General solutions of Maxwell's equations for signals in a lossymedium. I. Electric and magnetic field strengths due to electricexponential ramp function excitation

Modification of Maxwell's equations to obtain general solutions for a lossy medium is reviewed. It is done by adding an extra term, referred to as the fictitious magnetic charge density. The solutions, which are in integral form, are solved numerically by computer for an exponential ramp function excitation. Computer plots for the electric and magnetic field strengths as functions of time at different locations in a lossy transmission medium are presented     

Correction of Maxwell's Equations for Signals II

In the first of two companion papers it was shown that the addition of a magnetic current density to Maxwell's equations is a sufficient condition to obtain solutions in lossy propagation media for waves that are not infinitely extended periodic waves. The solutions obtained represented transients that may be used to represent signals having a beginning and an end. This second paper shows that the addition of a magnetic current density is also a necessary condition for the existence of transient solutions in lossy media. The modification of Maxwell's equations is thus necessary and sufficient for the study of the propagation of signals in lossy media.     

Solutions of Maxwell's equations for general nonperiodic waves inlossy media

Solutions of Maxwell's equations in lossy media for signals excited by a general applied source at the boundary plane are given. The excitation at the boundary plane can be through either electric or magnetic functions of any general time variation. No additional terms need be added to Maxwell's equations to obtain the solutions. Excitations by an electric step, exponential, and finite duration sinusoidal; functions of time are given as examples     

Remark on Harmuth's `Correction of Maxwell's equations for signals.I' [and response]

The commenter maintains that the claims made by H.F. Harmuth in the above-titled paper (ibid., vol.EMC-28, p.250-8, Nov. 1986) that a satisfactory condition for the existence of solutions for transients in lossy media is the modification of Maxwell's equations by the addition of a magnetic current density is not valid. In his reply, Harmuth states that if the commenter's claim holds true regardless of the method of solution, he will have contributed an important simplification to the problem of transient solutions of Maxwell's equations. He provides further discussion of the commenter's point     

On Harmuth's `correction' of Maxwell's equations for pulsetransmission in lossy media

It is shown that H.F. Harmuth's magnetic conductivity term is not needed to predict the transient response for plane-wave transmission in a homogeneous lossy medium. Thus, his repeated harsh criticisms of S.A. Stratton's classic analysis using Laplace transforms are not justified     

General solutions of Maxwell's equations for signals in a lossymedium. III. Electric and magnetic field strengths due to electric andmagnetic sinusoidal pulse excitation

For pt.II see ibid., vol.30, no.1, p.37-40 (1988). The representation of a function with a general time variation by a series expansion of time-shifted transients is discussed. On the basis of this representation, numerical solutions of Maxwell's equations are presented for the electric and magnetic field strengths in a lossy medium due to electric and magnetic excitation functions consisting of a finite number of sinusoidal cycles. The solutions are derived by means of a time-series expansion of the available solutions for the electric and magnetic exponential ramp function excitations     

Signal solutions of Maxwell's equations for charge carriers withnon-negligible mass

Discusses signal solutions to Maxwell's equations for charge carriers with non-negligible mass. In order to find solutions the authors add information to Maxwell's equations by means of a physical assumption to obtain a defined solution. The authors' assumption is that magnetic dipoles and magnetic dipole currents should be represented by a magnetic (dipole) current density term just as electric dipoles and electric dipole currents-or electric polarization currents-have always been represented by an electric current density term. It is perfectly possible that other physical assumptions can be made that yield defined solutions and that will withstand public scrutiny     

General solutions of Maxwell's equations for signals in a lossymedium. II. Electric and magnetic field strengths due to magneticexponential ramp function excitation

For pt.I see ibid., vol.30, no.1, p.29-36 (1988). Solutions for the electric and magnetic field strengths in a lossy medium due to a magnetic exponential ramp function excitation are presented. The solutions are in integral form and are evaluated by numerical integration methods using a digital computer. Computer plots for the electric and magnetic field strengths at different locations in the propagation medium are given. The plots obtained for the transients can be used to represent solutions in lossy media for signals that can be represented in terms of a time-series expansion of the transients     

Comments on `Riemann-Green function solution of transientelectromagnetic plane waves in lossy media'

In commenting on the above-named work by O.R. Asfar (see ibid., vol.EMC-32, no.3, p.228-31, Aug. 1990), the commenter notes that one can write infinitely many solutions for the associated magnetic field strength that will all satisfy Maxwell's equations, but Maxwell's equations cannot tell which one of these infinitely many solutions is the right one. It is further pointed out that the physical significance of the magnetic current density term used became clear when transients in lossy media were investigated with Lorentz's equations of electron theory, which allow for the fact that electric charges are always connected with particles having a mass, whereas Maxwell's original equations do not contain the concept of mass. A physical explanation for this is offered, and attention is given to the creation of the singularity in Maxwell's equations that make sit impossible to obtain the associated magnetic field strength without some limit process     

Integral equation solution of Maxwell's equations from zerofrequency to microwave frequencies

We develop a new method to precondition the matrix equation resulting from applying the method of moments (MoM) to the electric field integral equation (EFIE). This preconditioning method is based on first applying the loop-tree or loop-star decomposition of the currents to arrive at a Helmholtz decomposition of the unknown currents. However, the MoM matrix thus obtained still cannot be solved efficiently by iterative solvers due to the large number of iterations required. We propose a permutation of the loop-tree or loop-star currents by a connection matrix, to arrive at a current basis that yields a MoM matrix that can be solved efficiently by iterative solvers. Consequently, dramatic reduction in iteration count has been observed. The various steps can be regarded as a rearrangement of the basis functions to arrive at the MoM matrix. Therefore, they are related to the original MoM matrix by matrix transformation, where the transformation requires the inverse of the connection matrix. We have also developed a fast method to invert the connection matrix so that the complexity of the preconditioning procedure is of O(N) and, hence, can be used in fast solvers such as the low-frequency multilevel fast multipole algorithm (LP-MLFMA). This procedure also makes viable the use of fast solvers such as MLFMA to seek the iterative solutions of Maxwell's equations from zero frequency to microwave frequencies     

A vector diagram of Maxwell's equations

A vector diagram illustrating Maxwell's equations is derived. Upon equating various components of vectors in the diagram, a number of common relationships between field quantities are obtained. The potentials and their relationships to field quantities may also be represented. Justification of the procedure used to construct the diagram is based on Fourier transformation of Maxwell's equations. While the diagram is mostly of interest for its novelty, it has found use in clarifying certain gauge choices in dealing with electromagnetic potentials     

Comments on `Some comments on Harmuth and his critics' by J.E. Grayand S.P. Bowen

J.E. Gray and S.P. Bowen (see ibid., vol.30, no.4, p.586-9, Nov. 1988) claim to have developed the formalism necessary to solve the propagation of pulses in a lossy medium for both the magnetic and electric fields using the Laplace transformation and generalized functions. They claim that Harmuth's introduction of the magnetic current s, is neither necessary nor a sufficient reason to insure consistency and that their method permits calculating both the electric and magnetic fields uniquely for a wide variety of pulses. The commenter claims, however, that Gray and Bowen introduce covert assumptions that deny their claim     

Finite element implementation of Maxwell's equations for image reconstruction in electrical impedance tomography

Traditionally, image reconstruction in electrical impedance tomography (EIT) has been based on Laplace's equation. However, at high frequencies the coupling between electric and magnetic fields requires solution of the full Maxwell equations. In this paper, a formulation is presented in terms of the Maxwell equations expressed in scalar and vector potentials. The approach leads to boundary conditions that naturally align with the quantities measured by EIT instrumentation. A two-dimensional implementation for image reconstruction from EIT data is realized. The effect of frequency on the field distribution is illustrated using the high-frequency model and is compared with Laplace solutions. Numerical simulations and experimental results are also presented to illustrate image reconstruction over a range of frequencies using the new implementation. The results show that scalar/vector potential reconstruction produces images which are essentially indistinguishable from a Laplace algorithm for frequencies below 1 MHz but superior at frequencies reaching 10 MHz.     

Third response to comments by M.J. Neatrour on Maxwell's equations

For original papers by Harmuth see ibid., vol.28, no.4, p.250, p.259, p.267 (1986). For Neatrour's article see ibid., vol.29, no.3, p.258 (1987). The question of whether the authors' previous modification of Maxwell's equations adds anything new to physics is addressed. Several plots of electric field strength under various conditions are presented, of which the authors claim about half could not have been derived from Maxwell's original equations. Since they claim these plots are accurate, the modified equations do, indeed, add something to physics     

Comments on `Solutions of Maxwell''s equations for generalnonperiodic waves in lossy media by M.E. El-Shandwily

The commenter notes that the intention of M.E. El-Shandwily's paper (see ibid., vol.30, no.4, p.577-82, Nov. 1988) is to demonstrate that the linear Maxwell's equations can be applied to the case of a pulse or step-function field change. If this can be demonstrated, then, it is claimed, the Harmuth Ansatz (see ibid., vol.EMC-28, no.4, p.250-8, Nov. 1986) for solving Maxwell's equations to obtain the magnetic field for impulse excitation is unnecessary. The commenter seeks to show that El-Shandwily actually implements the Harmuth Ansatz, which explains the concurrence of Harmuth's and El-Shandwily's predictions     

Third response to Kuester's comments on Maxwell's equations

Additional comments on an ongoing controversy on Maxwell's equations are presented. Kuester's (see ibid., vol.EMC-29, no.2, p.187, 1987), relation for the magnitude of the electric field strength for a planar TEM wave with any polarisation excited by an electric step function is discussed     

Unconditionally stable Crank-Nicolson scheme for solving two-dimensional Maxwell's equations

The Crank-Nicolson method is an unconditionally stable, implicit numerical scheme with second-order accuracy in both time and space. When applied to solve Maxwell's equations in two-dimensions, the resulting matrix is block tri-diagonal, which is very expensive to solve. The Douglas-Gunn algorithm is used to subdivide the update procedure into two sub-steps. At each sub-step only a tri-diagonal matrix needs to be solved for one field component. The other two field components are updated explicitly in one step. The numerical dispersion relations are given for the original Crank-Nicolson scheme and for the Douglas-Gunn modification. The predicted numerical dispersion is shown to agree with numerical experiments, and its numerical anisotropy is shown to be much smaller than that of the ADI-FDTD.     

Frequency dependence of scattering and extinction of dense media based on three-dimensional simulations of Maxwell's equations with applications to snow

The frequency dependence of scattering by geophysical media at microwave frequencies is an important topic because multifrequency measurements are used in remote sensing applications. In this paper, we study rigorously the frequency dependence of scattering by dense media using Monte Carlo simulations of the three-dimensional solutions of Maxwell's equations. The particle positions are generated by deposition and bonding techniques. The extinction, scattering, and absorption properties of dense media are calculated for dense media of sticky and nonsticky particles. Numerical solutions of Maxwell's equations indicate that the frequency dependence of densely packed sticky small particles are much weaker than that of independent scattering. Numerical results are illustrated using parameters of snow in microwave remote sensing. Comparisons are made with extinction measurements as a function of frequency.     

In defense of J.A. Stratton [pulse transmission in lossy media]

The author shows that Maxwell's equations can be employed to predict pulse transmission in lossy media. He concludes that Harmuth and company are wrong in stating that Stratton's classic treatise contains a `fundamental misunderstanding' in the application of the Laplace transform technique     

A technique for solution of Maxwell's equations in a moving dielectric medium

A simple technique is presented for converting a known solution for the electric and magnetic vector fields in a dielectric medium at rest into the corresponding fields in a moving dielectric medium. The technique combines methods presented by Tai [1] with a scaling procedure developed by Clemmow [2]. Tai's work reduces the moving medium problem to the solution of Maxwell's equations in a uniaxial medium, and Clemmow's procedure enables one to convert a known solution in an isotropic medium to the corresponding solution in a uniaxial medium. Thus by first solving for the fields in the medium at rest, then following Clemmow's procedure to obtain the fields in Tai's uniaxial medium, and finally applying Tai's reasoning, one may easily obtain the solution of Maxwell's equations in the moving medium.