Page 459 - Engineering Electromagnetics, 8th Edition

CHAPTER 12 Plane Wave Reflection and Dispersion 441

the complex fields:
e
E c,net (z, t) = E 0 [e − jβ a z jω a t + e − jβ b z jω b t ]
e
Note that we must use the full complex forms (with frequency dependence retained)
as opposed to the phasor forms, since the waves are at different frequencies. Next,
we factor out the term e − jβ 0 z jω 0 t :
e
e
E c,net (z, t) = E 0 e − jβ 0 z jω 0 t [e j βz − j ωt + e − j βz j ωt ]
e
e
= 2E 0 e − jβ 0 z jω 0 t cos( ωt− βz) (81)
e
where
ω = ω 0 − ω a = ω b − ω 0
and

β = β 0 − β a = β b − β 0
The preceding expression for β is approximately true as long as ω is small. This
can be seen from Figure 12.12 by observing how the shape of the curve affects β,
given uniform frequency spacings.
The real instantaneous form of (81) is found through

E net (z, t) = Re{E c,net }= 2E 0 cos( ωt − βz) cos(ω 0 t − β 0 z) (82)
If ω is fairly small compared to ω 0 ,we recognize (82) as a carrier wave at fre-
quency ω 0 that is sinusoidally modulated at frequency ω. The two original waves
are thus “beating” together to form a slow modulation, as one would hear when the
same note is played by two slightly out-of-tune musical instruments. The resultant
wave is shown in Figure 12.13.

net

Figure 12.13 Plot of the total electric field strength as a function of z (with
t = 0) of two co-propagating waves having different frequencies, ω a and ω b ,
as per Eq. (81). The rapid oscillations are associated with the carrier
frequency, ω 0 = (ω a + ω b )/2. The slower modulation is associated with the
envelope or ‘‘beat’’frequency, ω = (ω b − ω a )/2.