176 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 30, NO. 2, MAY 1988

and

dp = (dP/dw) dw = ( 1 / u g ) do (12)

where ug is the group velocity [3, p. 2181, which one obtains from (6) with some effort:

1 2 4 { x [ x + ( x 2 + 1)1’2](x2+ 1)}”2 u,(w) = - = c

dP/dW 2 [ x + ( x 2 + 1)1’2](x2+ 1 ) ” Z - 1

x= w d u . (13)

We infer from (6) that w equals zero for /3 = 0 and that w approaches infinity when p does. Hence, the integration limits in (10) remain unchanged:

E,(O, t ) = 1; [ Vc(w)/ug(w)] cos w t dw

- 1” [ V,(w)/u,(w)] sin ut dw. (14)

On the right side we have the even function cos w t and the odd function sin at o f t , which suggests that we write the left side as the sum of an even and an odd function too:

1 1 Eg(0, f ) = ~ [Eg(O, f ) + E g ( O , - t ) l + i IEg(0, f ) -Eg(O, -01.

(15)

The even function on the left of (14) must equal the even function on the right, and the same must hold for the odd functions:

1 - [E,(o, t ) +E,(o, - t)] = Sa v,(w)/u,(w)] cos ut dw (16) 2

[E,(O, t)-E,(O, -[)I= - 1” [ V,(w)/ug(w)] sin w t dw. (17)

We recognize the Fourier cosine and the Fourier sine integrals. Using the Fourier transform pairs

1

I ” g,(w)= lm fs(t) sin ut dt, f,(t)=; J, gs(w) sin ut dw

-m

we obtain from (16) and (17)

1 Vc(w)=- ug(w) 1” [Eg(O, t)+E,(O, - t ) ] cos w t d t (20) 2T - m

We see that the boundary condition Eg(O, t ) determines the functions V,(w) and V,(w). These functions in turn determine the initial condition E,( y , 0) through (1 1). The initial condition thus cannot be chosen independently from the boundary condition, and E, (y , t ) of (9) can never be a physically meaningful solution that satisfies the causality law.

Two proofs that the Fourier transform (9) can never represent a causal solution of (1) were recently published [4]; one of the proofs is equal to the one given here, but was presented with less detail.

The use of the nonphysical solution (9) in electromagnetic theory and communications is practically universal. The mistake seems to have originated in the 19th century before boundary and initial conditions of partial differential equations were understood. In the literature around 1900 we find it connected with some of the most famous scientists of that time, and today it is solidly entrenched in textbooks on applications of the Fourier transform, filter theory, etc. For the sake of peace and tranquility, we cannot cite any references. However, we can reference one book that specifically points out that (9) or equivalent equations “become meaningless unless we restrict ourselves to special time-dependent components” [5, paragraph 58, p. 2351, [6, paragraph 58, p. 1641. This book has been a definitive work on electromagnetic theory through 18 editions and for more than half a century; the connection with the causality law does not seem to have been made.

REFERENCES

H. F. Harmuth, R. N. Boules, and M. G. M. Hussain, “Response to comments by M. J. Neatrour on Maxwell’s equations,” IEEE Trans. Electromagn. Compat., vol. 30, no. I , pp. 90-91, Feb. 1988. M. J. Neatrour, “Comments on ‘Correction of Maxwell’s equations for signals I,’ ‘Correction of Maxwell’s equations for signals 11,’ and ‘Propagation velocity of electromagnetic signals,’ ” IEEE Trans. Electromagn. Compat., vol. EMC-29, no. 3, pp. 258-259, Aug. 1987. H. F. Harmuth, Propagation of Nonsinusoidal Electromagnetic Waves. New York: Academic, 1986.

J . E. Gray and R. N. Boules, “Comments on a letter by E. F. Kuester,” IEEE Trans. Electromagn. Compat., vol. EMC-29, no. 4,

R. Becker, Electromagnetic Fields and Interactions (trans. by A. W. Knudsen of Theorie der Elektrizitaet, 16th ed.), vol. 1. New York: Blaisdell, 1964 (reprinted, New York: Dover, 1982). R. Becker, Theorie der Elektrizitaet, 18th ed. Stuttgart, FRG: Teubner, 1964.

pp. 317-318, NOV. 1987.

Third Response to Comments by M. J. Neatrour on Maxwell’s Equations

HENNING F. HARMUTH, MEMBER, IEEE, AND MALEK G. M. HUSSAIN, MEMBER, IEEE

Key Words-Bernoulli’s product method, homogeneous anisotropic

Index Code-J3d/e. time-invariant medium, nonsinusoidal waves.

Two previous responses to Neatrour’s comments [ 11 have clarified the difficulty of using a book by Stratton as reference [2] and the

Manuscript received December 4, 1987. H. F. Harmuth is with the Department of Hectrical Engineering, Catholic

M. G. M. Hussain is with the Department of Electrical and Computer

IEEE Log Number 8820091.

University of America, Washington, DC 20064. Tel. (202) 635-5193.

Engineering, Kuwait University, 13060 Safat, Kuwait. Tel. 4834506.

0018-9375/88/0500-0176$01 .00 0 1988 IEEE

IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 30, NO. 2. MAY 1988 177

Fig. 1. The function EE([ , O)/Eo representing the magnitude of the electric field strength due to an electric step function excitation [4, eq. (2.1-go)]. The normalization of the variables is 0 = 2d/u and E = ( P I E ) %y/2. The same plot holds for the function H H ( ( , 0)/Ho representing the magnitude of the magnetic field strength due to a magnetic step function excitation [4, p. 811.

distinction between (abstract) mathematical and physical solutions due to the causality law [3 ] or other laws imposed by physics in addition to the mathematical laws. A third problem that requires a response is the question of whether the modification of Maxwell’s equations adds anything new to physics.

This question is readily answered by the plots of Figs. 1-6, which are advanced versions of previously published plots [4, figs. 5.2-1, 2.4-1, 2.6-1, 2.3-2, 2.9-1, 2.10-11. Consider Fig. 1. It is possible that it can also be derived from Maxwell’s equations. We have neither a proof that this is possible nor one that it is impossible [5], but the difference between plots derived from Maxwell’s equations and from the modified Maxwell equations is at most on the order of 1 percent. A simple proof was given that the plots of Fig. 2 cannot be derived from Maxwell’s equations [5]. Complicated proofs that the plots of Figs. 2, 5 , and 6 cannot be derived from Maxwell’s equations were published earlier [4]. The plots of Fig. 4 may be derivable from Maxwell’s equations, but we have not yet seen such a derivation. Fig. 3 presents a problem since it deviates from the pattern of Figs. 2 , 5, and 6 by being derivable from Maxwell’s equations [4, p. 881.

Hence, something new is gained even in the limit of a vanishing magnetic current density, but the main interest is in nonvanishing magnetic currents or charges, since they would explain the quantiza- tion of electric charges. This task has eluded quantum mechanics for 70 years; the repeated references to Dirac make the point clear [4, p.

On a more mundane level, the theory that produced Figs. 1-6 makes i t possible to calculate the propagation of electromagnetic signals through seawater and through absorbing materials. We do not need to explain who is interested in these tasks and why; previously, only the transmission of power, but not signals, could be studied. Furthermore, the theory has shown that the period of sinusoidal pulses is increased when passing through absorbing media. As a result, the Doppler effect may not be responsible for the total

421, WI.

Fig. 2. The function Hz(E, 0)Z/Eo representing the magnetic field strength due to an electric step function excitation [4, eq. (2.446)].

4.75 4.25 3.75 3.25 2.75 2.25 1.75 1.25

.75

.25 .0

Fig. 3. The function E H ( [ , O)/ZHo representing the electric field strength due to magnetic step function excitation [4, eq. (2.6-25)].

- I ~ i ~ , 1 shows a dip below close to ( / l o = 0.15 for 0 0, which is

due to the limited accuracy of the computation. Corresponding deviations may be observed for small values of [ and 0 in Figs. 3, 4, and 6. The computation of the plots requires a considerable amount of computer time, which made it impossible to use greater accuracy. The authors want to thank Kuwait University for providing this computer time.

Fig. 4. The function E&, 0)/Et representing the magnitude of the electric field strength due to an electric exponential ramp function excitation 14, eq. (2.3-37)l. The Same Plot holds for the function HH(lr @/HI representing the magnitude of the magnetic field strength due to a magnetic exponential ramp function excitation 14, P. ‘031.

178 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 30, NO. 2, MAY 1988

a c .9

Fig. 5 . The function HE(( , O)Z/El representing the magnetic field strength due to an electric exponential ramp function excitation [4, eq. (2.9-28)].

Fig. 6. The function EHB. f3)/ZHl representing the electric field strength due to magnetic exponential ramp function excitation [4, eq. (2.10-31)].

observed frequency reduction of sinusoidal (light) pulses reaching us from great distances in the universe, and the distances inferred from the frequency shift may need correction.

physical results derivable from the modified Maxwell equations.

Comments on “Correction of Maxwell’s Equations for Signals I and 11”

We hope that this brief outline shows that there is no lack of R. K. ROSICH, SENIOR MEMBER, IEEE

REFERENCES

[l] M. J. Neatrour, “Comments on ‘Correction of Maxwell’s equations for signals I,’ ‘Correction of Maxwell’s equations for signals 11,’ and ‘Propagation velocity of electromagnetic signals,”’ IEEE Trans. Electromagn. Compat., vol. EMC-29, no. 3, pp. 258-259, Aug. 1987. H. F. Harmuth, R. N. Boules, and M. G. M. Hussain, “Response to comments by M. J. Neatrour on Maxwell’s equations,” IEEE Trans. Electromagn. Compat., vol. 30, no. 1, pp. 90-91, Feb. 1988. H. F. Harmuth, R. N. Boules, and J. E. Gray, “Second response to comments by M. J. Neatrour on Maxwell’s equations,” IEEE Trans. Electromagn.’ Compat., this issue, pp. 175-176. H. F. Harmuth, Propagation of Nonsinusoidal Electromagnetic Waves. New York: Academic, 1986. H. F. Harmuth, R. N. Boules, and M. G. M. Hussain, “Reply to Kuester’s comments on the use of a magnetic conductivity,” IEEE Trans. Electromagn. Compat., vol. EMC-29, no. 4, pp. 318-320, Nov. 1987. H. F. Harmuth, “Reply to comments on ‘Correction of Maxwell’s equations for signals I and 11’ by J. R. Wait,” IEEE Trans. Electromagn. Compat., vol. EMC-29, no. 3, p. 257, Aug. 1987.

[2]

[3]

[4]

[5]

[6]

Key Words-Lossy media, nonsinusoidal waves, pulse propagation. Index Code--J3d/e.

I have been following with fascination the written thrust and parry relative to the question of whether Maxwell’s equations need any “corrections” for pulse propagation in lossy media [l]. The discussions have brought forth many interesting points, and I believe that the electromagnetic side of the case has been very aptly presented by the reviewers and all of those who have so far chosen to comment. Consequently, I have nothing further of merit to add to this part of the discussion.

However, I do wish to take issue with Harmuth’s characterization [3] of K. Gadel’s work [2] on formal logic in his reply to the comments of J. R. Wait [ 5 ] . First, Harmuth’s remarks relative to

Manuscript received January 6, 1988. The author is with Hughes Aircraft Co., Denver, CO. Tel. (303) 793-5200. IEEE Log Number 8820092.

0018-9375/88/0500-0178$01 .OO 0 1988 IEEE