202 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 31, NO. 2. MAY 1989

is introduced, then [4, p. 1931

u,,-c2u,- [(CY - p)/2)2u =o (14)

is obtained. These authors caution that this equation represents the dispersionless case if and only if

C Y = & (15)

These authors state [4, p. 1931

“If condition (15) holds, the telegraph equation possesses damped, yet “relatively” undistorted, progressing wave solutions of the form

U =exp [ (1/2)(a+ b ) t ] f (xrt ct)

with arbitrary f, progressing in both directions of the cable.”

However, it is not necessary that CY = p. Therefore, procedure I requires justification. In effect, Gray and Bowen have arbitrarily “preprocessed” Maxwell’s equations, before Laplace transforma- tion. A simple Laplace transformation method this decidedly is not.

The central issue addressed by the Harmuth procedure [2], [3] is not whether linear Maxwell theory can be used together with mathematical “preprocessing” in order to leave the linear Maxwell theory unchanged no matter what. It is, rather, whether the linear Maxwell’s equations are the simplest representation of electromag- netic fields under certain boundary conditions.

Maxwell’s theory, in one sense, is a theory of transformation rules. In this sense, given an input field to a system or another field, (even if this input is considered a distribution rule), the question is: are the linear Maxwell equations necessary and sufficient to describe the output from that transforming system or field? If not, then modified Maxwell equations must be formulated for the particular boundary conditions considered. The question is not: given the absolute unalterability of the linear Maxwell equations under all conditions, can the input be preprocessed in any way possible so that the absolute unalterability is upheld? For if it were, why stop with a special case assumption and the Laplace transform? Any cascade of assumptions and transforms will accomplish the same aim.

The real point is that Occam’s razor must apply to the search for the transformation rules described by Maxwell’s equations, whether linear or nonlinear. The correct equations must stand unaided as a self-sufficient set, not as a set among a series of others unjustified. As it is with any scientific theory or model, ad hoc mathematical hybridicity is ruled out.

Therefore, the claim [ l , p. 5871 that

“. . . with distributions as part of the solution space and the integrals interpreted in the sense of Lebesgue, the modifying s is unnecessary.”

is both disarming in its frankness concerning the mathematically incorrect mixing of distributions and functions, and, because of the observations above, unjustified in its conclusion.

REFERENCES

J. E. Gray and S. P. Bowen, “Some comments on Harmuth and his critics,” IEEE Trans. Electromagn. Compat., vol. 30, no. 4, pp.

H. F. Harmuth, “Corrections of Maxwell’s equations for signals I,” IEEE Trans. Electromagn. Cornpat., vol. EMC-28, no. 4, pp. 250- 256, Nov. 1986. -, “Corrections of Maxwell’s equations for signals U,” IEEE Trans. Electromagn. Cornpat., vol. EMC-28, no. 4, pp. 259-265, Nov. 1986. R. Courant and D. Hilbert, Methods of Mathematical Physics, vol. II.

586-589, NOV. 1988.

New York: Interscience, 1953, pp. 192-193.

Comments on Wait’s “In Defense of J. A. Stratton”

MALEK G. M. HUSSAIN, MEMBER, IEEE

The problem of obtaining general solutions of electrical and magnetic field strengths for a transient planewave propagation in a lossy medium is not a trivial one. The Laplace transform technique fails in providing general solutions of Maxwell’s equations when applied to the above problem. “For a given physical situation,” general solutions of Maxwell’s equations must define the produced fields at the source, not only away from the source as given in [ 1, eq. (4)]. In other words, initial and boundary conditions must be satisfied if an excitation function exists. According to [l, eq. (4)], H,(O, t ) is undefined for a transient excitation function, e.g., unit step function, exponential ramp function, sinusoidal pulse, etc. Detailed discussions related to this issue are given in [2].

Since the solution H,( y , t ) given [ l , eq. (4)] does not satisfy the physical condition H,(O, t ) = finite, it cannot be regarded as a general solution. However, one may consider it as excellent “evidence” of why “Maxwell’s equations cannot be employed to predict pulse transmission in lossy media.” The question that needs a definite answer is: Did Stratton consider the problem of obtaining solutions of the complete electromagnetic fields for transients in lossy media trivial?

REFERENCES

[ I ] J. R. Wait, “In defense of J. A. Stratton,” IEEE Trans. Electro- magn. Cornpat., vol. 30, no. 4, p. 590, Nov. 1988.

[2] M. G. M. Hussain, “Comments on ‘Solutions of Maxwell’s equations for general nonperiodic waves in lossy media,’ ” IEEE Trans. Electromagn. Cornpat., vol. 31, no. 2, pp. 202-204, May 1989.

Manuscript received December 2, 1988. The author is with the Radiation Laboratory, Department of Electrical

Engineering and Computer Science, The University of Michigan, Ann Arbor, MI 48109.

IEEE Log Number 8926412.

Comments on “Solutions of Maxwell’s Equations for General Nonperiodic Waves in Lossy Media’’

MALEK G. M. HUSSAIN, MEmER, IEEE

The objective of El-Shandwily’s paper [l] is to show that the correction of Maxwell’s equations introduced by Harmuth [2], [3], for the propagation of signals in lossy media, is unnecessary. El- Shandwily starts with the solution of Stratton [4, sec. 5.131 for

Manuscript received December 2, 1988. The author is with the Radiation Laboratory, Department of Electrical

Engineering and Computer Science, The University of Michigan, Ann Arbor, MI 48109.

IEEE Log Number 8926415.

OO18-9375/89/0500-0202$01 .OO 0 1989 IEEE