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Classical and extended electrodynamics
Lee M. Hively
1,a)
and Andrew S. Loebl
2
1
4947 Ardley Drive, Colorado Springs, Colorado 80922, USA
2
9325 Briarwood Blvd., Knoxville, Tennessee 39723, USA
(Received 21 November 2018; accepted 8 February 2019; published online 25 February 2019)
Abstract: Classical electrodynamics is modeled by Maxwell’s equations, as a system of eight
scalar equations in six unknowns, thus appearing to be overdetermined. The no-magnetic-
monopoles equation can be derived from the divergence of Faraday’s law, thus reducing the number
of independent equations to seven. Derivation of Gauss’ law requires an assumption beyond
Maxwell’s equations, which are then overdetermined as seven equations in six unknowns. This
overdetermination causes well-known inconsistencies. Namely, the interface matching condition
between two different media is inconsistent for a surface charge and surface current. Also, the irrota-
tional component of the vector potential is gauged away, contrary to experimental measurements.
These inconsistencies are resolved by extended electrodynamics (EED), as a provably unique system
of 7 equations in 7 unknowns. This paper provides new physical insights into EED, along with
preliminary experimental results that support the theory. V
C2019 Physics Essays Publication.
[http://dx.doi.org/10.4006/0836-1398-32.1.112]
R
esum
e: L’
electrodynamique classique (CED) est mod
elis
ee par les
equations de Maxwell,
comme un syste`me de 8
equations scalaires dans 6 inconnues, apparaissant ainsi
a^
etre
surd
etermin
ee. L’
equation des monop^
oles non-magn
etiques peut ^
etre d
eriv
ee de la divergence de la
Loi de Faraday, r
eduisant ainsi le nombre d’
equations ind
ependantes
a sept. La d
erivation de la Loi
de Gauss exige une hypothe`se au-del
a des
equations de Maxwell, qui sont alors surd
etermin
ees
comme sept
equations dans six inconnues. Cette surd
etermination provoque des incoh
erences bien
connues.
A savoir, la condition de correspondance d’interface entre deux moyens diff
erents est
incoh
erente pour une charge de surface et un courant de surface. De plus, la composante irrotation-
nelle du potentiel vectoriel ne peut ^
etre mesur
ee, contrairement aux mesures exp
erimentales. Ces
incoh
erences sont r
esolues par l’
electrodynamique
elargi (EED), comme un syste`me provablement
unique de 7
equations dans 7 inconnues. Cet article fournit des nouvelles id
ees physiques sur
l’EED, ainsi que des r
esultats exp
erimentaux pr
eliminaires qui appuient la th
eorie.
Key words: General Physics; Classical Field Theory; Electromagnetism; Extended Electrodynamics.
I. INTRODUCTION
Classical electrodynamics (CED) is the cornerstone of
modern physics. CED provides the foundation for models
of nonlinearity, chaos, complexity, and statistical effects
in electrodynamic systems; bioelectrodynamics; polymers;
plasmas; conductive fluid dynamics; piezo-electric and
piezo-magnetic solids; and computational models thereof.
Maxwell
1
wrote CED as 20 partial differential equations
in Cartesian coordinates. Heaviside
2
rewrote Maxwell’s
equations in vector calculus form with solutions in terms of
the scalar (U) and vector (A) potentials. Lorenz
3
recognized
that wave equations for Uand Acan be obtained via an
additional constraint (the Lorenz gauge). However,
the Lorenz gauge does not completely eliminate the
arbitrariness in CED, which allows an infinitude of gauge
transformations.
4–6
This paper is inspired by the apparent overdetermination
in CED and is organized as follows. Section II summarizes
work on CED overdetermination, which is improperly
“resolved” by flawed circular logic and unwarranted assump-
tions. Section III elucidates CED inconsistencies: disparities
in the interface matching conditions and gauging away of
the gradient component of the vector potential. Section IV
introduces extended electrodynamics
7
(EED), together with
previous work on EED. Section Vgives basic EED
predictions: compatibility with CED, resolution of the incon-
sistencies in Section III, and charge balance in the classical
limit. Section VI explains EED prediction of the scalar-
longitudinal wave (SLW). Section VII derives the EED
conditions for the scalar wave. Section VIII describes the
revised energy and momentum balance equations under
EED. Section IX discusses preliminary experimental evi-
dence for the SLW: no constraint by the skin effect, 1=r2
attenuation in free space, and isotropic SLW transmission
from a monopole antenna. Section Xprovides testable pre-
dictions and discussion. Section XI has our conclusions.
a)
lee.hively314@comcast.net.us
ISSN 0836-1398 (Print); 2371-2236 (Online)/2019/32(1)/112/15/$25.00 V
C2019 Physics Essays Publication112
PHYSICS ESSAYS 32, 1 (2019)
II. CED OVERDETERMINATION
The vector-calculus form of Maxwell’s equations is
8
r•E¼q
e;(1)
r•B¼0;(2)
rE¼@B
@t;(3)
rBle @E
@t¼lJ:(4)
SI units are used. Bold symbols denote vectors; Band Eare
the magnetic and electric field vectors, respectively. The
source terms are the electric charge density ðqÞand the elec-
trical current density ðJÞ;eand lare the electrical permittiv-
ity and magnetic permeability, respectively (not necessarily
vacuum). Time is denoted by t:Equations (1)–(4) seem over-
determined, involving six unknowns (three scalar compo-
nents for each of Band EÞand eight equations [one from
each of Eqs. (1) and (2), and three from each of Eqs. (3)
and (4)].
Stratton
9
introduced the divergence-curl redundancy to
resolve the overdetermination in CED. Namely, the diver-
gence of Eq. (3) yields
10
r•rE¼0:Then, the partial-
time derivative of r•Bis zero, which has the solution as
r•B¼FðrÞ;where F is an arbitrary scalar function of spa-
tial location ðrÞ:The only physically meaningful result is
F¼0;as in Eq. (2). The divergence of Eq. (4) uses
10
r•rB¼0 to obtain the form
r•E¼1
eðdtr•J:(5)
Stratton
9
then assumed charge conservation, allowing
replacement of r•Jin Eq. (5) with the partial-time deriva-
tive of q;to obtain Eq. (1). While this assumption seems
reasonable, Section Vdescribes the first-principles, prov-
ably unique derivation of EED
7
that yields a modified form
of charge balance. Moreover, charge balance is typically
derived by substituting Eq. (1) into the divergence of
Eq. (4). Liu
11
noted this circular logic fallacy, but assumed
second derivatives in Eqs. (1)–(4) to resolve the overdeter-
mination. Arminjon
12
also addressed this problem, relying
on the well-known approach of adding ad hoc constraint(s)
or dummy variable(s) to the formulation to avoid charge
nonconservation.
13–17
These assumptions
9,11,17
hide the fact
that CED is overdetermined, as Sousa and Shumlak
17
explicitly state.
III. INCONSISTENCIES IN CED
Equations (1) and (2) have well-known solutions,
8
where
Aand Uare the electrodynamic vector and scalar potentials,
respectively,
B¼rA;(6)
E¼rU@A
@t:(7)
Substitution of Eqs. (6) and (7) into Eqs. (1) and (4), and
use of the Lorenz gauge, yields the Uand Awave
equations
8
r2U@2U
@c2t2¼q
e;(8)
r2A@2A
@c2t2¼lJ:(9)
Here, the speed of light is c¼ðelÞ1=2(not necessarily
in vacuum). Equation (1) has an interface matching
condition
8
e2E2ne1E1n¼qA:(10)
Substitution of Eq. (7) into Eq. (10) yields
e2rU@A
@t
2ne1rU@A
@t
1n¼qA:(11)
The subscript, “n,” denotes the normal component. The
subscripts “1” and “2” identify the two media. Equation (8)
also has a matching condition by taking a Gaussian pill box
with the end faces parallel to the interface in regions 1 and 2.
Noting that r2U¼r•rU, one can use the divergence the-
orem in the limit of zero pill-box height to obtain
erU
ðÞ
2nþerU
ðÞ
1n¼qA:(12)
Here, qAis the surface-charge density at the interface.
The discord between Eqs. (11) and (12) is well-known,
18
and
is not due to writing the equations in terms of Aand U, since
Eand Bare gauge invariant.
4–6
Section Vresolves this
disparity.
The interface matching condition for the tangential com-
ponent (“t”) in Eq. (4) is
8
l1B1tl2B2t¼l1rA
ðÞ
1tl2rA
ðÞ
2t¼JA:
(13)
Equation (9) has a matching condition by taking a
Gaussian pill box with the end faces parallel to the interface
in regions 1 and 2. Noting that r2A¼r•rA, one can use
the divergence theorem in the limit of zero pill-box height to
obtain
n
_•r
A
l
"#
1þn
_•r
A
l
"#
2¼JA:(14)
Here, JAis the surface-current parallel to the interface; n
_
is the unit vector normal to the interface. As before, the dis-
parity between Eqs. (13) and (14) is not due to writing the
equations in terms of Aand U, since Eand Bare gauge
invariant.
4–6
Section Valso resolves this disparity.
Maxwell’s equations have an arbitrariness in Uand A
for Eqs. (6) and (7), under the transformation
4
A!AþrK;(15a)
U!U@K
@t:(15b)
Physics Essays 32, 1 (2019) 113
“Gauge” is used to describe Eqs. (15a) and (15b), origi-
nally arising from the nonstandard width of railroad track in
the 1800 s (a synonym for “arbitrary”). Equations (15a) and
(15b) leave Band Eunchanged, which is termed “gauge
invariance.” A @
2
-infinitude of choices exists for the gauge
function
19
ðKÞ. For example, the velocity gauge is
r•Aþbel@U=@t¼0. If b¼1, a charge source propagates
at the speed of light (Lorenz gauge). For b¼0, Upropagates
at infinite speed (Coulomb gauge). For 0 <b<1;Upropa-
gates at a speed, c=b. Other conditions are equivalent to the
Lorenz gauge with different physical meanings.
5,6
CED invariance under Eqs. (15a) and (15b) can be cast
in four-vector form as A!Aþ@K. Here, A¼ðU=c;AÞ
and @¼ð@=@ct;rÞ with a metric signature of (–,þ,þ,þ).
Kis a harmonic, scalar function of space and time that satis-
fies @@K¼0. Thus, the four-gradient component of Ais
gauged away under CED.
20,21
The assumption of no four-gradient in Ais contrary to
experiments that have measured an irrotational vector poten-
tial.
22–25
Section VI shows that an irrotational vector poten-
tial implies an irrotational current density. Moreover, an
irrotational current been observed in (for example): arc dis-
charges,
26
ion-concentration-gradient-driven current across
living-cell walls,
27
and irrotational, human, electroencepha-
logram current.
28
This section has demonstrated three incon-
sistencies in CED: (a) the interface matching condition
between two different media is inconsistent at the theoretical
level for a surface charge (1) and surface current (2); and (b)
the irrotational component of the vector potential is gauged
away, contrary to experimental measurements (3). These
three inconsistencies may not seem compelling in the light
of the success of Maxwell theory in modern physics. How-
ever, falsifiability
29
states that a hypothesis (theory) cannot
be proved by favorable evidence, but can only be disproved,
even by a single failure. These disparities require resolution
via EED, as discussed next.
IV. EXTENDED ELECTRODYNAMICS
The Helmholtz theorem
8
uniquely decomposes any
three-vector into irrotational and solenoidal parts. For exam-
ple, electrical current density has the form, J¼rjþr
G, with jand G, as scalar and vector space-time func-
tions, respectively. Smooth, Minkowski four-vector fields
also can be uniquely decomposed into four-irrotational and
four-solenoidal parts with tangential and normal components
on the bounding three-surface.
30
Woodside
31
subsequently
used the Stueckelberg Lagrangian density
32
L¼ec2
4FlFlþJlAlcec2
2@lAl
2
ec2k2
2AlAl
:(16)
Flis the Maxwell field tensor; cis the speed of light (not
necessarily vacuum); Jl¼ðcq;JÞis the 4-current; the four-
potential is Al¼ðU=c;AÞ; the Compton wave number for a
photon with mass ðmÞis k¼2pmc=h; and his Planck’s con-
stant. The fully relativistic Stueckelberg Lagrangian
32
includes both Aand U, and resolves many issues with previ-
ous CED Lagrangians. For c¼0 and m>0, Eq. (16) yields
the Maxwell-Proca theory, for which a 2012 test measured
m1054 kg (equivalent to 1018 eV), consistent with
massless photons.
33
Equation (16) for c¼1 and m¼0is
31
L¼ec2
2
1
c2rUþ@A
@t
2
rA
ðÞ
2
"#
qUþJ•Aec2
2r•Aþ1
c2
@U
@t
2
:(17)
Equation (17) allows only two classes of four-vector
fields.
31
One class of fields has zero four-curl of Al:
Fl¼@lA@Al¼0, with a solution,
31
Al¼@lK,
together with a nonzero, dynamical, scalar field, C¼@lAl
¼@l@lK.Kis a scalar function of space-time. The second
solution
31
has zero four-divergence of Al,C¼@lAl¼0, as
the Lorenz gauge, consistent with CED.
4
Woodside
7
later
proved the uniqueness of Eqs. (18)–(22) [7 equations in 7
unknowns ðB;C;EÞ] that form EED:
E¼rU@A
@t;(18)
B¼rA;(19)
C¼r•Aþ1
c2
@U
@t;(20)
rB1
c2
@E
@trC¼lJ;(21)
r•Eþ@C
@t¼q
e:(22)
Equation (18) is equivalent to Faraday’s law; Eq. (19) is
equivalent to the no magnetic-monopoles equation. Equation
(21) uniquely decomposes Jinto solenoidal ðr BÞ
and irrotational ðrCÞparts, in accord with the Helmholtz
theorem.
8
Equation (17) implies
34
Eqs. (18)–(22). Moreover,
Eqs. (18)–(22) imply
7,30,31
Eq. (17). Equation (17) is then
necessary and sufficient for Eqs. (18)–(22). The Lagrangian
for curved space-time
35
reduces to Eq. (17) in Minkowski
four-space.
A long history of work exists on EED.
7,14,30,31,34–44
Fock
and Podolsky
36
wrote the new Lagrangian in 1932 with a
dynamical, scalar field, C¼r•Aþel@U=@t, without
deriving the resultant equations. Ohmura
37
first wrote the
dynamical equations in 1956. Aharonov and Bohm
34
gave
the revised Lagrangian and Hamiltonian, and derived the
dynamical equations therefrom in 1963. Munz et al.
14
showed the use of EED in computational simulations in
1999. Van Vlaenderen and Waser
38
used EED to derive a
wave equation for C, and revised forms for momentum and
energy conservation in 2001. Woodside
7,30,31
rigorously
derived EED (1999–2009), assuming only Minkowski
four-space. Jim
enez and Maroto
35
used Eq. (16) with c¼1
and m¼0 to model quantum, curved space-time, electrody-
namics for an expanding universe in 2011. These
papers
7,14,30,31,34–38
cited no previous EED work, and
serve as seven independent verifications of EED theory.
114 Physics Essays 32, 1 (2019)
Modanese
44
studied a nonrelativistic, nonlocal, EED quan-
tum source.
Equations (18)–(22) use the least-action principle,
7
requiring a finite, lower bound on the Lagrangian, Eq. (17).
The Planck scale
45
provides such a bound. Another term
ðr AÞ2=2l, in Eq. (17), has the same requirement for a
finite, lower bound, and has been well validated.
4,8
V. BASIC EED PREDICTIONS
The Bwave equation arises from the curl of Eq. (21).
Note that
10
rrB¼rðr•BÞr
2B, which for
Eq. (19),B¼rA, gives rrB¼r
2B. Faraday’s
law, rE¼@B=@t, along with
10
rrC¼0 yields
the EED Bwave equation, identical to CED
8
r2B@2B
@c2t2w2B¼lrJ:(23)
The Ewave equation relies on the equivalence of
Eq. (18) to Faraday’s law, to which the curl is applied with
10
rrE¼rðr•EÞr
2E; replacing rBfrom
Eq. (21); substituting for r•Evia Eq. (22); and noting that
@rC=@tr@C=@t¼0. The EED Ewave equation is the
CED result
46
r2E@2E
@c2t2¼rq
eþl@J
@t:(24)
The A-wave equation is obtained by: replacing B;E, and
Cin Eq. (21) with Eqs. (18)–(20); using the vector calculus
identity,
10
rrA¼rðr•AÞr
2A; and noting that
@rC=@tr@C=@t¼0. The result is the CED Awave
equation,
8
Eq. (9), without the use of a gauge condition.
The Uwave equation can be obtained by: substituting
Eand Cfrom Eqs. (18) and (20) into Eq. (22); and noting
that @r•A=@tr•@A=@t¼0. The result is the CED
Uwave equation,
8
Eq. (8), without the use of a gauge
condition. The usual wave equations for Aand Uare then
rigorously derivable without a gauge condition. Thus, EED
is gauge-free and predicts the same wave equations for
A;B;E, and Uas CED.
Section III described inconsistent interface matching
conditions, which are resolved by this gauge-free theory.
Namely, Eqs. (21) and (22) are alternative forms of the wave
equations for Aand U, respectively, as shown above. Thus,
the interface boundary conditions for the Aand Uwave
equations are the appropriate forms: Eqs. (12) and (14).
A wave equation for Ccan be obtained from the diver-
gence of Eq. (21); application of elð@=@tÞto Eq. (22); and
summing the two results with
10
r•rB¼0. The result
is
18,38
r2C@2C
@c2t2¼l@q
@tþr•J
:(25)
The rigorous derivation of Eq. (25) eliminates the ad hoc
assumptions that were described in Section II to avoid the
overdetermination of Maxwell’s equations.
Equation (25) is an instantaneous equation. But, all
experiments are performed over a finite time, DT, i.e., a time
average. A long-time average gives @q=@tþr•J¼0on
the right-hand side (RHS) of Eq. (25), in accord with
long-standing experiments that validate classical charge
balance.
47
For example, the lower bound on electron lifetime
for charge balance has been carefully measured as
6:61028 years
48
(decay into two c-rays, each at
mec2=2).
Long-time charge conservation is not inconsistent with
charge nonconservation over short-time scales, DTDt, per
the Heisenberg uncertainty relation, DEDth=2. Here, DE
is the charged-quantum-fluctuation energy; and h is Planck’s
constant divided by 2p. Equation (25) can be interpreted as
charge nonconservation driving C, and vice versa, not unlike
energy fluctuations driving mass fluctuations in quantum
theory and vice versa.
18
Thus, Eq. (25) predicts charge con-
servation on long time-scales (consistent with CED), and
exchange of energy between Cand quantum fluctuations
for DTDt. Confirmation of these quantum charge fluctua-
tions involves tests, consistent with the Heisenberg uncer-
tainty relation. One possible test could use the electron
[DE(electron) ¼mec2¼0.51MeV] corresponding to a time,
Dt61022 s. Subzeptosecond dynamics have been mea-
sured,
49
so a direct measurement of this prediction is feasi-
ble. Moreover, quantum fluctuations can control charge
quantization,
50
in accord with Eq. (25).
The homogeneous solution to Eq. (25) is wavelike, with
the lowest-order form in a spherically symmetric geometry,
4
C¼Coexp ½jðkr xtÞ=r. Here, j¼ffiffiffiffiffiffiffi
1
p;kis the wave
number ð2p=kÞfor a wavelength, k;x¼2pffor a fre-
quency, f; and ris the spherical radius. Boundary conditions
for Eq. (25) include Cðr!1Þ!0, which is trivially
satisfied. Equation (44) predicts that the energy density of
the Cfield is ðC2=2lÞ, yielding a constant energy,
4pr2ðC2=2lÞ, through a spherical boundary around a source
in arbitrary media, as required.
18
The interface matching
condition for Eq. (25) uses a Gaussian pill box with the end
faces parallel to the interface in regions 1 and 2. Noting that
r2C¼r•rC, use of the divergence theorem in the limit of
zero pill-box height yields continuity in the normal compo-
nent (‘n’) of rC=lfor long times
rC
l
1n¼rC
l
2n
:(26)
The subscripts, 1 and 2, denote medium 1 and medium 2 for
lnot necessarily in vacuum.
VI. EED PREDICTION OF SLW
Section III showed that the four-gradient component of
Ais gauged away under CED,
20,21
which is inconsistent
with experiments.
22–28
Gauge-free EED eliminates this dis-
parity by explicitly including solenoidal (or transverse,
denoted by superscript, “T”) and irrotational (or longitudinal,
denoted by superscript, “L”) parts in Eq. (21). Then, a longi-
tudinal vector potential, AL¼ra, yields
10
Physics Essays 32, 1 (2019) 115
B¼rAL¼rra¼0;or AL¼ra)B¼0:
(27)
Here, ais a scalar function of space-time. The inverse is
BT¼0¼rAL)AL¼ra;or
BT¼0)AL¼ra:(28)
Combining Eqs. (27) and (28) gives
AL¼ra() BT¼0:(29)
A longitudinal vector potential from Eq. (27) also
implies
10
rrAL¼rrra¼0
¼rðr•ALÞr
2AL)rðr•ALÞ
¼r
2AL¼r
2ra¼rr
2a:(30)
Insertion of Eqs. (29) and (30) into the Awave equation
gives
w2AL¼w2ra¼rw2a¼lJ)JL¼rj;or
AL¼ra)JL¼rj:(31)
A corollary to Eq. (31),rw2a¼lrj,is
w2a¼lj. The inverse is also true; J¼rjþrGT
allows decomposition of Eq. (21) into transverse and longitu-
dinal parts
rBT1
c2
@ET
@t¼lrGT)r BTlGT
þ@2AT
@c2t2¼0;(32a)
1
c2
@EL
@trC¼lrj)@2AL
@c2t2
¼r 1
c2
@U
@tþCþlj
:(32b)
Rearrangement of Eq. (32b) with substitution of Cfrom
Eq. (20) and cancellation of terms, gives the same corollary
to Eq. (31) as cited above. Since time and spatial derivatives
commute, the right-hand portion of Eq. (32b) results in
JL¼rj)AL¼ra:(33)
Combining Eqs. (31) and (33) yields
AL¼ra() JL¼rj:(34)
The combination of Eqs. (29) and (34) relates B¼0and
JL
JL¼rj() BT¼0:(35)
The net result of Eqs. (29),(34), and (35) is
AL¼ra() BT¼0() JL¼rj:(36)
Equation (36) is consistent with the above-cited
tests
22–28
and drives the SLW, which is also called an
electro-scalar wave.
38,51,52
Clearly, Eq. (36) holds only
where Jis nonzero. We prefer the more descriptive phrase,
“scalar-longitudinal wave,” or SLW, which is used through-
out this paper. The explanation for JLdriving a SLW is as
follows. The resultant electric field is EL¼JL=r¼rj=r
for media with a linear conductivity, r.ELand JLfor the
SLW are curl-free. Faraday’s law becomes rEL¼_
B¼
rrj=r¼0:The overdot denotes a partial-time deriva-
tive. Thus, no eddy currents occur, so the SLW is unimpeded
by the skin effect for propagation through linearly conduc-
tive media. See also Graham et al.
53
and Appendix A.
C;EL;and JLare all related by Eq. (32b) with
54
r¼eoe00x
and e¼eoðe0je00Þfor a SLW impedance, Z,
Z¼jErj
jC=lol0j¼ffiffiffiffiffi
lo
eo
rffiffiffiffi
l0
e0
r11=ðjkrÞ
1jtan ðdeÞ:(37)
Equation (37) assumes spherical waves in linear conduc-
tive media: EL¼Er^
rexp ½jðkr xtÞ=r, together with
C¼Coexp ½jðkr xtÞ=r. The unit vector in the radial
direction is ^
r;eoand loare the free-space permittivity and
permeability, respectively; e0and l0are the relative permit-
tivity and permeability (not necessarily vacuum), respec-
tively; tan ðdeÞ¼e00=e0. Here, the same definitions are used
for k;r;t;and xas in Section V. From Eq. (20),Chas the
same dimensions as B¼lH. Consequently, Eq. (37) uses
jEj=ðC=lÞto obtain the SLW impedance, like the CED
form, Z¼jEj=jHj. The Cand Efield energies from
Eq. (44),4pr2ðeE2=2Þand 4pr2ðC2=2lÞ, are constant through
a spherical boundary and Cðr!1Þ¼jELðr! 1Þj ! 0.
Equation (37) predicts Zo¼ffiffiffiffiffiffiffiffiffiffiffi
lo=eo
p, in free-space
(e0¼l0¼1 and e00 ¼l00 ¼0). The SLW radiation pattern
from a monopole antenna is isotopic, and attenuates as r2in
free space; see Appendix B,
POUT ¼I2^
r
4pr
ðÞ
2ffiffiffi
l
e
r:(38)
The SLW has not been previously observed, because
transverse electromagnetic (TEM) antennas detect only
waves that produce a circulating current. A TEM transmitter
produces only circulating currents, yielding C¼0 and EL¼
0 with no SLW power output or reception; see Appendix C.
The wave equations for A;B;E;and Uare unchanged
under time reversal. A sign change occurs on both sides of
Eq. (25) for t!tthat also gives time invariance, and indi-
cates the pseudo-scalar nature of C. Time-reversibility of
EED implies that reciprocity holds. Then, a SLW transmitter
can act as a receiver, and vice versa.
VII. EED PREDICTION OF SCALAR WAVE
EED also predicts a scalar wave (that has only a scalar
field, and is distinct from the SLW) under the two condi-
tions: E¼0 and Eq. (36).
18
E¼0 corresponds to zero on
the left-hand side (LHS) of Eq. (18). Then, the condition
(AL¼ra) from Eq. (36) can be combined with the RHS of
116 Physics Essays 32, 1 (2019)
Eq. (18) giving U¼_
a. Equation (20) for the scalar field
can subsequently be rewritten by the replacements, U¼_
a
and AL¼ra:
C¼r•Aþ1
c2
@U
@t¼r
2a1
c2
@2a
@t¼w2a:(39)
Substitution of Eq. (39) into Eq. (25) yields
18
w2w2a¼w4a¼l@q
@tþr•J
:(40)
Here, w2is the wave operator as defined in Eq. (23). Use of
E¼0 in Eq. (22) yields
@C
@t¼q
e)q¼e_
C¼ew2_
a:(41)
The last form in Eq. (41) comes from the partial-time
derivative of Eq. (39). Equation (42) arises from Eq. (40) by
replacing @q=@twith the partial-time derivative of Eq. (41),
use of JL¼rjfrom Eq. (36) to evaluate r•J, and rear-
rangement of the terms, resulting in the left-hand form
w4aþ1
c2w2€
aþlr2j
¼w2r2a€
a
c2þ€
a
c2
þlr2j
¼r
2w2aþlj
¼0:(42)
The second line of Eq. (42) arises from expansion of
the wave operator. The third line of Eq. (42) results from
cancellation of the positive and negative €
a-terms along with
interchange of the wave- and Laplacian (r2) operators,
which commute. One solution to Eq. (42) involves setting
the term inside the parentheses to zero, which is another
form for the longitudinal component of the Awave equa-
tion [Eq. (9)]withAL¼raand JL¼rj:This solution
arises, because the gradient and Laplacian operators com-
mute for AL,perEq.(30). A second solution to Eq. (42)
involves setting the entire left-hand form to zero, which
results in the ALwave equation being set equal to a har-
monic function
55
[HðrÞejxt]
w2ar;t
ðÞ
¼HðrÞejwt lj r;t
ðÞ
:(420Þ
Here, the space and time dependences are shown explic-
itly. r2HðrÞ¼0 is typically solved by separation of
variables.
56
For example, in Cartesian coordinates, HðrÞ
involves the sum HðrÞ¼PXnðxÞYwðyÞZfðzÞ,wheren2þ
w2þf2¼0 and [XnðxÞ;YwðyÞ;ZfðzÞ] are an appropriate set of
scalar functions of (x,y,z), respectively. The time-dependent
term (ejxt) that is associated with HðrÞis unaffected by the
Laplacian operator. Nonhomogeneous solutions to Eq. (420)
are beyond the scope of the present work, being dependent on
the specific boundary conditions, geometry, and form for
jðr;tÞ. The lowest-order, homogeneous solution to Eq. (420)is
a¼ao
rejðxtkrÞ:(43)
Substitution of Eq. (43) into Eq. (39) gives the same 1=r
dependence for Cin free space, as for the SLW. Revised
energy balance, Eq. (44), shows that the scalar-wave energy
is C2=2l1=r2. Thus, the total scalar-wave energy,
4pr2ðC2=2lÞ, is constant through a spherical boundary
around a source and Cðr!1Þ!0, as expected. Revised
momentum balance, Eq. (45), shows that the scalar wave has
a pressure of rC2=2l, but no momentum density.
VIII. REVISED BALANCE EQUATIONS
EED predicts a revised energy balance,
18,38
Eq. (44),
from the sum of: ðC=lÞtimes Eq. (22);ðB=lÞ• applied to
Faraday’s law; and application of ðE=lÞ• applied to
Eq. (21)
@
@t
B2
2lþC2
2lþeE2
!
þr•EB
lþCE
l
þJ•E¼qC
el :(44)
Equation (44) has new energy density terms: scalar field
energy ðC2=2lÞ, SLW energy ðCE=lÞ, and a power source
ðqC=elÞ.
EED predicts revised momentum balance,
18,38
Eq. (45),
as the sum of: ðB=lÞr•B¼0; the cross product of ðeEÞ
with Faraday’s law; Eq. (21) ðB=lÞ;ðC=lÞ Eq. (21);
and ðeEÞ Eq. (22). Equation (45) has new density terms:
SLW momentum flux ðCE=lÞ, TEM-SLW mixed mode flux
½ðr BCÞ=l, a force ðJCÞparallel to the current density,
and scalar-field pressure ðrC2=2lÞ. The last term in Eq. (45)
is the divergence of the CED Maxwell stress tensor
4
el @
@t
EB
lCE
l
þqEþJBþrBC
l
¼JCþrC2
2lþr•T
$:(45)
Equation (44) predicts a power gain ðþCE=lÞwith
momentum loss ðCE=lÞin Eq. (45), and vice versa. This
sign difference means that a SLW emission (power loss) drives
a momentum gain in a massive object that emits the SLW.
IX. PRELIMINARY SLW EXPERIMENTS
The present work eliminates sources of error in previous
tests,
51,52
which are discussed in Appendix D. High fre-
quency (8 GHz) experiments facilitate an indoor, con-
trolled test environment. Figure 1shows the test layout. The
transmitter and receiver are identical (inverted triangles in
the lower left of Fig. 1), since time-reversal symmetry
allows the transmitter to act as a receiver, as discussed in
Section VI. The directional couplers act as 45-dB isolators.
Grounding to a single point avoids current loops. Modern
digital instrumentation allows accurate measurement of sig-
nal amplitudes and distances for comparison of test results to
EED predictions.
Figure 2shows the linear, monopolar, SLW antenna
with the coaxial center conductor as the radiator. The outer
Physics Essays 32, 1 (2019) 117
coaxial conductor is electrically connected to the top of the
skirt balun.
57,58
The skirt balun length ðk=4Þcauses a phase
shift in the current flow along the guided path from the bot-
tom (inside surface) of the skirt balun conductor (0) to the
top (inside surface) of the skirt balun (90) and back down
the outer surface of the coax outer conductor to the end of
the skirt balun (180). The 180-phase shift attenuates the
return current along the outside of the outer coaxial conduc-
tor to form a monopole antenna, thus eliminating the image
charge and image current of previous tests.
51,52
The resultant
far-field contours of constant jEjfrom an HFSS electrody-
namic simulation are essentially spherical, as expected for a
monopolar antenna (top of Fig. 2). The RG-405/U coaxial
cabling uses a solid, outer conductor to minimize stray fields;
the presence or absence of an outer insulating jacket makes
no difference in the results of the electrodynamic simulation.
(A 3k-diameter ground-plane disk at the feed-point gives
essentially the same jEjcontours, thus confirming the linear
monopolar, counter-poise design.)
The return-current attenuation (23 dB) of the previous
paragraph is quantified in Fig. 3, as a sharp null at 7.94 GHz
(shown in red online). Return-current attenuation allows the
monopole antenna to draw charge from the ground plane
(top of the skirt balun) and also creates an impedance match
between the antenna-balun (49.76–j0.24 X) and the source
(50 X). Thus, the skirt balun reduces the return current along
the outside of the outer coax conductor, so that essentially all
of the electrical current goes into charging and discharging
the antenna (an irrotational current) to drive the SLW, as pre-
dicted in Eq. (36). The test result for a single skirt balun
(shown in green online in Fig. 3) shows the same trend as the
HFSS simulation with a minimum of 23 dB at 8.00 GHz;
the test result for a double balun was 42 dB (not shown).
The difference in null depth arises from inaccuracies in the
antenna fabrication. Variation in the return loss with fre-
quency arises from the tuned balun geometry.
One experiment tested two critical EED predictions.
Namely, the SLW is unconstrained by the skin effect, and
the free-space attenuation of the SLW has a 1=r2depen-
dence. The test measured signal attenuation (dB) versus dis-
tance (r) with the transmitting-antenna location fixed, and
the receiving antenna moved in a straight line horizontally
via a linear positioner with 1-mm accuracy. The source
frequency was 8.00 GHz with a free-space wavelength,
FIG. 1. (Color online) Test layout, with numbered items in the diagram corresponding to the tabular description above. This figure does not show Items 3,
and 6–8, which are discussed in the text.
118 Physics Essays 32, 1 (2019)
k¼3:75 cm. Figure 4shows the results for two, facing, col-
linear SLW-antennas inside an anechoic, Faraday chamber
(90 dB isolation).
In addition, each antenna was inside its own separate
Faraday enclosure. Each antenna enclosure consisted of a
copper pipe (Nibco copper coupling without C x C – Wrot,
601) and two hemispherical end caps (outer diameter ¼
15.85 mm, inner diameter ¼14.34 mm). The pipe length was
28.89 mm with a wall thickness of 1.02 mm. The hemispheri-
cal end caps (Nibco Cap C – Wrot, 617) had an outer diame-
ter of 17.80 mm, an inner diameter of 15.80 mm, and a wall
thickness of 1.05 mm. One end cap had a central hole slightly
larger than the outer diameter of the coaxial outer conductor.
This end cap was carefully soldered to the coax outer con-
ductor to create a strong structural bond and a tight, 360
electrical seal between the Faraday enclosure and the coax
outer conductor. Likewise, the hemispherical caps were sol-
dered to each end of the copper pipe to create a strong struc-
tural bond and a tight, 360electrical seal. Figure 4shows
the SLW signal attenuation through both antenna enclosures.
The straight-line, least-squares-fit, log-log slopes are
2.6301 and 2.2273 for the top and bottom plots, respec-
tively. The measurements are presently too noisy to distin-
guish these slopes from 2. The interior of the Faraday
enclosures for each antenna did not have any RF attenuating
foam, allowing resonance-cavity effects that are not included
FIG. 2. (Color online) Cross-sectional view of constant |E| contours for the SLW antenna. Heavy lines (shown in purple online) are conductors for the linear
monopole antenna (top label), skirt balun (middle label), and outer conductor of the coaxial cable (bottom label).
Physics Essays 32, 1 (2019) 119
in the present theory. SLW attenuation in conductive media
is the subject of future work. The combined solid-copper
thickness (2 1.02 mm) is 2914 skin depths in the result of
Fig. 4, which should produce a classical attenuation of
101265 in TEM waves. The test measurement yielded an
attenuation between 115 dB at r¼2 cm separation and
137 dB at r¼30 cm. Extrapolating the straight-line fit to a
separation of 2 mm (where the enclosed antennas are barely
separated) gives 79 dB and 86 dB, for an attenuation of
104. The difference between the measured value and the
FIG. 3. (Color online) Attenuation of the return current as a function of frequency, showing a minimum in the HFSS-model results with a sharp null to 49 dB
at 7.94 GHz (shown in red online) in comparison to the measured balun effectiveness with a shallow null to 23 dB at 8.00 GHz (shown in green online).
FIG. 4. (Color online) SLW attenuation (S21 in the top plot and S12 in the bottom plot) in dB versus the transmitter-to-receiver distance in r(meters).
120 Physics Essays 32, 1 (2019)
classical prediction is >1261 orders of magnitude. This dis-
parity far exceeds the criterion for a scientific discovery (five
standard deviations or a probability of <3107). These
results show that the SLW is unconstrained by the skin effect
with 1=r2attenuation in free-space. These results are con-
sistent with EED and cannot be explained by CED.
The extraordinary result in the previous paragraphs
requires substantial discussion. We emphasize that the above
results are from preliminary measurements and may have
confounding factors that have yet to be identified. One issue
may be TEM signals being picked up by the coaxial cables,
either inside or outside the Faraday cage. This issue was
addressed by the use of RG405/U cables with a solid copper
outer conductor everywhere, which eliminated TEM interfer-
ence. In this light, we did not use a TEM receiving antenna
inside the Faraday cage to confirm the absence of such a sig-
nal. A second issue is that the SLW is unconstrained by the
skin effect (Appendix A) and can therefore penetrate the
solid copper outer conductor of the RG405/U cabling, which
moved with the motion of the linear and rotational position-
ers. This cable motion probably accounts for the large vari-
ability in the results of Figs. 4and 5. A third issue is the
distance between the transmitter and receiver in comparison
to the wavelength, Namely, a longitudinal E-field component
occurs in the near-field under classical (Maxwell) electrody-
namics at a distance of 3kfor the largest dimension of the
antenna small in comparison to the wavelength.
59
However,
a significant signal was received 30 cm (8 wavelengths) from
the transmitter, which is well beyond the near-field. More-
over, detection of this far-field signal required the SLW to
penetrate a 1-mm thick Faraday enclosure that surrounded
the transmitter, as well as a 1-mm thick Faraday enclosure
around the receiver. Wave penetration of the two 1-mm thick
Faraday enclosures (2-mm total) demonstrates the low-loss
nature of the SLW in conductive media. A more accurate
and compelling result clearly requires a much better experi-
mental design, which is the subject of future work.
Equation (38) predicts isotropic transmission from the
SLW monopole antenna. Figure 5shows the variation
between a bare, single-balun transmitter and a Faraday-
enclosed, single-balun receiver with a fixed separation of
r¼0:75 m. Both antenna axes were in the horizontal plane
inside the anechoic Faraday chamber. The receiving antenna
was rotated 360in the horizontal plane (polar angle, hÞby a
rotational positioner in 1increments. The variability over
the entire angular range is 61 dB, showing isotropy within
the measurement accuracy from the linear, monopole
antenna, in accord with the EED prediction.
X. DISCUSSION
EED allows spherical symmetry, Eq. (38), in the electrical
current density (J) and longitudinal fields, which CED for-
bids.
60
Table Ishows testable EED predictions. Starred items
in Table Iare consistent with tests in Section IX.Thestarfor
FIG. 5. (Color online) SLW power variation (dB) versus polar angle (h)
for a bare SLW antenna (transmitter) and a Faraday-boxed SLW antenna
(receiver). Two repetitions of the test (green and blue) show reproducibility
within 61 dB.
TABLE I. Summary of testable EED predictions.
Item Brief description of testable prediction Reference
1 The interface matching condition for qAis… Equation (12)
2 The interface matching condition for JAis… Equation (14)
3 The SLW has a scalar field, C¼r•Aþel@U=@t. Equation (20)
4 The scalar field is also charge-fluctuation driven. Equation (25)
5 The interface matching condition for Cis… Equation (26)
6 The SLW has drivers: AL¼ra() BT¼0() JL¼rj:Equation (36)*
7 The SLW has a longitudinal Efield. Section VI
8 The SLW is unconstrained by the skin effect. Section VI*
9Cis a pseudo-scalar field. Section VI
10 The SLW has a power comparable to the TEM wave. Equation (37)*
11 The SLW free-space attenuation goes like 1=r2. Equation (38)*
12 The SLW monopole radiation is isotropic. Equation (38)*
13 The scalar wave arises from U¼_
aand… Equation (42)
14 The scalar-field energy density is C2=2l. Equation (44)
15 The SLW power density vector is CE=l. Equation (44)
16 Energy balance has a new source, qC=el. Equation (44)
17 The SLW momentum density is CE=l. Equation (45)
18 Momentum balance has a mixed-mode term, rBC=l. Equation (45)
19 Momentum balance also has source term, JC. Equation (45)
20 The scalar-field pressure density is rC2=2l. Equation (45)
Physics Essays 32, 1 (2019) 121
Item 6 occurs, because the linear, monopolar SLW antenna
with a balun imposes an irrotational current, as discussed in
Section IX. The star for Item 10 arises, because the SLW atten-
uation is measured with standard instrumentation and has a
free-space impedance that is identical to TEM waves.
Haag’s theorem
61
states that two Hilbert solutions may
be unitarily inequivalent within quantum field theory (QFT).
The “proper” representation must then be chosen from an
infinite set of inequivalent forms. Seidewitz
62
showed that
Haag’s theorem does not apply to Eq. (16). Equations
(18)–(22) are a necessary and sufficient condition for
Eq. (16) with c¼1 and m¼0, resolving the problem of
inequivalent unitary QFT forms. We note that the relativistic
field equations arising from the Stueckelberg Lagrangian
32
are taught in advanced physics texts, e.g., Ref. 63.
XI. CONCLUSIONS
The conclusions of this work follow. Section II shows
that CED is overdetermined. Section III identifies inconsis-
tencies in CED that arise from this overdetermination: incor-
rect interface matching conditions and gauging away of the
four-gradient in A. Section IV shows the provably unique
basis for EED,
7
which has a new term ðrCÞin Ampere’s
law and a new term ð@C=@tÞin Gauss’ law. Section Vrigor-
ously derives the CED wave equations from EED without a
gauge condition. EED also eliminates the inconsistencies
that are identified in Section III, and predicts a revised form
for charge conservation without any ad hoc assumption(s).
In Section VI, EED further predicts a free-space, SLW.
Equation (36) shows that the SLW arises from an irrotational
current, a longitudinal vector potential, or antennas that have
a null magnetic field. Section VII derives the conditions for a
scalar wave under EED. Section VIII shows the EED deriva-
tion of revised energy and momentum balance. Section IX
presents preliminary experimental results for the SLW (five
of the testable predictions in Table I) that are consistent with
EED and cannot be explained by CED. Specifically, the tests
show that the SLW: (a) is unconstrained by the skin effect
via propagation through Faraday enclosures with a disparity
of 1261 orders of magnitude between CED and EED; (b) can
be transmitted and received by a monopolar antenna (not to
be confused with magnetic monopoles) with an isotropic
radiation pattern; and (c) has a free-space attenuation consis-
tent with 1=r2. The disparity under item (a) far exceeds the
criterion for a scientific discovery {five standard deviations
[probability 3107(7.5 orders of magnitude)]. These
measurements are in accord with the EED theory developed
in Sections Vand VI. Clearly, much additional work is
needed to strengthen and replicate these measurements.
ACKNOWLEDGMENTS
Insightful suggestions by Aly Fathy, David Fugate, David
Froning, George Hathaway, Roger Kisner, Giovanni Moda-
nese, Mike Pascale, Theophanes Raptis, Donald Reed, Igor
Smolyaninov, John Wilgen, and Dale Woodside are gratefully
acknowledged. Additional comments by two anonymous
reviewers led to changes that substantially strengthened this
essay. Professor Aly Fathy kindly provided the use of his
experimental equipment and facility at the University of Ten-
nessee (Knoxville). LMH did the theoretical analysis, wrote
the paper, designed the experiments, and analyzed test results;
ASL performed the tests and provided numerous editorial
comments. This work was partially funded by ScalarWave
LLC, and its successor (Gradient Dynamics, LLC), a privately
owned, Delaware company. Further details are disclosed in
US Patent #9,306,527 by the first author, dated 05 April 2016.
APPENDIX A: SLW IN CONDUCTIVE MEDIA
A necessary and sufficient condition for the SLW,
Eq. (36),isB¼0, which gives w2B¼lrJ¼0. The
curl of the (RHS) of this last equation is
10
rrJ¼
rðr•JÞr
2J¼0:The resultant form is
rðr•JÞ¼r
2J:(A1)
The gradient of the classical charge-conservation equa-
tion with substitution from Eq. (A1) yields
@rq
@t¼rr•JÞ¼r
2J:
(A2)
The partial time-derivative of the E-wave equation, Eq.
(24), with substitution from Eq. (A2) is
w2_
E¼l@2J
@t2þ1
e
@rq
@t¼l@2J
@t2r2J
e¼w2J
e:(A3)
Equation (A3) can be rewritten, using Ohm’s law,
J¼rE;for linearly conductive media
w2_
EþJ
e
¼w2_
EþrE
e
:(A4)
One solution to Eq. (A4) sets the terms inside the paren-
theses to zero, as a transient solution
4
that typically decays
in 1019s
E¼Eoeet=r:(A5)
Eoin Eq. (A5) is the initial value of E. A second solution to
Eq. (A4) uses the nontransient, separable form for Eafter the
transient decay
E¼EoðrÞejxt:(A6)
Here, ris the position vector. Substitution of Eq. (A6) into
Eq. (A4) with elimination of common terms gives
w2E¼0:(A7)
The lack of a source term on the RHS of Eq. (A7) means
that the SLW propagates without loss in isotropic, homoge-
neous conductive media (r¼constant). That is, the SLW is
unconstrained by the skin effect. This prediction can be veri-
fied by substitution of the spherical wave forms for Eand C
into the energy balance equation, Eq. (44), with B¼0 and
the ratio of jErj=jC=ljfrom Eq. (37); the same result arises
for plane waves.
122 Physics Essays 32, 1 (2019)
APPENDIX B: SLW TRANSMISSION POWER
The SLW power output (POUT ) is obtained via the
method in Chapter 9 of Jackson.
4
A linear, thin-wire
antenna is assumed on the z-axis over the interval,
0zLkr.Jand qare initially written in Cartesian
coordinates
JL¼^
zIdðxÞdðyÞejxtðcos kz cos kLÞ
ð1cos kLÞ:(B1)
q¼jIdðxÞdðyÞejxtsin kz
cð1cos kLÞ:(B2)
The irrotational current density ðJLÞis maximal at the
feed point, z¼0 (where the center conductor exits the coax-
ial cable), and zero at the end of the antenna ðz¼LÞ. The
Dirac delta function is denoted by dð…Þ; the current is I.
The charge density ðqÞis determined from Jby classical
charge balance. Uand Aare obtained by Green’s-function
solutions to Eqs. (8) and (9)
U¼IejðkrxtÞ
4pcekrð1cos kLÞ
ejkL cos hðjcos kL þcos hsin kLÞþj
sin2h
;
(B3a)
A¼lI^
zejðkrxtÞ
4pkrð1cos kLÞ
ejkL cos hðsin kL jcos hcos kLÞþjcos h
sin2h
jcos kL
cos hejkL cos h1
ðÞ
:(B3b)
Equations (B3a) and (B3b) are now written in cylindri-
cal coordinates. As before, eand lare the permittivity and
permeability of the propagation medium, respectively, (not
necessarily vacuum). Equations (B3a) and (B3b) assume
kL 1 (small antenna) and kr 1 (far field), allowing
terms on the order of ðkrÞ1and higher to be neglected
in comparison to unity. Cand Ethen are obtained from
Eqs. (18) and (20) in spherical coordinates with ^
z¼^
rcos h
^
hsin h:
C¼r•Aþ1
c2
@U
@t¼lIejðkrxtÞ
4pr;(B4a)
E¼rU@A
@t¼clIejðkrxtÞ^
r^
hfðhÞ
hi
4pr:(B4b)
Equation (B4b) has fðhÞ;which is irrelevant to this deri-
vation, because the radiated power flows only in the radially
outward (þ^
r) direction). POUT for the SLW is determined
from the time-averaged, radial component, <CE=l>from
Eq. (44)
POUT ¼I2^
r
4pr
ðÞ
2ffiffiffi
l
e
r:(B5)
APPENDIX C: SLW FROM TEM ANTENNA
The SLW power output (POUT ) is obtained by the proce-
dure in Chapter 9 of Jackson.
4
A linear, thin-wire, dipolar
antenna is assumed along the z-axis over the interval,
LzLkr. The current (J) and charge densities
(q) are the same as in Appendix B; the notation is also identi-
cal to Appendix B.Uand Aare obtained from the Green’s-
function solutions to Eqs. (8) and (9)
A¼lI^
zejðkrxtÞ
2pkr
sin kL cos kL cos h
ðÞ
cos hcos kL sin kL cos h
ðÞ
ð1cos kLÞsin2hcos h
"#
;
(C1a)
U¼IejðkrxtÞ
2pe xr
sin kL cos kL cos h
ðÞ
cos hcos kL sin kL cos h
ðÞ
ð1cos kLÞsin2h
"#
:
(C1b)
Equations (C1a) and (C1b) are written in cylindrical
coordinates. As in Appendix B,eand lare the permittivity
and permeability of the propagation medium, respectively,
(not necessarily vacuum). Equations (C1) assume kL 1
(small antenna) and kr 1 (far field), allowing terms on the
order of ðkrÞ1and higher to be neglected in comparison to
unity. Cis then obtained from Eqs. (C1) with ^
z¼^
rcos h
^
hsin h, using spherical coordinates on the RHS
C¼r•Aþ1
c2
@U
@t¼1
r2
@r2Ar
ðÞ
@rþ1
c2
@U
@t:(C2)
The resultant expression in the far-field for Cbecomes
C¼ð11ÞjlIejðkrxtÞ
2pr
sinkL cos kL cosh
ðÞ
coshcos kL sin kLcosh
ðÞ
ð1coskLÞsin2h
"#
¼0:
(C3)
Moreover, the radial component of Ein the far field is
E¼ð11ÞjI^
rejðkrxtÞ
2prffiffiffi
l
e
r
sinkL cos kL cosh
ðÞ
coshcos kL sin kLcosh
ðÞ
ð1coskLÞsin2h
"#
¼0:
(C4)
Consequently, POUT (SLW)¼<CE=l>is zero. This
result explains the nondetection of the SLW by TEM anten-
nas, which can detect waves that generate only a circulating
Physics Essays 32, 1 (2019) 123
current (no gradient-driven current). Again using ^
z¼
^
rcos h^
hsin hfor TEM waves, the theta-component of E
from Eq. (18) is
E¼jI^
hejðkrxtÞ
2prffiffiffi
l
e
r
sin kL cos kL cos h
ðÞ
cos hcos kL sin kL cos h
ðÞ
ð1cos kLÞsin hcos h
"#
:
(C5)
The magnetic field from the TEM antenna from
Eq. (19) is
B¼jlI^
uejðkrxtÞ
2pr
sin kL cos kL cos h
ðÞ
cos hcos kL sin kL cos h
ðÞ
ð1cos kLÞsin hcos h
"#
:
(C6)
POUT (TEM)¼<EB=l>for the TEM wave then is
POUT ¼I2^
r
2pr
ðÞ
2
sinkLcos kLcos h
ðÞ
coshcoskLsin kL cosh
ðÞ
ð1coskLÞsinhcosh
"#
2
:
(C7)
Equation (C7) is zero in the limits of h!0;p
via L’Hospital’s rule. POUT is maximal for h!p=2 via
L’Hospital’s rule
POUT ¼I2^
r
2pr
ðÞ
2
kL cos kL
ðÞ
sin kL
ðÞ
ð1cos kLÞ
"#
2
:(C8)
Equations (C7) and (C8) are consistent with Jackson’s
Eq. (9.28), which assumed a constant current density, rather
than a more realistic sinusoidal current density,
4
as done
here.
APPENDIX D: DISCUSSION OF PREVIOUS SLW TESTS
A test by Monstein and Wesley
51
used a 6-cm diameter,
center-fed aluminum spherical transmitter and receiver at
433.59 MHz (k¼69:2 cm). A three-by-three array of half-
wavelength, linear, electric-dipole antennas was placed
between the transmitter and receiver. The signal was strongly
attenuated for the dipole-array parallel to the transmitter-
receiver direction. No attenuation occurred for an array ori-
entation perpendicular to the propagation direction, showing
that the Ewave polarization was longitudinal. The results
agreed with image-charge theory for separation distances
less than 100 m (144 wavelengths), including two interfer-
ence minima. The outdoor tests were performed near the
bank of the Rhein River in Switzerland. The outer conductor
of the coaxial cable was grounded, inducing unquantified
electrical ground currents. The transmitter-receiver distance
was measured via GPS (accuracy of 65m). Bray and
Britton
64
note that the solution for Uis inconsistent with
CED, which Monstein and Wesley
65
accept as a failure in
the classical Maxwell theory. These tests were poorly con-
trolled, and not reproducible.
Butterworth et al.
52
unsuccessfully attempted replication
of the tests by Monstein and Wesley
51
[TMW], as shown in
Table II. The test-to-theory match in Table II refers to the
image-charge theory in the previous paragraph. Specifically,
both tests placed the antenna at a height ðþHÞabove a large,
conductive ground plane. An image-charge model
4
has a
negative-charge source at the same distance below the
ground plane ðHÞ. However, an image-current flows in the
same (opposite) direction as the real current for vertically-
(horizontally-) oriented dipole antennas.
46
TMW and the
Butterworth
52
omitted the image current, leading to poor
agreement between the test results and the models.
66
The last
line of Table II refers to measurements of the Ewave
polarization, also as discussed above.
Table III summarizes the inadequacies in these tests and
gives suggestions for improvements for the present work.
Both experiments have noise that confounds the results. The
coaxial feed-line into the antenna creates an asymmetry in
the spherical geometry. Lacking a balun, capacitive coupling
(displacement current) between the antenna and the coaxial
cable’s outer conductor induces a return current on the out-
side of the outer conductor of the coaxial cable, creating
TEM radiation (items c-d in Table III). Impedance matching
TABLE II. Comparison of tests by Monstein and Wesley
51
and Butterworth et al.
52
Feature in Monstein and Wesley (2002) Feature in Butterworth et al. (2013)
Aluminum-sphere diameter, D¼6 cm Aluminum-sphere diameter, D¼7:62 cm
Antennas on 4.3 m and 4.7 m high stanchions Antennas on 2 m high stanchions
f¼433:59 MHz, k¼69:2cm f¼446 MHz, k¼67:3cm
Signal on and off for calibration purposes No mention of on/off signal for calibration
Outdoor, north-south test on Rhein River bank Indoor-hallway/outdoor tests (east-west)
Use of ball antennas only Ball and half-wave-dipole antennas
No mapping of ball-antenna radiation pattern Radiation pattern vs angle from ball apex
Transmitter-to-receiver distance, r¼13 700 m Transmitter-to-receiver distance, r¼290 m
Test-to-theory match: minima at r¼24;40 m Test-to-theory match: minimum at r30 m
Longitudinal Ewave from dipolar polarizer Ewave polarization shift by p=2 radians
124 Physics Essays 32, 1 (2019)
between the source and antenna and signal-to-noise ratio are
unaddressed by Monstein and Wesley
51
who used a custom-
made transmitter and receiver, instead of a standard signal
source and spectrum analyzer. Without a phase-locked signal
from a network analyzer, external/stray signals can confound
the measurements. The Fresnel zone of the polarizer-
analyzer (from diffraction by the circular aperture) is not
addressed; the Fresnel radius ðFÞis 11 m for the TMW. A
small-diameter polarizer-analyzer ðk=F1Þdoes not pro-
vide 99% attenuation of the TMW signal in their Fig. 3,
unless the polarizer-analyzer is very close to either the trans-
mitter or receiver; that location was unspecified. A definitive
test requires better control of the experimental conditions.
Thus, these tests do not provide clear SLW evidence.
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Inadequacy in previous experiments Ways to avoid inadequacies in present work
(a) Frequency too low (433-446 MHz) Frequency of 8 GHz for lab test ðk3:75 cmÞ
(b) Poorly controlled, test environment Use of well-controlled, laboratory environment
(c) Image charge due to conductive grounding Elimination of return charge by balun
(d) Image current from conductive grounding Elimination of return current by balun
(e) Imprecise measurements Use of modern digital instrumentation
(f) Longitudinal polarization from dipole array Measurement with modern instrumentation
(g) Transmitter-receiver distance error of 65m Positional measurements with sub-mm error
(h) No statistical analysis of test-versus-theory Statistical comparison: experiment with theory
Physics Essays 32, 1 (2019) 125
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