Page 428 - Engineering Electromagnetics, 8th Edition

410 ENGINEERING ELECTROMAGNETICS

where we have let jk 1 = 0 + jβ 1 in the perfect dielectric. These terms may be
combined and simplified,

E xs1 = (e − jβ 1 z − e jβ 1 z ) E x10
+
=− j2 sin(β 1 z) E + (11)
x10
Multiplying (11) by e jωt and taking the real part, we obtain the real instantaneous
form:
E x1 (z, t) = 2E x10 sin(β 1 z) sin(ωt) (12)
+
We recognize this total field in region 1 as a standing wave, obtained by combining
two waves of equal amplitude traveling in opposite directions. We first encountered
standing waves in transmission lines, but in the form of counterpropagating voltage
waves (see Example 10.1).
Again, we compare the form of (12) to that of the incident wave,

E x1 (z, t) = E + cos(ωt − β 1 z) (13)
x10
Here we see the term ωt − β 1 z or ω(t − z/ν p1 ), which characterizes a wave traveling
in the +z direction at a velocity ν p1 = ω/β 1 .In (12), however, the factors involving
time and distance are separate trigonometric terms. Whenever ωt = mπ, E x1 is zero
at all positions. On the other hand, spatial nulls in the standing wave pattern occur
for all times wherever β 1 z = mπ, which in turn occurs when m = (0, ±1, ±2,...).
In such cases,
2π
z = mπ
λ 1
and the null locations occur at

λ 1
z = m
2
Thus E x1 = 0at the boundary z = 0 and at every half-wavelength from the boundary
in region 1, z < 0, as illustrated in Figure 12.2.
Because E xs1 = η 1 H + and E − =−η 1 H ys1 , the magnetic field is
−
+
xs1
ys1
E +
H ys1 = x10 (e − jβ 1 z + e jβ 1 z )
η 1
or
E +
H y1 (z, t) = 2 x10 cos(β 1 z) cos(ωt) (14)
η 1
This is also a standing wave, but it shows a maximum amplitude at the positions
where E x1 = 0. It is also 90 out of time phase with E x1 everywhere. As a result, the
◦
average power as determined through the Poynting vector [Eq. (77), Chapter 11] is
zero in the forward and backward directions.
Let us now consider perfect dielectrics in both regions 1 and 2; η 1 and η 2 are
both real positive quantities and α 1 = α 2 = 0. Equation (9) enables us to calculate