
˜
D = ˜ ¯ · E + ξ · H,
˜
˜
˜
˜ ¯
B = ζ · E + ˜ ¯µ · H,
into Maxwell’s curl equations (4.7) and (4.8) to get the coupled differential equations
˜
˜ ¯
˜
˜
˜
∇× E =− jω[ζ · E + ˜ ¯µ · H] − J m ,
˜
˜ ¯
˜
˜
˜
∇× H = jω[ ˜ ¯ · E + ξ · H] + J,
˜
˜
˜
for E and H. Here we have included magnetic sources J m in Faraday’s law. Using the
¯
dyadic operator ∇ defined in (2.308) we can write these equations as
˜ ¯
˜
˜
˜
¯
∇+ jωζ · E =− jω ˜ ¯µ · H − J m , (4.199)
˜ ¯
˜
˜
˜
¯
∇− jωξ · H = jω ˜ ¯ · E + J. (4.200)
˜
˜
We can obtain separate equations for E and H by defining the inverse dyadics
−1 −1
¯
¯
˜ ¯ · ˜ ¯ = I, ˜ ¯ µ · ˜ ¯µ = I.
−1
Using ˜ ¯µ we can write (4.199) as
−1
−1
˜
¯
˜
˜ ¯
˜
− jωH = ˜ ¯µ · ∇+ jωζ · E + ˜ ¯µ · J m .
Substituting this into (4.200) we get
−1 2 −1
˜ ¯
˜
˜ ¯
˜
˜
¯
¯
˜ ¯
¯
∇− jωξ · ˜ ¯µ · ∇+ jωζ − ω ˜ ¯ · E =− ∇− jωξ · ˜ ¯µ · J m − jωJ. (4.201)
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