DISCUSSIONS AND CO?\IRIUNICATIONS PROPAGATION OF TRANSIENT ELECTROMAGNETIC WAVES IN A CONDUCTING MEDIUM

Wait (1951) has calculated the transient electric fields for several types of step-function current sources placed inside a conducting medium. Now any generated pulse will require a finite build-up time to reach its final magnitude from its initial value of zero. In most cases, this type of pulse may be very well approximated by a ramp-function pulse (Figure I). Expressions for the electric field of this type of pulse will be deduced in the following analysis. Let us assume that an electric dipole is immersed in a homogeneous, isotropic medium having conductivity Q, dielectric constant E, and permeability p. If displacement currents be neglected in FIG. I. Plot of ramp function. comparison with conduction currents, we may write the expressions for the non-vanishing field com-ponents in sp6 erical coordinates as follows (Wait, rggr): e,(s) = g $ (I + cd~2)e-Q~1’2 cos e eeb) = %(I + cd2 + &)e aa’/ * sin 0 and (3) where and i(s), G(S), ee(s), and /Q,(S) are the Laplace transforms of i(t), er(t), ee(t), and h,(l). The Laplace transform of a function representing a ramp-function current pulse i(t) which begins at time t=o and flattens out at time t=tr (Figure I), is given by 960 DISCUSSIONS AND COMMUNICATIONS Substituting (4) into (I) and normalizing by the substitutions 1 E 4) &)- &). zam3) h a2 I & cos e and II=-? c2 we obtain. [I- e-a+]. (5) The inverse Laplace transform of (5) may be obtained by utilizing the following two important results: +~.Stli(~)‘-~.~(~>“+...] (6 and where L-r denotes the inverse Laplace transform. FIG. 2. Radial electric field component as function of time for various build-up times. With the help of equations (6) and (7), equation (5) may be expressed in the following way, after some algebraic manipulations, for the two separate time-domains: (i) o$/jl, E,(f) =;.jr(t) (8) and (ii) rlt, where u(t) and u(t-t,) are step-functions beginning at times t=o and t=t, respectively and jr(t) = (1- +)-f + [ (2- 4t) erf (5~) + zt” * erf ’ (;+)I