ELECTROMAGNETIC FUNDAMENTALS FOR SIGNAL INTEGRITY - Advanced Signal Integrity for High-Speed Digital Design - page 28

the extraneous e jωt terms, the time-harmonic form of Maxwell's equations , where
time has been eliminated, is shown.

∇× E jω B

+ = 0 (2-33)

∇× H =

J + jω D

(2-34)

∇· D = ρ (2-35)

∇· B = 0 (2-36)

Note that the time variation of the fields can be restored by multiplying by e jωt
and taking the real part:

F(x, y,z,t) = Re [ F(x, y,z)e jωt ]

(2-37)

2.3.4 Propagation of Time-Harmonic Plane Waves

As will be demonstrated in subsequent chapters, the propagation of time-harmonic
plane waves is of particular importance for the study of transmission-line or other
guided-wave structures. This allows us to study a simplified subset of Maxwell's
equations where propagation is restricted to one direction (usually along the
z -axis) and time is removed as described in Section 2.3.3. A plane wave is
defined so that propagation occurs in only one direction ( z ) and the fields do
not vary with time in the x - and y -directions. If the fields were observed at an
instant in time, they would be constant in the x - y plane for any given point z
and would change for different values of z or t . Figure 2-11 depicts a plane wave
propagating in the z -direction.
To study the behavior of time-harmonic plane waves, it is necessary to
re-derive the wave equation from the time-harmonic form of Maxwell's

x

Direction of
propagation

z

y

Figure 2-11 Plane wave propagating in the z -direction.