Page 281 - Electromagnetics

ˆ
But k · d = 0,so
ˆ
k · r = r 0 , (4.246)
which is a spatial constant, hence (4.245) holds for any t. The planar surfaces described
by(4.245) are wavefronts.
Note that surfaces of constant amplitude are determined by
ˆ
α(k · r) = C A
ˆ
where C A is some constant. As with the phase term, this requires that k · r = constant,
and thus surfaces of constant phase and surfaces of constant amplitude are coplanar.
This is a propertyof uniform plane waves. We shall see later that nonuniform plane
waves have planar surfaces that are not parallel.
The cosine term in (4.243) represents a traveling wave.As t increases, the argument of
ˆ
the cosine function remains unchanged as long as k·r increases correspondingly. Thus the
ˆ
planar wavefronts propagate along k. As the wavefront progresses, the wave is attenuated
because of the factor e −α( ˆ k·r) . This accounts for energytransferred from the propagating
wave to the surrounding medium via Joule heating.

Phase velocity of a uniform plane wave. The propagation velocityof the progress-
ing wavefront is found bydifferentiating (4.245) to get
dr
ˆ
ˇ ω − βk · = 0.
dt
By(4.246) we have
dr 0 ˇ ω
v p = = , (4.247)
dt β
where the phase velocity v p represents the propagation speed of the constant-phase sur-
faces. For the case of a lossymedium with frequency-independent constitutive parame-
ters, (4.227) shows that
1
v p ≤ √ ,
µ
hence the phase velocityin a conducting medium cannot exceed that in a lossless medium
with the same parameters µ and . We cannot draw this conclusion for a medium with
c
frequency-dependent ˜µ and ˜ , since by(4.224) the value of ˇω/β might be greater or less
√

than 1/ µ ˜ , depending on the ratios ˜µ / ˜µ and ˜ /˜ .
c

˜
c
c
Wavelength of a uniform plane wave. Another important propertyof a uniform
plane wave is the distance between adjacent wavefronts that produce the same value of
the cosine function in (4.243). Note that the field amplitude maynot be the same on
these two surfaces because of possible attenuation of the wave. Let r 1 and r 2 be points
on adjacent wavefronts. We require
ˆ
ˆ
β(k · r 1 ) = β(k · r 2 ) − 2π
or
ˆ
λ = k · (r 2 − r 1 ) = r 02 − r 01 = 2π/β.
We call λ the wavelength.

© 2001 by CRC Press LLC