Page 391 - Engineering Electromagnetics, 8th Edition

CHAPTER 11 The Uniform Plane Wave 373

√
where again, k 0 = ω/c = ω µ 0 0 . Equation (28) is known as the vector Helmholtz
1
equation in free space. It is fairly formidable when expanded, even in rectangular
coordinates, for three scalar phasor equations result (one for each vector component),
and each equation has four terms. The x component of (28) becomes, still using the
del-operator notation,

2 2
∇ E xs =−k E xs (29)
0
and the expansion of the operator leads to the second-order partial differential equation
2 2 2
∂ E xs ∂ E xs ∂ E xs 2
0
∂x 2 + ∂y 2 + ∂z 2 =−k E xs
Again, assuming a uniform plane wave in which E xs does not vary with x or y, the
two corresponding derivatives are zero, and we obtain

2
d E xs 2 (30)
0
dz 2 =−k E xs
the solution of which we already know:

E xs (z) = E x0 e − jk 0 z + E e jk 0 z (31)

x0
Let us now return to Maxwell’s equations, (23) through (26), and determine the
form of the H field. Given E s , H s is most easily obtained from (24):
∇× E s =− jωµ 0 H s (24)
which is greatly simplified for a single E xs component varying only with z,
dE xs
=− jωµ 0 H ys
dz
Using (31) for E xs ,wehave
1

H ys =− (− jk 0 )E x0 e − jk 0 z + ( jk 0 )E e jk 0 z
x0
jωµ 0

0 − jk 0 z 0 jk 0 z − jk 0 z jk 0 z

= E x0 e − E x0 e = H y0 e + H e (32)
y0
µ 0 µ 0
In real instantaneous form, this becomes

0 0
H y (z, t) = E x0 cos(ωt − k 0 z) − E x0 cos(ωt + k 0 z) (33)
µ 0 µ 0
where E x0 and E x0 are assumed real.

1 Hermann Ludwig Ferdinand von Helmholtz (1821–1894) was a professor at the University of Berlin
working in the fields of physiology, electrodynamics, and optics. Hertz was one of his students.