Page 275 - Electromagnetics

ˆ
Figure 4.10: Surface of constant k · r.

Differentiation gives
dω
dr 0
= v g = . (4.240)
dt dβ
ˆ
So the envelope propagates along k at a rate given by the group velocity v g . Associated
with this propagation is an attenuation described by the factor e −α(ω 0 ) ˆ k·r . This accounts
for energy transfer into the lossy medium through Joule heating.
Similarly, we can identify a plane over which the phase of the carrier is constant; this
will be parallel to the plane of constant envelope described above. We now set
ˆ

ω 0 t − k · r/v p (ω 0 ) = C
and differentiate to get
ω
dr 0
= v p = . (4.241)
dt β
ˆ
This shows that surfaces of constant carrier phase propagate along k with velocity v p .
Caution must be exercised in interpreting the two velocities v g and v p ; in particular, we
must be careful not to associate the propagation velocities of energy or information with
v p . Since envelope propagation represents the actual progression of the disturbance, v g
has the recognizable physical meaning of energy velocity. Kraus and Fleisch [105] suggest
that we think of a strolling caterpillar: the speed (v p ) of the undulations along the
caterpillar’s back (representing the carrier wave) may be much faster than the speed (v g )
of the caterpillar’s body (representing the envelope of the disturbance).
In fact, v g is the velocity of energy propagation even for a monochromatic wave (§ ??).
However, for purely monochromatic waves v g cannot be identified from the time-domain
field, whereas v p can. This leads to some unfortunate misconceptions, especially when
v p exceeds the speed of light. Since v p is not the velocity of propagation of a physical
quantity, but is rather the rate of change of a phase reference point, Einstein’s postulate
of c as the limiting velocity is not violated.
We can obtain interesting relationships between v p and v g by manipulating (4.237)
and (4.241). For instance, if we compute
dβ
d ω β − ω

dv p dω
= =
dω dω β β 2

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