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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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