Page 387 - Engineering Electromagnetics, 8th Edition

CHAPTER 11 The Uniform Plane Wave 369

Equations (5) and (6) can be more succinctly written:

∂E x ∂ H y
=−µ 0 (7)
∂z ∂t
∂ H y ∂E x
=− 0 (8)
∂z ∂t
These equations compare directly with the telegraphist’s equations for the lossless
transmission line [Eqs. (20) and (21) in Chapter 10]. Further manipulations of (7)
and (8) proceed in the same manner as was done with the telegraphist’s equations.
Specifically, we differentiate (7) with respect to z, obtaining:
2 2
∂ E x ∂ H y
=−µ 0 (9)
∂z 2 ∂t∂z
Then, (8) is differentiated with respect to t:
2 2
∂ H y ∂ E x
=− 0 (10)
∂z∂t ∂t 2
Substituting (10) into (9) results in

2 2
∂ E x ∂ E x
= µ 0 0 (11)
∂z 2 ∂t 2
This equation, in direct analogy to Eq. (13) in Chapter 10, we identify as the wave
equation for our x-polarized TEM electric field in free space. From Eq. (11), we
further identify the propagation velocity:

1
8
ν = √ = 3 × 10 m/s = c (12)
µ 0 0
where c denotes the velocity of light in free space. A similar procedure, involving
differentiating (7) with t and (8) with z, yields the wave equation for the magnetic
field; it is identical in form to (11):

2 2
∂ H y ∂ H y (13)
∂z 2 = µ 0 0 ∂t 2

As was discussed in Chapter 10, the solution to equations of the form of (11) and
(13) will be forward- and backward-propagating waves having the general form [in
this case for Eq. (11)]:

E x (z, t) = f 1 (t − z/ν) + f 2 (t + z/ν) (14)

where again f 1 amd f 2 can be any function whose argument is of the form t ± z/ν.
From here, we immediately specialize to sinusoidal functions of a specified fre-
quencyandwritethesolutionto(11)intheformofforward-andbackward-propagating