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On Energy Flow in Standing Waves

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This paper presents, analyzes, and explains the propagation of energy in a variety of standing waves. Schelkunoff noted the duality of the impedance concept. In one sense, impedance defines the ratio of electric to magnetic field. In another sense, impedance describes the properties of a transmission line or medium that give rise to the same ratio. Usually, these two ways of looking at impedance yield identical results. For instance, the ratio of electric to magnetic field intensity for a pure wave is 50 ohm in a well-matched 50 ohm line. When standing waves superimpose, however, the actual local instantaneous impedance diverges from that of the medium. Just as electromagnetic waves reflect from discontinuities in the impedance of a medium or a transmission line, so also does the energy associated with electromagnetic waves reflect from the impedance discontinuities resulting from their own superpositions. This physical picture yields the same time average or macroscopic results as a conventional analysis. However a detailed time-domain analysis shows exactly how energy bounces or recoils from the impedance discontinuities associated with superimposed waves. This paper concludes that electromagnetic phenomena are almost universally "near-field" in nature with typical energy velocities much less than the speed of light.
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On Energy Flow in Standing Waves
Hans G. Schantz
Q-Track Corporation
2223 Drake Avenue SW 1st Floor
Huntsville, AL 35805
Abstract: This paper presents, analyzes, and explains the propagation of energy in a
variety of standing waves. Schelkunoff noted the duality of the impedance concept.
In one sense, impedance defines the ratio of electric to magnetic field. In another
sense, impedance describes the properties of a transmission line or medium that give
rise to the same ratio. Usually, these two ways of looking at impedance yield
identical results. For instance, the ratio of electric to magnetic field intensity for a
pure wave is 50 ohm in a well-matched 50 ohm line. When standing waves
superimpose, however, the actual local instantaneous impedance diverges from that
of the medium. Just as electromagnetic waves reflect from discontinuities in the
impedance of a medium or a transmission line, so also does the energy associated
with electromagnetic waves reflect from the impedance discontinuities resulting
from their own superpositions. This physical picture yields the same time average or
macroscopic results as a conventional analysis. However a detailed time-domain
analysis shows exactly how energy bounces or recoils from the impedance
discontinuities associated with superimposed waves. This paper concludes that
electromagnetic phenomena are almost universally “near-field” in nature with
typical energy velocities much less than the speed of light.
1 Introduction
This paper applies conventional electromagnetic energy flow concepts to the problem of
standing waves. The conclusions that logically follow from this well established physics
may appear counterintuitive at an initial glance.
First, this paper reviews basic concepts in electromagnetic energy and describes the
development of the impedance concept in electromagnetics. Then, this paper reviews
recent work on electromagnetic energy flow in superpositions and extends the results to
examine additional cases including standing waves on a transmission line and the
interaction of orthogonal waves in free space. This paper argues that electromagnetic
energy bounces or reflects from impedance discontinuities caused by the interactions of
electromagnetic waves. The conventional point-of-view in electromagnetics holds that
near fields are relevant only in close proximity to sources, sinks, and scatterers of
electromagnetic energy, and that “far-field” electromagnetics describes the much more
commonly observed electromagnetic behavior in free space. This paper offers a contrary
conclusion, that electromagnetic phenomena are almost universally “near-field” in nature,
and far-field electromagnetics is an approximation rarely encountered in reality.
2 Electromagnetic Energy
When the pioneers of mathematical electromagnetics first derived equations to describe
electromagnetic phenomena, they were heavily influenced by the success of Newton’s
Universal Law of Gravity. Many interpreted Newtonian physics as describing
gravitational interactions in terms of an action at a distance between massive objects.
Early mathematical electromagnetics described interactions between charges and currents
similarly. Michael Faraday (1791-1867) challenged this orthodoxy with his concept of
fields. Faraday held that electromagnetic interactions depended not only on the charges
and currents, but also on “lines of force” pervading the intervening spaces. William
Thompson, 1st Baron Kelvin (1824-1907) and James Clerk Maxwell (1831-1879) placed
Faraday’s nebulous concept on a firm mathematical foundation. In Maxwellian
electromagnetics, the energy density (u) is proportional to the sum of the squares of the
electric (E) and magnetic (H) fields:
2
0
2
1
2
0
2
1HE
u
(1)
where
0 = 8.85410-12 F/m and
0 = 410-7 H/m [
1
].
John Henry Poynting (1852-1914) [
2
] and Oliver Heaviside (1850-1925) [
3
] noted that
the flow of electromagnetic energy, or the electromagnetic power per area
HES
(2)
Heaviside further noted that electromagnetic energy flow for a plane wave may be further
characterized by an energy velocity [
4
]:
2
0
2
1
2
0
2
1HE
HES
v
u
(3)
The conception of electromagnetic energy due to Poynting and Heaviside led to some
counterintuitive conclusions regarding the behavior of electromagnetic energy. For
instance, Poynting-Heaviside theory holds that the energy conveyed from battery to load
in a simple electric circuit propagates, not through the wires, but rather through the space
around the wires [
5
].
Fig. 1a (left): A simple electric circuit. Fig. 1b (center): Hydraulic model of energy flow in a circuit, and
Fig. 1c (right): Field model of energy flow in a simple circuit (courtesy William Beaty) [
6
].
In one view, power is the product of current and voltage, and energy is associated with
the movement of the charges. In the other view, the energy is associated with the fields
and the energy dissipated in the resistors follows from integrating the Poynting Vector
over the surface of the resistor. Although the physical interpretation is different, both
approaches will yield the same correct answer for power. Why then does it matter which
model is right and which interpretation we adopt?
The reason is that the correct interpretation that of Poynting and Heaviside laid the
groundwork for further understand and discovery of electromagnetic phenomena. The
idea that energy is carried by the fields and enters the surface of a wire helps explain the
skin effect and AC resistance. Furthermore, understanding that signals and energy are
conveyed by fields propagating at or near the speed of light helps us understand how the
current is actually composed of charges moving at a much slower drift velocity.
The other key idea deserving of a brief review in this context is the concept of
impedance.
3 Impedance
The discovery of the relationship between voltage (V) and current (I) by Georg Simon
Ohm (17891854) was a giant leap forward in understanding electrical physics [
7
]. Ohm
placed the concept of electrical resistance (R) on a firm mathematical footing by
identifying it as the ratio of voltage to current:
I
V
R
(4)
Oliver Lodge (1851-1940) and Oliver Heaviside (1850-1925), generalized Ohm’s
concept to the case of alternating current (AC) and dubbed it “impedance” [
8
,
9
,
10
].
Their ideas faced considerable resistance, however. Paul Nahin vividly describes the epic
battle between the “practical” electrical engineers, exemplified by William Henry Preece
(1834-1913), Engineer-in-Chief of the British Government Post Office (GPO), and the
“theoretical” side championed by the two great Olivers [
11
]. Heaviside and Lodge
defined impedance (Z) as the ratio of voltage and current in a circuit characterized by
resistance (R), capacitance (C), and inductance (L) operating at an angular frequency (
):
2
21
L
C
R
I
V
Z
(5)
The concept spread quickly in electrical engineering practice. Within just a few years the
implications and applications were well and thoroughly explored, resulting in rigorous
time domain analysis of AC systems [
12
]. Arthur Kennelly (1861-1939) and Charles
Proteus Steinmetz (1865-1923) both contributed to the development and popularization
of the phasor approach to AC circuit analysis, extending the impedance concept by
employing complex analysis to describe the behavior of AC phase as well as amplitude
[
13
,
14
].
In the following decades, Sergei Schelkunoff (1897-1992) extended the impedance
concept to include the impedance of fields in free space [
15
]. Schelkunoff identified the
field impedance (Z) as the ratio of the electric (E) to the magnetic (H) field intensities:
(6)
Furthermore, Schelkunoff introduced five basic ideas with respect to impedance:
Use analogies between dynamical fields in which impedance is commonly applied
and those in which it is not,
Extend application of impedance from circuits to fields,
Regard impedance as an attribute of the field as well as the medium,
Assign direction to impedance, and
Generalize the 1-D transmission line concept to higher dimensional problems in
which some of the dimensions may be neglected.
Schelkunoff’s discoveries opened a new perspective on antennas: “From this point of
view, the arms of an antenna form the banks of a channel in which the waves excited by
the source are confined before they emerge into unlimited space. In this sense antennas
are similar to waveguides” [
16
]. Schelkunoff demonstrated that the impedance concept
bridges the gap between fields guided by transmission lines on the one hand and fields
propagating in free space on the other.
We may normalize energy velocity to the speed of light
00
1
c
as follows:
 
   
 
 
 
he
he
he
HE
HESv ˆ
ˆ
12
ˆ
ˆ
2
ˆ
ˆ
222
1
22
2
0
2
1
2
0
2
1
0
0
0
0
z
z
Z
HE
HE
c
cucS
H
E
Z
H
E
S
(7)
where we express the energy velocity in terms of the normalized impedance:
H
E
Z
Z
zH
E
S0
0
0
0
(8)
The normalized impedance is the ratio of the actual instantaneous impedance Z = E/H and
the characteristic impedance of the medium or free space. For a “pure” wave in which the
instantaneous impedance is the same as the characteristic impedance, z = 1 where
positive denotes forward propagation and negative denotes reverse propagation along a
particular direction. A pure wave has equal proportions of electric and magnetic energy.
Wave superposition or interference upsets this equal balance and the resulting “impure”
wave exhibits a normalized impedance z 1. William Suddards Franklin (1863-1930)
explained the distinction between pure and impure waves in 1909 [
17
]. The fact that
electromagnetic energy velocity may in general be less than c appears to have been first
noted by Harry Bateman (1882-1946) in 1915 [
18
], and was recently re-examined by
Gerald Kaiser who interprets energy velocity as a local time dependent characteristic of
electromagnetic fields [
19
].
Fig. 2a (left): Constructive interference of electric fields yields destructive interference of magnetic fields
and a virtual open. Fig. 2b (left): Destructive interference of electric fields yields constructive
interference of magnetic fields and a virtual open.
4 Superposition of Waves on Transmission Lines
This section summarizes a recent study on the physics of electromagnetic superposition
[
20
]. When two identical electromagnetic waves interact so that the electric fields add
constructively, the resulting electric field is twice that of the original waves. Because
energy goes as the square of the field intensity, the resulting superposition has four times
the electric energy of the individual waves. This apparent violation of the conservation of
energy becomes clear by examining the behavior of the magnetic field. A constructive
interference of electric fields results in a destructive interference of magnetic fields.
There is no magnetic energy associated with the superposition, because the magnetic
fields cancel destructively. The excess electric energy is due to the total conversion of
electric energy to magnetic energy. Figure 2a summarizes this constructive interference.
Similarly when two identical electromagnetic waves interact so that the electric fields add
destructively, the resulting electric field is zero and the magnetic field is twice that of the
original waves. Because energy goes as the square of the field intensity, the resulting
superposition has four times the magnetic energy of the individual waves. A destructive
interference of electric fields results in a constructive interference of magnetic fields. The
excess magnetic energy is due to the total conversion of magnetic energy to electric
energy. Figure 2b depicts this destructive interference.
In each case, the superposition disrupts the balance of electric and magnetic energy
characteristic of a pure electromagnetic waves. The Poynting vector is zero at the point of
interaction. The forward and reverse propagating waves exchange energy. The energy
recoils or bounces off the virtual short or virtual open caused by the superposition.
Fig. 3a (top left) EM signals from a discone antenna are incident on a PEC plane.
Fig. 3b (top right) By image theory, we may replace the PEC plane with an inverted source.
Fig. 3c (bottom) Equal and opposite sources cancel out the E-field on the plane of symmetry. The solution
is identical to what we would expect if there were a PEC plane on the plane of symmetry. The identical
solutions should have identical interpretations signals reflect from the plane of symmetry.
One aid to understanding this process is to think in terms of image theory. In image
theory, we consider the problem of a signal reflecting from a conducting plane with
boundary condition
0
ˆnE
, as in Figure 3a, where
n
ˆ
is normal to the conducting
plane. By replacing the conducting plane with an inverted virtual source at an equal and
opposite distance behind the plane, as in Figure 3b, we recover the same boundary
condition. Suppose now that instead of an inverted virtual source we have an inverted
real source. With the conducting plane in place, signals from each source bounce off the
plane. If we remove the plane the boundary conditions remain exactly the same, and the
signals or at least the energy flows associated with them bounce or reflect from the
virtual short formed by the superposition of the signals.
5 Examples of Energy Flow in Standing Waves
This section presents a variety of standing-wave energy-flow examples, including short
Gaussian impulses colliding on a 1-D transmission line, Gaussian impulses interacting at
an angle, and standing waves of harmonic signals.
5.1 Gaussian Waveforms
Gaussian functions provide a useful model for short impulse waveforms. Taking the
derivative of successive Gaussian functions and their derivative yields a family of short
waveforms with increasing numbers of zero crossings. An “nth order” Gaussian has n
zero crossings and n + 1 lobes. Figure 4(a) presents the first five Gaussian waveforms
(n = 0, 1, …, 4) where
is a time constant. Figure 4(b) presents their mathematical
descriptions. The waveforms of Figure 4(a) are in the domain of retarded or advanced
time,
c
z
tu
. The zeroth order Gaussian must be assumed to propagate on a
transmission line since it has DC content. Otherwise, we may assume the waveforms
propagate either on an ideal transmission line or as plane waves in free space. In each
case, we assume equal and opposite electric or voltage waves superimpose destructively
at position z = 0 and time t = 0. Thus, the corresponding magnetic or current waves
superimpose constructively. The mathematical descriptions may be swapped to observe a
constructive superposition of electric or voltage waves and a destructive superposition of
magnetic or current waves.
Assume forward and reverse propagating electric and magnetic waves:
   
   
yyHHH xxEEE
ˆˆ
,,,
ˆˆ
,,,
00
00 uFHuFHtztztz
uFEuFEtztztz
nn
nn
(9a, b)
Each of the following Figures present space-time diagrams for the electric and magnetic
waves, their energy, power, normalized impedance, and normalized velocity:
Figure 4a (left): First five Gaussian waveforms. Figure 4b (right): Mathematical descriptions.
3
2
1
1
2
3
1.0
0.5
0.5
1.0
Gaussian Signals
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
3
124
exp
3
1
6
4
exp
4
1
2
exp2
exp
exp
exp
!!
2
2
4
4
2
4
3
3
2
3
2
2
2
2
2
1
2
0
2
2
uu
uF
u
u
uF
u
uF
u
uF
uF
un
uF
u
u
u
u
u
u
n
n
n
n
n
Figure 5: Interaction of two equal and opposite Gaussian waves with n = 0.
3
2
1
1
2
3
u
1.0
0.5
0.5
1.0
F0u
Gaussian Signal n 0
Figure 6: Interaction of two equal and opposite Gaussian waves with n = 1.
3
2
1
1
2
3
u
1.0
0.5
0.5
1.0
F1u
Gaussian Signal n 1
Figure 7: Interaction of two equal and opposite Gaussian waves with n = 2.
3
2
1
1
2
3
u
1.0
0.5
0.5
1.0
F2u
Gaussian Signal n 2
Figure 8: Interaction of two equal and opposite Gaussian waves with n = 3.
3
2
1
1
2
3
u
1.0
0.5
0.5
1.0
F3u
Gaussian Signal n 3
Figure 9 Interaction of two equal and opposite Gaussian waves with n = 4.
3
2
1
1
2
3
u
1.0
0.5
0.5
1.0
F4u
Gaussian Signal n 4
Figure 10 Two examples of the interactions between unlike Gaussian impulses with different amplitudes.
5.2 Interaction of Arbitrary Gaussian Impulses
When identical waveforms interact, there is a perfect reflection of energy off the virtual
shorts or opens caused by the superposition of the waveforms. In the more general case of
waveform interaction, energy from the smaller waveform accretes to the leading edge of
the larger waveform. The smaller waveform displaces and delays the larger waveform’s
energy as the two waves interact. Finally, the larger waveform contributes the energy to
reconstitute the smaller waveform as the two waves separate. Figure 10 shows two
examples. In each case, the wave propagating in the forward or +z direction is larger than
the wave propagating in the reverse or z direction. Figure 10 plots the energy density on
a log scale for emphasis.
On may readily plot electromagnetic energy flow
patterns by following a simple rule don’t cross the
streams [
21
]. The behavior inherent in the Poynting-
Heaviside theory of electromagnetic energy requires
that the order or arrangement of electromagnetic
energy remain fixed. Note in both of the one-
dimensional interactions of Figure 10, the order of the
energy from right to left is preserved. A Newton’s
Cradle like that of Figure 11 serves as a model for
visualizing the interactions. Although a particular wave
motion propagates through the system, the energy
(balls) preserve their order.
Figure 11 A Newton’s Cradle
(courtesy Wikimedia).
5.3 Interaction of Harmonic Signals
This section looks at standing waves of harmonic signals. The forward and reverse
propagating voltage and current signals are:
     
     
tkzItkzItzItzItzI
tkzVtkzVtzVtzVtzV
sinsin,,,
sinsin,,,
00
00
(10a, b)
where k = 2/
is the wave number,
= 2
f is the angular frequency, and is the
reflection coefficient, the ratio of the reverse wave amplitude relative to the forward wave
amplitude. The power then is:
             
               
   
 
tkztkzIV
tzItzVtzItzVtzItzVtzItzV
tzItzItzVtzVtzItzVtzP
222
00 sinsin
,,,,,,,,
,,,,,,,
(11)
and the energy density is:
     
 
   
2
2
0
2
1
2
0
2
1
2
2
1
2
2
1sinsin,,, tkztkzLICVtzLItzCVtzu
(12)
where C is the capacitance per length and L is the inductance per length. The voltage
standing wave ratio (VSWR) follows from the reflection coefficient:
1
1
VSWR
(13)
By analogy to (7), the normalized energy velocity is:
     
 
tzcu tzP
ctzv
tz ,
,,
,
(14)
The normalized energy velocity provides a local measure of energy velocity. We may
obtain an average energy velocity to characterize the overall behavior of a particular
system by employing an energy-weighted average:
 
   
   
 
2 2 2
0 0 0 0
22
11 2
00
22
00
2
22
sin sin
sin sin
12
11
k
k
S
cu
kz t kz t dt
uS VI
u cu c CV LI kz t kz t dzdt
VSWR
VSWR




 
   


 


(15)
We perform a spacetime average of the periodic functions over a half period and a half
wavelength, although equivalently, a time average at a fixed location or a space average
at a fixed time would yield equivalent results.
VSWR
<S>/<cu>
LogMag S11 (dB)
1.000000
infinite
0.000
0
0.707107
5.83:1
0.333
-3.01
0.500000
3.00:1
0.600
-6.02
0.333333
2.00:1
0.800
-9.54
0.200000
1.50:1
0.923
-13.98
0.100000
1.22:1
0.980
-20.00
0.031623
1.07:1
0.998
-30.00
0.010000
1.02:1
1.000
-40.00
Table 1: Voltage Standing Wave Ratio and Average Energy Velocity as a function of “
.
Although in general the average of the ratio is not the ratio of the average, the energy-
weighted normalized velocity works out to the ratio of the average power to the average
energy. Table 1 presents results for a variety of voltage standing wave ratios.
In a perfect standing wave, = 1. The trig-product identities allow for a considerable
simplification in (10a, b) in this case:
 
 
 
 
tkzI
tkztkztkztkzI
tkzItkzItzI
tkzV
tkztkztkztkzV
tkzVtkzVtzV
sincos2
sincoscossinsincoscossin
sinsin,
cossin2
sincoscossinsincoscossin
sinsin,
0
0
00
0
0
00
(17a, b)
Then, the power in this special case is given by:
     
tkzIVttkzkzIV
tzItzVtzP
2sin2sincossincossin4
,,,
0000
(18)
The power goes to zero at quarter-wave intervals: z = 0, ¼
, ½
, ¾
,
,… This
bounds energy within each quarter-wave segment of the transmission line. Power also
goes to zero at quarter period intervals: t = 0, ¼T, ½T, ¾T, T,… This means energy
is momentarily static at those instants. By inspection of the voltage and current, electric
energy concentrations occur at voltage maxima: z = 0, ½
,
,… and t = ¼T, ¾T,…
and magnetic energy maxima appear offset a quarter wavelength and a quarter period at
the current maxima: z = ¼
, ¾
,… and t = 0, ½T, T,… Thus the electric and
magnetic energies are in space-time quadrature. Gerald Kaiser examined this same
problem, demonstrating that energy oscillates back and forth between successive nodes in
a standing wave, recoiling from the resulting superpositions [19]. Figure 12 shows space-
time energy flow diagrams along a transmission for the first four examples of Table 1
where k =
=
/2 so that a half wavelength and a half period are both of dimension one.
Figure 12 Energy flow in standing waves for various values of VSWR.
For
< 1, a net energy flow arises. At
= 0.707 for instance, the voltage standing wave
ratio (VSWR) is 5.83:1. The energy-weighted average velocity is 0.333c. The reflected
energy is half the forward energy, or in other words, the forward energy is two-thirds the
total while the reverse energy is one-third. One way to interpret the result is to note that
the reverse energy flow cancels out half the forward energy flow leaving one-third
forward propagating energy and two-thirds static energy. In this interpretation, energy is
either static or propagates at the speed of light, and the energy velocity is the ratio of
propagating to total energy.
5.4 Interaction of Gaussian Signals at Right Angles
This section looks at Gaussian waves interacting at right angles. Consider two
electromagnetic waves, one propagating in the +x-direction and the other in the +y-
direction with Gaussian doublet (n = 1) time dependence:
   
 
   
 
2
2
cty
ctx
ectyctyg
ectxctxf
(19a,b)
The electric and magnetic fields are:
   
     
tyxtyxtyx
tyxtyxtyx
,,,,,,
,,,,,,
21
21 HHH EEE
(20a,b)
where:
 
 
 
 
2
2
10
20
ˆˆ
,,
ˆˆ
,,
x ct
y ct
x y t f x ct E x ct e
x y t g y ct E y ct e


 
 
E y y
E x x
(21a,b)
and:
 
 
 
 
2
2
10
20
ˆˆ
,,
ˆˆ
,,
x ct
y ct
x y t f x ct H x ct e
x y t g y ct H y ct e


 
 
H z z
H z z
(22a,b)
where E0 and H0 are electric and magnetic field intensity per length, respectively. Figure
13 plots the electric field and electric energy on the x-y plane. The fields are orthogonal at
the intersection, so the net field is
2
the individual field intensity, and the electric
energy doubles.
Figure 13 Electric field and electric energy for two waves interacting at right angles at time t = 0.
Figure 14 Magnetic field and magnetic energy for two waves interacting at right angles at t = 0.
Figure 14 plots the magnetic field and magnetic energy on the x-y plane. The fields add
constructively for part of the intersection and destructively for part. The magnetic energy
rearranges itself, concentrating at the constructive interference and dispersing away from
the destructive interference. The magnetic field goes to zero along the line x = y. The
superposition of the two waves creates a virtual perfect magnetic conductor along this
diagonal. Figure 15 shows power and total energy. The magnetic energy concentrations
propagate tangent to the x = y diagonal plane at an apparent (phase) velocity of
2
c. No
energy propagates through the x = y plane. On one side, energy from the +x-direction
wave reflects off the x = y plane to reconstitute the +y-direction wave. On the other side
of the plane, energy from the +y-direction wave reflects to reconstitute the +x-direction
wave. Oliver Heaviside described this behavior in 1912 [
22
].
Figure 15 Power and total energy for two waves interacting at right angles at t = 0.
6 The Big Picture: Near-Field or Far-Field?
The superposition principle allows us to reduce complicated electromagnetic problems to
more manageable component pieces. By allowing us to focus on each particular aspect of
interest in turn, while ignoring less relevant characteristics of the system, the
superposition principle vastly simplifies our investigations and analysis.
However, the great benefit of superposition leads to a similarly great misconception the
notion that aspects of a system not relevant to an analysis are not relevant to
understanding the system as a whole. Superposition allows us to ignore other aspects of
reality as irrelevant to a particular problem. While those other aspects may be safely
ignored in the narrow context of a particular calculation, they are nevertheless present in
reality. To understand the big picture of an electromagnetic system, one must step back
and take a look at how all the components fit together to contribute to how the system as
a whole really works.
Consider a typical electromagnetic calculation involving a cell phone link. We envision
the problem as involving a direct propagation of energy from the phone to the tower. We
might also consider the effect of one or more alternate propagation paths, for example, a
signal reflecting off the ground. For purposes of a detailed calculation we might look at
near-field effects close to the transmit and receive antennas. But most of the problem is
the essentially far-field calculation of understanding how the cell phone energy
propagates through free space from phone to the tower.
While the picture of Figure 16a may lead to a correct calculation, the physical depiction
misleads. In reality, signals and energy propagate from the phone in all directions, setting
up the complicated interference pattern of signals in Figure 16b. However, this “optical”
picture of the situation similarly misleads. Applying the principle “don’t cross the
streams,” leads to the more correct physical picture of Figure 16c. Each point at which
rays intersect actually denote mutual recoil or reflection of energy. The energy
trajectory from the phone to the tower does not involve the ground at all, only myriad
meanders as the energy bounces off and interacts with other energy in the system.
Although signals and fields propagate at the speed of light, the energy propagates at a
more leisurely pace depending on the amount and extent of the delays caused by the
overall standing wave environment.
Figure 16a (left) Two-ray model of direct and multipath energy from a phone to a tower.
Figure 16b (center) Multiple-ray model of propagation around a phone.
Figure 16c (right) Energy flow model of propagation around a phone.
Figure 17a (left): Map of downtown Huntsville, AL (1.9km x 1.4km) showing measurement locations (blue
squares). Figure 17b (right): Measured impedance for 1230kHz broadcast signal in an urban area.
Even the depiction of Figure 16c presents a gross oversimplification. The actual standing
wave reflections and reverberations of energy occur on a scale on the order of a half-
wavelength too small for a coherent presentation in the figure. These actually become
significant for frequencies low enough and wavelengths long enough that a half-
wavelength becomes a significant distance.
Consider, for instance, the propagation of an AM broadcast signal with frequency
1230kHz and wavelength 244m. Figure 17a shows an urban area throughout which we
sampled electric and magnetic field strength for the broadcast signal. Figure 17b shows
the measured impedance in deciBels relative to the free space impedance. Significant
swaths of the area are as much as 20dB higher (37.67kohm) or 20dB lower (3.767ohm)
than would be expected if the impedance of this region far from the transmitter were truly
characterized by a so-called far-field impedance. “Near-field” variations like these lend
themselves well to a precise indoor location system [
23
].
Even looking at the behavior of a particular frequency signal, the so-called far-field zone
exhibits substantial near-field behavior. The assumption that near-fields are relevant only
near to sources and that reality as a whole exhibits far-field behavior may be a helpful aid
to simplifying analysis, but this assumption presents a mistaken view of reality.
We may more closely approach an understanding of nature’s true behavior by
recognizing that reality comprises a vast superposition of electromagnetic signals from
different sources all mutually co-existing and interacting at the same time and the same
places.
Even an updated Figure 16c showing the detailed interactions of the energy flow from the
phone to the tower correctly capturing behavior on a scale of a half wavelength presents a
gross over-simplification, because it only considers the particular frequency of operation
of the cell phone. In reality, other frequency signals from DC to daylight pervade the
region between the phone and the tower. The 800mW or so phone signal contributes only
the faintest of whispers to the cacophony of other signals and other energy flows.
Figure 18: One-dimensional standing wave energy flow resulting from equal and opposite waves with a
factor of fifteen difference in frequency or wavelength as might be typical of sunlight in thermal
equilibrium with the ground.
The dominant energy flow during daylight hours is likely to be the approximately
1kW/m2 flux of energy from sun light. Later in the day as the region reaches thermal
equilibrium, there will be an equal and opposite flux of infrared energy [
24
]. Figure 18
depicts a notional standing wave energy flow as equal and opposite fields with a factor of
fifteen difference in wavelength interact. At night, the infrared radiation of the earth
dominates, supplemented by everything from the earth’s magnetostatic and electrostatic
fields, to lighting impulses, and high power broadcast signals.
In all likelihood, none of the energy actually radiated from the phone ever reaches the
tower. Instead, the energy originating from the phone quickly loses itself in the collective
superposition of energy near the phone. The fields from the phone send a faint ripple
through the collective superposition of electromagnetic region, slightly perturbing the
existing standing waves. In the vicinity of the tower, a small fraction of the local energy
decouples from the superposition and interacts with the receive antenna on the tower.
In this picture, electromagnetic energy behaves like current in a wire. Fields or signals are
disturbances that propagate at the speed of light. However, energy usually propagates at a
much more sedate drift velocity, far less than the speed of light.
The far-field representation of electromagnetics is a useful analytic approach to solve a
broad range of physical problems. As a physical model of reality however, the far-field
representation is only valid in a context in which there is one dominant energy flow.
Examples might include the propagation of sunlight within the solar system, the flow of
energy near a high power transmitter, or the behavior of a transmitter at power levels well
above thermal in an anechoic chamber.
Self-induction’s “in the air,”
Everywhere, everywhere;
Waves are traveling to and fro
Here they are, there they go.
Try to stop ‘em if you can
You British Engineering man!
Oliver Heaviside [
25
]
7 Conclusion
Schelkunoff described the dual nature of the impedance concept as the ratio of electric to
magnetic field intensity as well as a characteristic of media that typically gives rise to a
particular ratio. Just as an impedance discontinuity in a medium causes a reflection of
some of the propagating energy, so also does an impedance discontinuity arise in the
interaction or superposition of waves cause with similar reflections of energy.
When equal and opposite electromagnetic waves interact, energy bounces back and forth,
but there is no net flow of energy. As one wave becomes larger than the other, the
bouncing continues but a net energy flow emerges. As one wave dominates the other, the
dominant flow comes in half-wavelength scale packets with a slight half wavelength
backwash caused by the influence of the weaker wave. When two waves interact in a
two-dimensional setting, the point of intersection may generate impedance discontinuities
that reflect some or all of the interacting energy.
We may relate average electromagnetic energy velocity to conventional microwave
parameters like reflection coefficient, (normalized) impedance, and VSWR:
 
2
2 2 2
1 2 2
11
1
z VSWR
cu zVSWR

 
 
S
(23)
Superposition allows us to solve complicated electromagnetic problems by abstracting
away unnecessary complexities. We can ignore all other frequency components to focus
on a particular frequency of interest. We can even ignore most of what might be
happening at a particular frequency to consider only the particular components relevant to
the narrow purposes of our calculation. But just because we can ignore the complexity to
solve a particular problem doesn’t mean that the complexity isn’t lying in wait to
confound our understanding of other problems.
When one considers the myriad different electromagnetic waves interacting around us, it
becomes clear that near-field interactions and behavior are the rule and not the exception.
Reality itself may be best characterized as a predominantly near-field, not far-field,
phenomenon. The propagation of electromagnetic energy from source to destination
follows, not ideal optical rays, but rather a complicated meander or drift as particular
fields perturb the energy of the collective superposition one way or the other.
Electromagnetic energy very rarely propagates significant distances at the speed of light,
but instead ebbs and flows at an average drift velocity less than the speed of light.
The concepts presented in this paper are not new physical discoveries. Rather they are the
logical conclusion of applying time-tested and fundamental concepts of electromagnetics
and electromagnetic energy flow to the problem of understanding the large scale
interactions of electromagnetic waves with each other. The accomplishments of this
perspective are relatively modest to date helping the author and his colleagues
understand the behavior and propagation of long wavelength, low-frequency signals to
enable better location systems for urban and indoor environments.
But just as the insights of Poynting and Heaviside led to a clearer understanding of such
concepts as the drift velocity of electrons and the skin effect, so also can understanding
the large scale behavior of electromagnetic energy transfer help us to understand the
seemingly mysterious and counterintuitive ways in which electromagnetic entities act
across the intervening space.
8 Acknowledgements
The author acknowledges helpful discussions with Gerald Kaiser who has done much to
champion the view that electromagnetic waves may be characterized by a spacetime local
energy velocity.
9 References
[
1
] Maxwell, James Clerk, A Treatise on Electricity and Magnetism, Vol. 2, 3rd ed.,
Oxford: Clarendon Press, 1892, pp. 270-274. See Section 631 in particular.
[
2
] Poynting, John Henry, On the Transfer of Energy in the Electromagnetic Field,”
Philosophical Transactions, Royal Society, London, Vol. 175 Part II, 1885, pp. 334-
361.
[
3
] Heaviside, Oliver, Electrical Papers, Vol. 1, London: The Electrician Publishing
Company, 1892, pp. 449-450. Originally published as “Electromagnetic Induction
and Its Propagation,” in The Electrician, February 21, 1885.
[
4
] Heaviside, Oliver, Electromagnetic Theory, Vol. 1, The Electrician, London, 1893,
pp. 78-80. In particular see Equation (15) on p. 79: http://bit.ly/1fmBZsw
[
5
] Galili, Igal and Elisabetta Goihbarg, “Energy transfer in electrical circuits: A
qualitative account,” American Journal of Physics, Vol. 73, #2, Feb. 2005, pp. 141-
144. See: http://sites.huji.ac.il/science/stc/staff_h/Igal/Research%20Articles/Pointing-
AJP.pdf
[
6
] Beaty, William, “In a Simple Circuit, Where Does the Energy Flow? December,
2000. See http://amasci.com/elect//poynt/poynt.html . Figure courtesy of William
Beaty.
[
7
] Ohm, Georg Simon, Die galvanische Kette, mathematisch bearbeit (The Galvanic
Circuit Investigated Mathematically), Berlin: T.H. Riemann, 1827.
[
8
] Heaviside, Oliver, “On Induction Between Parallel Wires,” Journal of the Society of
Telegraph Engineers, vol. 9, 1881, p. 427. Collected in Electrical Papers, Vol. 1, pp.
116-141. See in particular p. 125.
[
9
] Heaviside, Oliver, “On Resistance and Conductance Operators, and Their
Derivatives, Inductance and Permittance, Especially in Connection with Electrical
and Magnetic Energy,” Philosophical Magazine, December 1887, p. 479. Collected in
Electrical Papers, Vol. 2, pp. 355-374.
[
10
] Lodge, Oliver, “On Lightning, Lightning Conductors, and Lightning Protectors,”
Electrical Review, May 3, 1889, p. 518.
[
11
] Nahin, Paul, Oliver Heaviside: The Life, Work, and Times of an Electrical Genius
of the Victorian Age, Baltimore: The Johns Hopkins University Press, 2002. See
Chapter 8: “The Battle With Preece,” and Chapter 10: “Strange Mathematics.”
[
12
] Bedell, Frederick and Albert Crehore, “Derivation and Discussion of the General
Solution for the Current Flowing in a Circuit Containing Resistance, Self-Induction
and Capacity, with Any Impressed Electromotive Force,” Journal of the AIEE,
Volume 9, 1892, pp. 303-374.
[
13
] Kennelly, A.E., “Impedance,” Transactions of the AIEE, Vol. X, January 1893, pp.
172-232.
[
14
] Steinmetz, Charles Proteus, Theory and Calculation of Electric Circuits, New
York: McGraw Hill, 1917.
[
15
] Schelkunoff, S. A., “The Impedance Concept and Its Application to Problems of
Reflection, Refraction, Shielding and Power Absorption,” The Bell System Technical
Journal, Vol. 17, No. 1, 1938, p. 17-48. See: http://www3.alcatel-
lucent.com/bstj/vol17-1938/articles/bstj17-1-17.pdf
[
16
] Schelkunoff, S. A., Advanced Antenna Theory, New York: John Wiley & Sons,
1952, p. viii.
[
17
] Franklin, William Suddards, Electric Waves: An Advanced Treatise on Alternating-
Current Theory, New York: The Macmillan Company, 1909, pp. 88-93. See Chapter
4. See: http://bit.ly/1rerFKe
[
18
] Bateman, Harry, The Mathematical Analysis of Electrical and Optical Wave-
Motion On the Basis of Maxwell’s Equations, Cambridge University Press, 1915, p.
6. See: http://bit.ly/1ykUfeE
[
19
] Kaiser, G., “Electromagnetic inertia, reactive energy and energy flow velocity,” J.
Phys. A: Math. Theor. 44, 8 August 2011. See: http://bit.ly/1w51bAB
[
20
] Schantz, Hans G., “On the Superposition and Elastic Recoil of Electromagnetic
Waves,” FERMAT, Vol. 4, No. 2, July-August 2014 [ART-2014-Vol4-Jul_Aug-002].
See also http://arxiv.org/abs/1407.1800
[
21
] Aykroyd, Dan, Harold Ramis, Ghostbusters, Columbia Pictures, 1984. According to
Egon Spengler, crossing streams is really bad: “Try to imagine all life as you know it
stopping instantaneously and every molecule in your body exploding at the speed of
light. His colleague, Ray Stantz, describes this as “total protonic reversal.
Fortunately, Poynting-Heaviside theory does not appear to allow electromagnetic
energy flow streams to cross.
[
22
] Heaviside, Oliver, Electromagnetic Theory, Vol. 3, London: The Electrician
Printing and Publishing Company, 1912, p. 15.
[
23
] Schantz, Hans et al, “Method and apparatus for determining location using signals of
opportunity, U.S. Patent 8,253,626, August 28, 2012.
[
24
] Neglecting convection, of course, which is probably a gross oversimplification.
[
25
] Heaviside Notebook 7, p. 113, OH-IEE; quoted by Bruce J. Hunt, The Maxwellians,
Ithaca, NY: Cornell University Press, 1991, p. 173.
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