Page 445 - Engineering Electromagnetics, 8th Edition

CHAPTER 12 Plane Wave Reflection and Dispersion 427

The position vector, r, can be similarly expressed:

r = xa x + za z

so that

k · r = k x x + k z z

Equation (49) now becomes

E s = E 0 e − j(k x x+k z z) (50)

Whereas Eq. (49) provided the general form of the wave, Eq. (50) is the form that
is specific to the situation. Given a wave expressed by (50), the angle of propagation
from the x axis is readily found through

θ = tan −1 k z
k x
The wavelength and phase velocity depend on the direction one is considering. In the
direction of k, these will be

2π 2π
λ =
k = k + k z 2
1/2
2
x
and
ω ω
ν p =
k = k + k z 2
1/2
2
x
If, for example, we consider the x direction, these quantities will be
2π
λ x =
k x
and

ω
ν px =
k x
Note that both λ x and ν px are greater than their counterparts along the direction of k.
This result, at first surprising, can be understood through the geometry of Figure 12.6.
The diagram shows a series of phase fronts (planes of constant phase) which intersect
k at right angles. The phase shift between adjacent fronts is set at 2π in the figure; this
corresponds to a spatial separation along the k direction of one wavelength, as shown.
The phase fronts intersect the x axis, and we see that along x the front separation is
greater than it was along k. λ x is the spacing between fronts along x and is indicated