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Figure 2.7: Propagation of a transient plane wave in a lossless medium.


                        From (2.329) we also have the solution for E(z, t):

                                                   vµH 0      z     vµH 0       z
                                         E(z, t) = ˆ p  f t −    − ˆ p    f t +   ,           (2.337)
                                                     2        v       2        v
                        where

                                                          ˆ p × ˆ q = ˆ z.
                        The boundary conditions E(0, t) = 0 and H(0, t) = H 0 f (t) are easily verified by substi-
                        tution.
                          This solution displays the quintessential behavior of electromagnetic waves. We may
                        interpret the term f (t +z/v) as a wave field disturbance, propagating at velocity v in the
                        −z-direction, incident from z > 0 upon the conductor. The term f (t − z/v) represents
                        a wave field disturbance propagating in the +z-direction with velocity v, reflected from
                        the conductor. By “propagating” we mean that if we increment time, the disturbance
                        will occupy a spatial position determined by incrementing z by vt. For free space where
                        v = 1/(µ 0 
 0 ) 1/2 , the velocity of propagation is the speed of light c.
                          A specific example should serve to clarify our interpretation of the wave solution.
                        Taking µ = µ 0 and 
 = 81
 0 , representing typical constitutive values for fresh water, we
                        can plot (2.335) as a function of position for fixed values of time. The result is shown in
                        Figure 2.7, where we have chosen

                                                        f (t) = rect(t/τ)                     (2.338)

                        with τ = 1 µs. We see that the disturbance is spatially distributed as a rectangular
                                                                            7
                        pulse of extent L = 2vτ = 66.6 m, where v = 3.33 × 10 m/s is the wave velocity,



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