Content uploaded by Hans G. Schantz

Author content

All content in this area was uploaded by Hans G. Schantz on Sep 21, 2014

Content may be subject to copyright.

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 = 410-7 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 (1789–1854) 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:

H

E

Z

(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.