Page 196 - Maxwell House

176 Chapter 4

4. Here the constant denoted by symbol =
0
= � ⋅ � [Ω] is called the characteristic or wave
0

0
0
impedance and completely defined by the medium
parameters. Therefore, in free space, electric and magnetic
vectors of propagating waves are orthogonal to each other
and both are normal to the -direction of propagation (see
Figure 4.1.6). Such waves are defined as transverse or TEM
waves and they are only ones that can propagate in free-of-
sources and free-of-loss inhomogeneous linear mediums.
Sun or stars light, radio waves carrying radio, TV, satellite
or cell phone signals to your house (but not inside the
Figure 4.1.6 Orientation building) are close to be exemplary.
of vectors , and
+
+
+
5. According to (4.42) and (4.43), Poynting’s
vector of the propagating wave points out the direction of propagation as shown in Figure
4.1.6 for the forward wave (plus sign) and backward wave and can be calculated as

(, ) = ± (, ) = ± (, )
2
2
±
⁄
0 0
0

0
It is worthwhile to note that we, in fact, uncovered quite a remarkable set of data that
characterizes not only TEM waves but all variety of propagating EM waves. In spite of an
enormous difference in their quantitative and structure characteristics, the general and
rigorous solution of wave equations in (4.35) can be written as the superposition (or more
precisely, the multi-dimensional integral) of plane TEM waves like we have just discovered. It
can be proved [4, 5] we could apply the multi-dimensional Fourier transform resembling (1.77)
to all three spatial variables in (4.35) as we have done it previously in time domain only.
4.1.9 General Solution of Wave Equations and Green’s Function
Now we are in good position to build solution of (4.22) and (4.23) in free-of-loss space-time
domain. Let us come back to (4.31) assuming that the infinitesimal charge is time dependable

( , ) ′
′ ′
∞

′
(, , ) = ∫ ∘ = (4.44)
′
4 0 |− |
Here t is the first moment of time when the potential aka energy can be measured at the
point P and is the moment of time when the charge source starts emitting the energy.
′
Undoubtedly, > since any physical phenomenon can never precede its cause in time.
′
Applying this casualty principle to (4.40) we can see that only the first term in (4.40) is
consistent with it. If so, the time delay must be equal to the temporal interval required for the
wave of retarding potential to run the distance | − | with speed of light
′
= − | − | (4.45)
′
′
⁄
Here we treated | − | as the positive direction over z-axis in (4.40). Therefore, the general
′
solution of wave equations (4.22) and (4.23) in the free-of-loss medium is