Page 388 - Engineering Electromagnetics, 8th Edition

370 ENGINEERING ELECTROMAGNETICS

cosines. Because the waves are sinusoidal, we denote their velocity as the phase ve-
locity, ν p . The waves are written as:

E x (z, t) = E x (z, t) + E (z, t)

x
=|E x0 | cos [ω(t − z/ν p ) + φ 1 ] +|E | cos [ω(t + z/ν p ) + φ 2 ]

x0
=|E x0 | cos [ωt − k 0 z + φ 1 ] +|E | cos [ωt + k 0 z + φ 2 ] (15)

x0

forward z travel backward z travel
In writing the second line of (15), we have used the fact that the waves are traveling in
free space, in which case the phase velocity, ν p = c. Additionally, the wavenumber
in free space in defined as
ω
k 0 ≡ rad/m (16)
c
In a manner consistant with our transmission line studies, we refer to the solutions
expressed in (15) as the real instantaneous forms of the electric field. They are the
mathematical representations of what one would experimentally measure. The terms
ωt and k 0 z, appearing in (15), have units of angle and are usually expressed in radians.
We know that ω is the radian time frequency, measuring phase shift per unit time;
it has units of rad/s. In a similar way, we see that k 0 will be interpreted as a spatial
frequency, which in the present case measures the phase shift per unit distance along
the z direction in rad/m. We note that k 0 is the phase constant for lossless propagation
of uniform plane waves in free space. The wavelength in free space is the distance
over which the spatial phase shifts by 2π radians, assuming fixed time, or
2π
k 0 z = k 0 λ = 2π → λ = (free space) (17)
k 0
The manner in which the waves propagate is the same as we encountered in
transmission lines. Specifically, suppose we consider some point (such as a wave
crest) on the forward-propagating cosine function of Eq. (15). For a crest to occur,
the argument of the cosine must be an integer multiple of 2π. Considering the mth
crest of the wave, the condition becomes

k 0 z = 2mπ
So let us now consider the point on the cosine that we have chosen, and see what
happens as time is allowed to increase. Our requirement is that the entire cosine
argument be the same multiple of 2π for all time, in order to keep track of the chosen
point. Our condition becomes
ωt − k 0 z = ω(t − z/c) = 2mπ (18)
As time increases, the position z must also increase in order to satisfy (18). The wave
crest (and the entire wave) moves in the positive z direction at phase velocity c (in
free space). Using similar reasoning, the wave in Eq. (15) having cosine argument
(ωt + k 0 z) describes a wave that moves in the negative z direction, since as time