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, M E m E R , 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

IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 31, NO. 2, MAY 1989 203

electric field strength in lossy media due to an electric excitation function and derives solutions for the associated magnetic field strength [ l , eqs. (18) and (20)]. Computer plots of the derived solutions are then compared to the plots obtained by Harmuth, based on corrections of Maxwell’s equations [ 3 , fig. 1 1 , to show equality. Although the numerical results may appear identical, they do not prove equality (or inequality) of the derived analytical solutions of Harmuth and El-Shandwily.

In this correspondence, we shall show that the obtained solutions in [ 11 are incorrect, and the steps followed by El-Shandwily in deriving them only prove that Maxwell’s equations in their usual forms cannot provide the desired solutions for signal propagation in lossy media.

The solutions of the magnetic field strength H(&, 0) given in [ l , eqs. (18) and (20)] are derived from the following two integrals [ l , eqs. (15) and (16)]:

H ( & , 0 ) = -(l/G) so !!?&& d0 ~ = t d &

Here, x = 240 - ( ) / a , p is permeability, E is permittivity, U is conductivity, O is normalized time variable, [ is normalized space variable, and E ( t , 0) is electric field strength due to an electric excitation function of the normalized form A240 - &)/a ] . The controversy is whether one can solve ( I ) and (2) for a transient excitation, i.e., a unit step function S(t), without encountering divergence. El-Shandwily misinterpreted the properties of the unit impulse function, or Dirac-delta function ti(t), and obtained incorrect solutions for H(&, 0) based on ( 2 ) ; the major error in the derivations will be highlighted shortly.

The unit impulse 6(t) is defined by an assignment rule or process since it is not a function in the strict mathematical sense

sl: x(t) t i ( t ) dt=x(O), t1<0<t2

=0, otherwise (3)

where x(t) is any ordinary function that is continuous at the point t = 0. For x( t ) = 1 in the interval - 00 < t < 00, (3) results in

Sm t i( t) dt= Iff t i( t) dt= 1 -m -a

(4)

where a is of an infinitesimal value. The interpretation of (4) is that ti(t) has a unit area concentrated at the discrete point t = 0. The relationship between the unit step function S(t) and the unit impulse ti(t) follows from ( 3 ) and (4), and it is as follows [ 5 ] :

( 1 , t>O

S ( t ) = [ ‘ 6 ( u ) du= 0 , t < O - m I

( undefined, t = 0

and

El-Shandwily misinterprets the definition in (5 ) and claims that for a unit step functionf(t) = S(t) , the integration in (2) yields a unit step function, -S[2t(O - &)/a], and the initial valuef(0) = 0. This is not correct. According to (5) , for a unit step functionf(t) = S(t) and df( t ) /dt = 6(t) , the integration in (2) is UNdefined, since the lower limit of the integral is x = 0. Hence, the derived solutions in [ l ] are not correct solutions of Maxwell’s equations, and they cannot be equal to the ones derived by Harmuth [2] , [ 3 ] , which are based on the correction of Maxwell’s equations.

Recently, solutions of electric and magnetic field strengths in lossy media due to exponential ramp function and sinusoidal pulse excitations were published [6]-[8]. The plots obtained in [ l , figs. 3- 51 for the above excitations are not correct solutions of Maxwell’s equations and cannot be equal to the ones presented in [6]-[8]. For the exponential-ramp function

r ( t ) = ( l - e-‘IT)S(t) (7)

one obtains

Insertion of (8) into (2) yields an undefined solution for the integral according to (5). Hence, the plots in [ l , fig. 31 are not the correct solutions of Maxwell’s equations for the exponential-ramp function given in (7), and they cannot be equal to the solution presented in [6] and [7] .

Consider now the sinusoidal pulse

f(t)=sin 2ut/T, O 1 t 1 2 T

= 0 , t > 2 T (9)

which was used to obtain the plots in [ 1 , figs. 4 and 51. The sinusoidal pulse in (9) can be represented as

f ( t ) = [ S ( t ) - S ( t - T ) ] sin 2x t /T (10)

where S(t) is the unit step function defined in (5) . The derivative of (10) yields

d f O = [ 6 ( t ) - 6 ( t - T ) ] dt sin 2ut/T

+ ( 2 s / T ) [ S ( t ) - S ( t - T ) ] COS 2 ~ t / T . ( 1 1 )

Again, insertion of ( 1 1 ) into (2) results in an undefined solution for the integral. Since the lower limit of the integral in (2 ) is x = 0, the assignment rule given in ( 3 ) is not applicable in this case and would result in undefined solutions according to (5). Hence, the plots in [ l , figs. 4 and 51 are not correct solutions of Maxwell’s equations, and they cannot be equal to those presented in [ 8 ] .

One must now be convinced that it is not possible to use Stratton’s solution of the electric field strength in a lossy medium to obtain a solution for the associated magnetic field strength. The obstacle encountered in [l, eq. (16)] reveals why Stratton himself or anyone else did not derive the magnetic field strength from the available solution of the electric field strength, for the case of transients in lossy media.

The correction of Maxwell’s equations introduced by Harmuth is indeed necessary for the study of signal propagation in lossy media.

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

REFERENCES

M. E. El-Shandwily, “Solutions of Maxwell’s equations for general nonperiodic waves in lossy media,” IEEE Trans. Electrornagn. Compat., vol. 30, no. 4, pp. 571-582, Nov. 1988. H. F. Harmuth, “Correction of Maxwell’s equations for signals I.,” IEEE Trans. Electromagn. Compat., vol. EMC-28, no. 4, pp. 250- 258, Nov. 1986. -, “Correction of Maxwell’s equations for signals II.,” ZEEE Trans. Electromagn. Compat., vol. EMC-28, no. 4, pp. 259-266, Nov. 1986. J. A. Stratton, Electromagnetic Theory. New York: McGraw-Hill, 1941. R. E. Ziemer and W. H. Tranter, Principles of Communications, 2nd ed. Boston: Houghton Mifflin, 1985. M. G. M. Hussain, “General solutions of Maxwell’s equations for signals in a lossy medium: I. Electric and magnetic field strengths due to electric exponential ramp function excitation,” IEEE Trans. Electromagn. Compat., vol. 30, no. 1, pp. 29-36, Feb. 1988. -, “General solutions of Maxwell’s equations for signals in a lossy medium: II. Electric and magnetic field strengths due to magnetic exponential ramp function excitation,” ZEEE Trans. Electromagn. Compat., vol. 30, no. 1, pp. 37-40, Feb. 1988. -,“General solutions of Maxwell’s equations for signals in a lossy medium: III. Electric and magnetic field strengths due to electric and magnetic sinusoidal pulse excitation,” IEEE Trans. Electromagn. Compat., vol. 30, no. 1, pp. 41-47, Feb. 1988.

Correction to “Relation Between Equivalent Antenna Radius and Transverse Line Dipole Moments of a

Narrow Slot Aperture Having Depth”

LARRY K. WARNE AND KENNETH C. CHEN, MEMBER, IEEE

Equation (50) of [l] should read

REFERENCES

[ l ] L. K. Warne and K. C. Chen, “Relation between equivalent antenna radius and transverse line dipole moments of a narrow slot aperture having depth,” ZEEE Trans. Electromagn. Compat., vol. 30, no. 3, pp. 364-370, Aug. 1988.

Manuscript received December 2, 1988. The authors are with Division 7553, Sandia National Laboratories,

IEEE Log Number 8926401. Albuquerque, NM 87185.

I