Comment on a Letter by EF Kuester

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IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. EMC-29, NO. 4, NOVEMBER 1987

Correspondence Comments on the Comments of A. R. Djordjvic and T. Sarkar on "Correction of Maxwell's Equations for Signals I and II" and "Propagation Velocity of Electromagnetic Signals" MALEK G. M. HUSSAIN, MEMBER, IEEE

addresses the difference between Harmuth's work and what is claimed by Djordjvic and Sarkar. REFERENCES [1] A. R. Djordjvic, T. Sarkar, and H. F. Harmuth, "Comments on 'Correction of Maxwell's equations for signals I,' 'Correction of Maxwell's equations for signals II,' and 'Propagation velocity of electromagnetic signals,' " IEEE Trans. Electromagn. Compat., vol. EMC-29, no. 3, pp. 255-256, Aug. 1987.

Index Code-J3d/e.

The claim of Djordjvic and Sarkar that the telegrapher's equations are identical to Maxwell's equations for plane wave propagation in a lossy medium is well answered by Harmuth's reply [1]. The procedure followed in obtaining their final solution is rather ambiguous, as can be pointed out by addressing the following points. 1) The equation for E ( y, t) has typographical errors which make it difficult for one to follow how E (, 0) can be derived "by simply convolving" with the unit step function h(t). It is mentioned that "Pipes has a typographical error in (83)." What is the correction? 2) No convolution relationship is directly apparent in E (y, t), which shows that causality will not be violated and boundary and initial conditions will be satisfied for both E and H. The justification for performing the convolution should have been presented. That is, the lossy propagation medium is modeled as a linear time-invariant system. One may ask these questions. Is it assumed that the linear system is causal prior to obtaining a solution by convolution? Is such an assumption valid?! 3) It is necessary to show how the limits for the integral (i) in the solution E( , 6) are obtained. It is understood that t is the space variable while 0 is the time variable. The integration is for the variable 0. How can the lower limit be a space variable while the upper limit is a time variable, especially when both appear in the function within the integral?!! 4) What is the justification for h(t - y/c) in the impulse response solution E (y, t)? Is the excitation function 6(t - y/c)h(t - y/c) applied? Would the solutions and causality hold for 6(t)h(t)? Harmuth's excitation is EOS(t), not EOS(t - y/c). 5) No attempt was made by Djordjvic and Sarkar to obtain a solution for the magnetic field due to the applied electric field. This is the focus of the whole issue. Did they know that their approach leads to a dead end, and did they knowingly withhold this fact from the

readers? 6) Harmuth specifies the boundary and initial conditions for E and H separately and provides solutions for them in a lossy medium. The solutions are obtained when the applied excitation is an electric field and when it is a magnetic field. The solutions satisfy boundary and initial conditions, as well as the causality law. To the best of my knowledge, such solutions of Maxwell's equations for plane wave propagation have never been reported in the literature. As pointed out by Harmuth, the transmission-line problem cannot be classified as wave propagation. This point is rather important since it directly Manuscript received May 14, 1987. The author is with the Department of Electrical and Computer Engineering, Kuwait University, 13060 Safat, Kuwait. Tel. 4834-506. IEEE Log Number 8717790.

Comment on a Letter by E. F. Kuester J. E. GRAY AND RAOUF N. BOULES

Index Code-J3d/e.

In this letter, we propose to discuss a mathematical comment made by Kuester [1] to a series of controversial papers written by Harmuth [2]-[5]. This comment is directed to the question of whether the initial conditions and the Fourier transform are compatible [5, section 2.2]. Harmuth makes a heuristic argument that they are not, while Kuester contends that they are. While Harmuth's proof is not correct as stated, his conclusion is correct. First note that equations (2-17) and (2-18) of [5] are not correctly written in the form they appear; instead they should be written in the form

E(O, t)=

0

[Eol(,3) cos ct-Eo22(3) sin wst] do

(la)

cos fy+E02(3) sin 3yle-a"y do (lb) E(y, 0)= J[Eo0(f) 0 so we will concern ourselves with these equations instead of those in Harmuth. Since ,B = f3(w), it is more convenient to write these equations as explicit functions of w. This can be done by making a change of variables in (la) and (lb) or by going back to the original equations and integrating them directly as a function of X to give

E(O, t)-

E(O, y)

0

[A I(cc) cos (wt) - A2(cc) sin (wt)] dcw

(2a)

[A1(c) cos (g(co)y)+A2 (co) sin (g(co)y)]e-h(')y dco (2b)

Manuscript received April 23, 1987; revised August 18, 1987. J. E. Gray is with the Naval Surface Weapons Center, Dahlgren, VA 22448-0489. Tel. (703) 6634211. R. N. Boules is with the Department of Mathematics, The Catholic University of America, Washington, DC 20064. Tel. (202) 635-5221. IEEE Log Number 8716609.

0018-9375/87/1100-0317$01.00 © 1987 IEEE

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IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. EMC-29, NO. 4, NOVEMBER 1987

where

g(w)

-3.

Note that we have two equations in two unknowns for AI (w) and A2(w) that should enable us to solve for these two variables; this is not the case, however. There are two different ways one can see this. The first way is to rewrite the equations as

R(w) sin (cwt -4()) dw

(3a)

R (co) sin (h (cw)y +4b(w))e - 9M@Y dco

(3b)

E(O, t)=

E

E(O, Y) =

00

0

[3] H. F. Harmuth, "Correction of Maxwell's equations for signals II," IEEE Trans. Electromagn. Compat., vol. EMC-28, no. 4, pp. 259266, Nov. 1986. [4] H. F. Harmuth, "Propagation velocity of electromagnetic signals," IEEE Trans. Electromagn. Compat., vol. EMC-28, no. 4, pp. 267272, Nov. 1986. [5] H. F. Harmuth, Propagation of Nonsinusoidal Electromagnetic Waves. New York: Academic, 1986. [6] A. Papoulis, The Fourier Integral and Its Applications. New York: McGraw-Hill, 1962, pp. 215-217.

where

R(w) = (A I + A 2

1,

4(Co) = arctan (A 1/ A2).

Then, if one realizes that both y and t are evolution parameters of the kernel equation that one gets by taking the limit as t approaches zero from the right in (3a) and as y approaches zero from the right in (3b), then one sees that they contradict initial conditions because of the minus sign in (3a). This is not a surprising result, since one does not expect the causality law to result strictly from mathematical formalism. There is, however, no mathematical reason why, with the proper choice of arbitrary functions, one could not get consistency in our evolution equations. This is a consequence of the Paley-Weiner theorem which gives necessary and sufficient conditions for a function to be the Fourier spectrum of a causal function [6]. The existence of a causal function H(co) does not imply the existence of a causal inverse, which is the source of our difficulties. The other method of noting the difficulties between the imposition of initial conditions and boundary conditions on the Fourier transform at the same time is to return to (2a) and note that it is a normal Fourier transform. Therefore, one can invert it for both the A terms since sine and cosine are orthogonal functions. The function E(0, t) can be written as a sum of even and odd parts, i.e.,

E(0, t)=EE(O, t)+Eo(0, t).

(4)

One can then write the Fourier transform as R (w) +jX(w) - FE(co) + Fo(c)

(5)

Reply to Kuester's Comments on the Use of a Magnetic Conductivity HENNING F. HARMUTH, MEMBER, IEEE, RAOUF N. BOULES, AND MALEK G. M. HUSSAIN, MEMBER, IEEE

Index Code-J3d/e.

Kuester's comments [1] have previously been shown [2] to be based largely on the steady-state equations of electron theory characterized by frequency-dependent conductivity a(w), permittivity e(w), and permeability ,u(w). The respective results were thus not applicable to the question of the existence of transient solutions in lossy media of Maxwell's equations, whose conductivity, permittivity, and permeability can only be tensor functions of location and time, but not of frequency. However, there is one important claim in Kuester's comments that is not affected by this misunderstanding. Kuester gives the following magnitude of the electric field strength for a planar TEM wave with any polarization excited by an electric step excitation function: EE(K) (., 0) 0, =

0