Henning F. Harmuth’s research while affiliated with Catholic University of America and other places

The direct numerical evaluation of Eq.(2.4-45) by computer runs into difficulties. To recognize the reason we list in Tables 6.1-1 typical values of λ1, λ2, λp1, and λp2. The magnitude of λl and λ2 lies between 1 and 2.51 x 10-3 for p ≥ 0.05 but the magnitude of c pλp1 for ŋ in the range from 1 to 100 and c p = 1.98 x 10-12 is in the order of 10-14 to 10-19 and c pλp2 is even smaller except when p is very close to 0.5. The computer must produce small differences of numbers of the order of 1020, which requires too much precision. Our way around this difficulty uses a series expansion in powers of c p.

We start with the modified Maxwell equations without electric and magnetic charges, having constant scalars for the permeability µ and the permittivity ϵ. The following equations apply to a medium at rest: $curlH = \in \frac{{\partial E}}{{\partial t}} + {{g}_{e}}$ (1)$- curlE = \mu \frac{{\partial H}}{{\partial t}} + {{g}_{m}}$ (2)$epsilon divE = \mu divH = 0$ (3)$\epsilon E = D,\mu H = B$ (4) We restrict the investigation to currents that are carried either by electric or magnetic dipoles and avoid any monopole currents. This assumption will be ideally satisfied in a medium with atomic hydrogen, if any ionization can be neglected. For the electric dipole current density we use Eq.(1.2-11), writing the partial derivative ∂ge /∂t since the current density may now be a function of location as well as of time, \begin{array}{*{20}{c}} {{{g}_{e}} + {{\mathcal{T}}_{{mp}}}\frac{{\partial {{g}_{e}}}}{{\partial t}} + \frac{{{{\mathcal{T}}_{{mp}}}}}{{\mathcal{T}_{p}^{2}}}\int {{{g}_{e}}dt = {{\sigma }_{p}}E} } \\ {{{\sigma }_{p}} = \frac{{{{N}_{0}}{{e}^{2}}{{\mathcal{T}}_{{mp}}}}}{m}} \\ \end{array} (5) but the use of Eqs.(1.3-26), (1.3-29), (1.3-31), and (1.3-32) for the magnetic dipole current density gm(t) runs into the difficulty of containing the function sin ϑ. There is little hope of finding an analytical solution of Maxwell’s modified equations (1)–(4) if the magnetic dipole current density gm is defined by a transcendental differential equation. We get around this difficulty by observing that the plots of Figs.1.3-3 and 1.3-5 for a hypothetical magnetic charge dipole are similar to the plots of Figs.1.3-11 and 1.3-10 for a bar magnet dipole.

In Section 2.5 we produced plots of the electric field strength for electric excitation functions with the time variations shown in Fig.2.5-1. The jump at θ = 0 makes these excitation functions less than desirable for physics. We had to use the functions according to Eq.(2.2-2) \begin{array}{*{20}{c}} {{{E}_{E}} = 0} & {for} & {\theta < 0} \\ { = {{E}_{{0e}}}^{{ - \theta r/pq}}} & {for} & {\theta 0} \\ \end{array} (1) for excitation since they are eigenfunctions of the differential equation Eq.(2.2-28). Two of the undesirable excitation functions according to Eq.(1) can be combined to a double exponential ramp function according to Fig.2.2-2 that avoids the jump at θ = 0: \begin{array}{*{20}{c}} {{{E}_{E}}\left( {0,\theta } \right) = 0} & {for} & {\theta < 0} \\ { = {{E}_{0}}\left( {{{e}^{{ - \theta r/pq}}} - {{e}^{{ - \theta s/pq}}}} \right)} & {for} & {\theta 0,s > r} \\ \end{array} (2)

Consider Maxwell’s equations for a medium at rest with scalar constants for the permeability µ permittivity ϵ and conductivity σ. Furthermore, we have the electric and magnetic field strengths E and H, electric charge and current density ρ e and g e as well as electric and magnetic flux densities D and B. Using the International System of units we get:

We rewrite Eqs.(2.1-45) and (2.1-60) in normalized form for the derivation of the associated magnetic field strength. First we get with the help of Eqs.(2.1-32) and (2.1-48): \begin{array}{*{20}{c}} {\frac{{{{\tau }_{{1m}}}{{\tau }_{{2m}}}}}{{{{\tau }_{{2m}}} - {{\tau }_{{1m}}}}} = - \frac{{{{\tau }_{{mp}}}}}{{{{{\left( {1 - 8\alpha '{{\tau }_{{mp}}}} \right)}}^{{{{1} \left/ {2} \right.}}}}}}} \\ {{{\tau }_{{1m}}} = {{\tau }_{{mp}}}\frac{{1 + {{{\left( {1 - 8{{{\alpha '}}_{1}}{{\tau }_{{mp}}}} \right)}}^{{{{1} \left/ {2} \right.}}}}}}{{4{{{\alpha '}}_{1}}{{\tau }_{{mp}}}}},{{\tau }_{{2m}}} = {{\tau }_{{mp}}}\frac{{1 + {{{\left( {1 - 8{{{\alpha '}}_{1}}{{\tau }_{{mp}}}} \right)}}^{{{{1} \left/ {2} \right.}}}}}}{{4{{{\alpha '}}_{1}}{{\tau }_{{mp}}}}}} \\ {t/{{\tau }_{{1m}}} = {{{v'}}_{1}}\theta /pq,t/{{\tau }_{{2m}}} = {{{v'}}_{2}}\theta /pq,\theta = t/\tau ,p = {{\tau }_{{mp}}}/{{\tau }_{p}},q = {{\tau }_{p}}/\tau } \\ {{{{v'}}_{1}} = \frac{{4{{{\alpha '}}_{1}}{{\tau }_{{mp}}}}}{{1 + {{{\left( {1 - 8{{{\alpha '}}_{1}}{{\tau }_{{mp}}}} \right)}}^{{{{1} \left/ {2} \right.}}}}}},{{{v'}}_{2}} = \frac{{4{{{\alpha '}}_{1}}{{\tau }_{{mp}}}}}{{1 - {{{\left( {1 - 8{{{\alpha '}}_{1}}{{\tau }_{{mp}}}} \right)}}^{{{{1} \left/ {2} \right.}}}}}}} \\ {\frac{{{{\tau }_{{3m}}}{{\tau }_{{4m}}}}}{{{{\tau }_{{3m}}} - {{\tau }_{{4m}}}}} = - \frac{{{{\tau }_{{mp}}}}}{{{{{\left( {1 - 8{{{\alpha '}}_{2}}{{\tau }_{{mp}}}} \right)}}^{{{{1} \left/ {2} \right.}}}}}}} \\ {{{\tau }_{{3m}}} = {{\tau }_{{mp}}}\frac{{1 + {{{\left( {1 - 8{{{\alpha '}}_{2}}{{\tau }_{{mp}}}} \right)}}^{{{{1} \left/ {2} \right.}}}}}}{{4{{{\alpha '}}_{2}}{{\tau }_{{mp}}}}},{{\tau }_{{4m}}} = {{\tau }_{{mp}}}\frac{{1 + {{{\left( {1 - 8{{\alpha }_{2}}{{\tau }_{{mp}}}} \right)}}^{{{{1} \left/ {2} \right.}}}}}}{{4{{\alpha }_{2}}{{\tau }_{{mp}}}}}} \\ {t/{{\tau }_{{3m}}} = {{v}_{3}}\theta /pq,t/{{\tau }_{{4m}}} = {{v}_{4}}\theta /pq,\theta = t/\tau ,p = {{\tau }_{{mp}}}/{{\tau }_{p}},q = {{\tau }_{p}}/\tau } \\ {{{v}_{3}} = \frac{{4{{\alpha }_{2}}{{\tau }_{{mp}}}}}{{1 + {{{\left( {1 - 8{{\alpha }_{2}}{{\tau }_{{mp}}}} \right)}}^{{{{1} \left/ {2} \right.}}}}}},{{{v'}}_{4}} = \frac{{4{{\alpha }_{2}}{{\tau }_{{mp}}}}}{{1 - {{{\left( {1 - 8{{\alpha }_{2}}{{\tau }_{{mp}}}} \right)}}^{{{{1} \left/ {2} \right.}}}}}}} \\ \end{array} ((1)) KeywordsField StrengthMagnetic Field StrengthElectric Field StrengthIntegration ConstantTime FunctionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Most of the information we have obtained about the universe beyond the planet Earth came from electromagnetic waves, and most of the information obtained from electromagnetic waves came from steady state waves. Whenever received electromagnetic waves are decomposed into sinusoidal waves with various frequencies one thinks and acts in terms of steady state wave theory. The Doppler shift of a frequency—implied every time the term red-shift is used—is a steady state concept even though the Doppler effect is not. A frequency spectrum of received waves is a steady state concept.

We start with the modified Maxwell equations of Section 1.1. They were written in Section 2.1 for Cartesian coordinates and planar TEM waves. The polarization was extracted by Eqs.(2.1–16), (2.1–17) and Eqs.(2.1–18), (2.1–19) were obtained. We can use these equations here but we leave out the indices i and 1 since we consider waves in one medium rather than at the boundary of two mediums:$\frac{{\partial E}}{{\partial y}} + \mu \frac{{\partial H}}{{\partial t}} + sH = 0$ \frac{{\partial E}}{{\partial y}} + \in \frac{{\partial E}}{{\partial t}} + \sigma H = 0

In Section 1.7 we discussed electromagnetic missiles or electromagnetic waves with moving focal points in terms of ray optics. We want to extend the analysis here to the wave theory

Consider an electromagnetic wave excited in the plane of excitation in Fig2.1-1, beginning at the time t = t 0 The ‘beginning’ or front (plane) of the wave reaches the origin of a Cartesian coordinate system with the axes x, y, z at the time t = 0. The x-axis is assumed to be in the wave front and to stick out perpendicularly from the paper plane. The plane z = 0 is the boundary between two media characterized by the parameters εl, µl, σ1, s1 and ε2, µ2, σ2, s2. The angle of incidence between the boundary plane and the wave front is denoted υ2 in Fig2.1-1. All points y ≤ 0 of the boundary plane have been reached at a time t≤ 0 by the wave, while none of the points y > 0 have been reached yet at the time t = O. A reflected cylinder wave Cr is excited at the boundary plane z = 0 for any value y ≤ 0 in medium 1 and a transmitted cylinder wave Ct in medium 2. Fig2.1-1 is the basic illustration for the reflection and transmission of an incident planar wave. We will see in Section 2.2 that this illustration can be modified by the introduction of a signal boundary to produce a model that is better suited for waves that represent signals.