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z 2 2 2
2 t+ J 1 z − (t − u) v
z H 0 − t v u v
− e f (u)e du (2.343)
2v z z − (t − u) v
2
2 2
t−
v v
where = σ/2
. The first two terms resemble those for the lossless case, modified
by an exponential damping factor. This accounts for the loss in amplitude that must
accompany the transfer of energy from the propagating wave to joule loss (heat) within
the conducting medium. The remaining term appears only when the medium is lossy, and
results in an extension of the disturbance through the medium because of the currents
induced by the passing wavefront. This “wake” follows the leading edge of the disturbance
as is shown clearly in Figure 2.8. Here we have repeated the calculation of Figure 2.7,
−4
but with σ = 2 × 10 , approximating the conductivity of fresh water. As the wave
travels to the left it attenuates and leaves a trailing remnant behind. Upon reaching
the conductor it reflects much as in the lossless case, resulting in a time dependence at
z = 0 given by the finite-duration rectangular pulse (2.338). In order for the pulse to
be of finite duration, the wake left by the reflected pulse must exactly cancel the wake
associated with the incident pulse that continues to arrive after the reflection. As the
reflected pulse sweeps forward, the wake is obliterated everywhere behind.
If we were to verify the Poynting theorem for a dissipative medium (which we shall
not attempt because of the complexity of the computation), we would need to include
the E·J term. Here J is the induced conduction current and the integral of E·J accounts
for the joule loss within a region V balanced by the difference in Poynting energy flux
carried into and out of V .
Once we have the fields for a wave propagating along the z-direction, it is a simple
matter to generalize these results to any propagation direction. Assume that ˆ u is normal
to the surface of a plane over which the fields are invariant. Then u = ˆ u · r describes the
distance from the origin along the direction ˆ u. We need only replace z by ˆ u · r in any
of the expressions obtained above to determine the fields of a plane wave propagating in
the u-direction. We must also replace the orthogonality condition ˆ p × ˆ q = ˆ z with

ˆ p × ˆ q = ˆ u.
For instance, the fields associated with a wave propagating through a lossless medium in
the positive u-direction are, from (2.336)–(2.337),

H 0 ˆ u · r vµH 0 ˆ u · r
H(r, t) = ˆ q f t − , E(r, t) = ˆ p f t − .
2 v 2 v
2.10.7 Propagation of cylindrical waves in a lossless medium
Much as we envisioned a uniform plane wave arising from a uniform planar source, we
can imagine a uniform cylindrical wave arising from a uniform line source. Although this
line source must be infinite in extent, uniform cylindrical waves (unlike plane waves) dis-
play the physical behavior of diverging from their source while carrying energy outwards
to infinity.
A uniform cylindrical wave has fields that are invariant over a cylindrical surface:
E(r, t) = E(ρ, t), H(r, t) = H(ρ, t). For simplicity, we shall assume that waves propagate
in a homogeneous, isotropic, linear, and lossless medium described by permittivity
and permeability µ. From Maxwell’s equations we find that requiring the fields to be
independent of φ and z puts restrictions on the remaining vector components. Faraday’s

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