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Figure 2.9: Propagation of a transient cylindrical wave in a lossless medium.
we can write the electric field in closed form as
√ √
x 2 + x 2 + 2ρ/v
E z (ρ, t) = 2E 0 ln √ √ , (2.349)
x 1 + x 1 + 2ρ/v
where x 2 = max[0, t − ρ/v] and x 1 = max[0, t − ρ/v − τ]. The field is plotted in Figure
2.9 for various values of time. Note that the leading edge of the disturbance propagates
outward at a velocity v and a wake trails behind the disturbance. This wake is similar to
that for a plane wave in a dissipative medium, but it exists in this case even though the
medium is lossless. We can think of the wave as being created by a line source of infinite
extent, pulsed by the disturbance waveform. Although current changes simultaneously
everywhere along the line, it takes the disturbance longer to propagate to an observation
point in the z = 0 plane from source points z
= 0 than from the source point at z = 0.
Thus, the field at an arbitrary observation point ρ arrives from different source points at
different times. If we look at Figure 2.9 we note that there is alwaysa nonzero field near
ρ = 0 (or any value of ρ< vt) regardless of the time, since at any given t the disturbance
is arriving from some point along the line source.
We also see in Figure 2.9 that as ρ becomes large the peak value of the propagating
disturbance approaches a certain value. This value occurs at t m = ρ/v+τ or, equivalently,
ρ m = v(t − τ). If we substitute this value into (2.349) we find that
τ τ
E z (ρ, t m ) = 2E 0 ln + 1 + .
2ρ/v 2ρ/v
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