Waves in a short cable at low frequencies, or just hand-waving?

Abstract

We address the question of low-frequency signals in a short cable, which are
often considered as waves in engineering calculations. Such an assumption
violates several laws of physics, but exact calculations can be carried out via
linear network theory.

Full text

Waves in a short cable at low frequencies, or just
hand-waving?
What does physics say?
(Invited paper)
L.B. Kish
Department of Electrical Engineering
Texas A&M University
College Station, TX 77843-3128, USA
S.P. Chen
Department of Electrical Engineering
Texas A&M University
College Station, TX 77843-3128, USA
C.G. Granqvist
Department of Engineering Sciences
The Ångström Laboratory, Uppsala University
P.O. Box 534, SE-75121 Uppsala, Sweden
J.M. Smulko
Department of Metrology and Optoelectronics
Faculty of Electronics, Telecommunications and Informatics
Gdansk University of Technology
Narutowicza 11/12, 80-233 Gdansk, Poland
Abstract—We address the question of low-frequency signals
in a short cable, which are often considered as waves in
engineering calculations. Such an assumption violates several
laws of physics, but exact calculations can be carried out via
linear network theory.
Keywords—waves; retarded potentials; thermal noise; black
body radiation.
I. INTRODUCTION
Recently, a, seemingly effective, new attack [1] against the
Kirchhoff-law–Johnson-noise (KLJN) secure key exchange
scheme [2,3] was published. Even though the KLJN scheme
has proven unconditional security [3], the claimed strong
information leak, if valid, would have severely limited its
practical applications. Soon it was shown that the attack [1] had
both theoretical [4] and experimental [5] flaws and that it is no
more efficient than the old wire resistance based attacks [6].
Moreover, even this negligible information leak can be
nullified by a recent defense protocol [6].
While the new attack [1] became obsolete, an interesting
question, which is unrelated to security, remained. The authors
behind the new attack [1] used a wave picture to deduce their
results. However, the KLJN scheme operates in the quasi-static
limit of electrodynamics, which means that the cable length is
much shorter than the wavelength. In a physical description of
cables with finite length D, the frequency-space of wave
solutions is quantized to discrete values so that integer
multiples of the half-wavelength fit in the cable. Thus the
longest allowed wavelength
λ
max
of the wave, and the lowest
allowed frequency fmin at phase velocity
v
, can be written as
λ
max =2D , fmin =v
2D
, (1)
respectively. Frequencies below fmin, down to zero frequency,
constitute a forbidden band of wave states. The KLJN
condition for the signal (noise) bandwidth B is
B<< v
2D
, (2)
which excludes the possibility of wave solutions in the band.
This is how physicists treat this problem.
However, we have learned from electrical- and microwave
engineers that, during everyday practice, such low-frequency
signals in small systems are usually treated as waves and that
those wave-based mathematical solutions are taken to be
exact.
A recent paper of ours [7] clarified the situation. Here we
summarize those arguments and also show how an
engineering-mathematical calculation can be exact in its final
result but, at the same time, unphysical in its internal steps.
The final conclusion is that the assumption of waves in the
low-frequency (quasi-static) regime violates several laws of
physics, yet it is fine to use waves in engineering calculations
provided the waves in these internal steps are not considered
physically existent. The observed propagation delays are not
physical waves but retarded potentials without significant
Invited paper at the 23'rd International Conference on Noise and Fluctuations, 2015

involvement of the conjugate physical quantities, which would
play an identical role in a real electromagnetic wave.
II. WAVE SOLUTIONS IN LINEAR RESPONSE THEORY
In linear time-invariant network analysis, the weighting
function
h(t)
is the response to a Dirac pulse at the input, as
shown in Fig. 1.
Fig. 1. The weighting function h(t) is the response to a Dirac pulse at the input.
Any signal x(t) with limited bandwidth at the input is then
divided into abstract zero-width pulses, and the time response
to these input pulses at the output are calculated and summed
up by a convolution to get the output signal Y(t); see Fig. 2.
Fig. 2. The output signal Y(t) is the convolution of elementary responses due
to the abstract input pulses with zero width and infinite bandwidth.
The zero-width input pulses have infinite bandwidth, even
though the total input signal may be very slow. Thus, if the
linear system is a short cable and the signal bandwidth
satisfies Rel. 2, the zero-width input pulses still satisfy Eq. (1)
(excepting an infinitesimally small fraction of their Fourier
spectrum). These pulses will hence propagate through, and
reflect in, the cable as waves. The propagation can be
described by waves and at the end, during the output
summation, all wave components disappear as a consequence
of their destructive interference so that we regain the slow,
non-wave output signal Y(t).
It is obvious that these abstract pulses and their propagation
are non-physical because, for example, we cannot detect X-
rays or impacts of the uncertainty principle due to the high
bandwidth and short time scales involved. In engineering and
physics, many similar mathematical contrivances are used to
calculate results in an easy way. Clearly, mathematics is a
much broader field than physics, and physics will ultimately
make use of those mathematical solutions that are physical. It
is important not to assign physical characteristics to elements
that are only non-physical mathematical constructs.
III. PROPAGATION OF RETARDED NON-WAVE POTENTIALS
A. What is a physical wave, and what makes it different from
a propagating slow, non-wave signal?
(i) A physical wave is a dynamical oscillation that takes
place via the propagation in space without energy decay in a
loss-free medium and without the need of an external generator
to keep it going.
(ii) It is also essential that there are two oscillating, dual
energy forms (with conjugate physical quantities), and the total
energy is the sum of these two energies, which are equally
distributed. These energies are manifest via a dynamical
transfer into the dual form, and back, during propagation: they
“induce” and “regenerate” each other during the propagation
without loss (in loss-free media).
Example 1: In the case of electromagnetic waves, the two
energy forms are electric and magnetic, and the way of
regenerating each other is via induction and displacement
currents.
Example 2: In the case of elastic waves, the two energy
forms are potential energy due to deformations and kinetic
energy due to motion, and the way of regenerating each other is
via Newton’s and Hook’s laws.
It is important that, in a loss-free medium, this energy
transfer must have 100% efficiency since otherwise the wave
would decay during propagation (without external generator
drive) and then it is not an electromagnetic wave but a non-
wave-type retarded potential, such as the near-field around an
antenna or the magnetic and electrical field around a wire.
Figures 3 and 4 show two situations where travelling
structures (or their fields) look like waves for the superficial
observer who does not execute a deeper investigation toward
the nature of these propagating oscillations.
Fig. 3. Non-wave-type retarded oscillation: a wave-shaped solid structure
(such as wire) is traveling with velocity v in positive direction along the x
axis. The observer sees it as a propagating wave. While the oscillation
satisfies d’Alembert’s equation for waves, this is not a wave. The dynamical
transfer between the two dual energy forms is missing.
Surprisingly, both non-wave oscillations in Figs. 3 and 4
satisfy d’Alembert’s equation [1] typically used for waves,
i.e.,
U t,x
( )
=U+t−x
v
⎛
⎝
⎜⎞
⎠
⎟
, (3)
thus indicating that d’Alembert’s equation is not necessarily
for waves but for propagating oscillations in general.

Fig. 4. Non-wave-type retarded oscillation: a wave-shaped frozen-in magnetic
field or electrical field structure (such as in a magnetic tape or electret tape) is
traveling with velocity v in positive direction along the x axis. The observer
sees it as a propagating wave. While the oscillation satisfies d’Alembert’s
equation for waves, this is not a wave. The dynamical transfer between the
two dual energy forms is missing.
B. Electric and magnetic energy balance in a driven short
cable at low frequencies
Consider now a short cable satisfying Rel. 2, which is
driven by an ac voltage generator
U(t)
at one end and, at the
other end, is terminated with the wave resistance Rw given as
Rw=Lu
Cu
, (4)
where
Lu
and
Cu
are the unit length (1 meter) cable
inductance and cable capacitance, respectively. Then the cable
has equal amounts of energy in the electric and magnetic
forms, as discussed below, and both of these energies are
oscillating. However, these energies are not “bouncing”
between the dual electric and magnetic forms but oscillate
separately between these forms and the generator. This fact is
obvious from the considerations that follow next.
Quasi-static conditions (Rel. 2) imply that the time-
dependent voltage and current are spatially homogeneous
along the wire, i.e.,
U(x,t)U(t)
, (5)
I(x,t)I(t)
. (6)
Thus the electrical and magnetic energies, denoted
Ee(t)
and
Em(t)
, respectively, in the cable can be written
I(x,t)I(t)
, (7)
Em(t)=1
2
LcI2(t)=1
2
Lc
U2(t)
Rw
2
, (8)
where
Cc=DCu
and
Lc=DLu
are the capacitance and
inductance of the whole cable, respectively, and the last term
of Eq. (8) originates from the well-known fact that a cable of
arbitrary length, closed with the wave impedance (resistance)
Rw, has an input impedance of exactly Rw. Thus
Ee(t)
Em(t)
=Cc
Lc
Rw
2=Cu
Lu
Rw
2=1
(9)
at each moment. Obviously, no “bouncing” phenomenon takes
place between the dual electric and magnetic forms of energy,
because the energy is bouncing rather between the generator
and the magnetic and electric fields.
Suppose now that RA and RB, rather than Rw, terminate the
cable ends. The only significant change that will occur in the
quasi-static limit is that the ratio of electrical and magnetic
energies will differ from unity, which further goes against the
wave hypothesis.
In conclusion, even the simple picture of wave modes given
above proves that waves do not exist in a cable in the quasi-
static limit.
IV. SHORT CABLE DRIVEN BY LOW-FREQUENCY JOHNSON
NOISE
Simple but fundamental thermodynamic considerations also
prove that waves cannot exist in a cable in the quasi-static
frequency range, as elaborated below.
A. Proof that the total energy of a cable is less than the
energy required for a single wave mode
As an example, we mathematically analyze how the no-
wave situation manifests itself in a lossless cable, which is
closed at both ends by resistors equal to the wave impedance
Rw at the temperature T. These conditions are not necessary
but serve to simplify the calculations. Suppose that, in
accordance with Rel. 2, the Johnson noise of the resistors has
an upper cut-off frequency at fc so that
fc<< fmin =v
2D
(10)
holds in order to satisfy Rel. 2. Thus partial homogeneity of
current and voltage is valid for the cable, and it is straight-
forward to calculate the electric and magnetic energies of
thermal origin, as shown next.
According to the Johnson–Nyquist formula, the thermal
electrical energy in the cable capacitance is
Ee,th =1
2CcU2(t)=1
2Cc
4kTRw/ 2
1+f2/f0C
2
0
fc
∫df ≅
≅kTCcRwfc=kT
2
fc
fmin
<< kT
2
(11)
and the thermal magnetic energy in the cable inductance is
Em,th =1
2LcI2(t)=1
2Lc
4kT / 2Rw
( )
1+f2/f0L
2
0
fc
∫df ≅
≅kT Lc
Rw
fc=kT
2
fc
fmin
<< kT
2
(12)
where the characteristic frequencies of the Lorentzian spectra
in Eqs. (11) and (12) are defined as

f0C=1
2
π
CcRw/ 2
=1
π
DCu
Cu
Lu
=1
π
D
1
LuCu
=
= 1
π
vc
D
=2
π
fmin
(13)
and
f0L=2Rw
2
π
Lc
=1
π
Lc
Lu
Cu
=1
π
D
1
LuCu
=1
π
vc
D
=2
π
fmin
. (14)
Similar calculations can be carried out for the general case
in which the cable ends are not terminated by Rw but with
different resistance values RA and RB. Specifically, the parallel
resultant resistance RA and RB (instead of Rw/2) enters in the
left-hand side of Eqs. (11) and (13), and the serial (sum) loop
resistance RA + RB (instead of 2Rw) enters in the left-hand side
of Eqs. (12) and (14), while the final inequalities shown by
Eqs. (11) and (12) remain.
If there is loss in the cable, and the cable is terminated by
RA and RB, the same situation holds provided RA and RB as well
as the cable have the same (noise) temperature, because the
system is still in thermal equilibrium. If the cable is cooler,
then there is an energy flow out of the cable, which further
strengthens the inequalities at the right-hand sides of Eqs. (11)
and (12).
In conclusion, Eqs. (11) and (12) prove that for a short
cable and within the frequency range of interest for the KLJN
scheme, the sum of electrical and magnetic energies in all of
the hypothetical “wave modes” is much less than the energy
needed for a single wave mode in thermal equilibrium.
B. Violation of the Energy Equipartition Theorem, the
Principle of Detailed Balance and the Second Law of
Thermodynamics
According to Boltzmann’s Energy Equipartition Theorem
[7] for thermal equilibrium at temperature T, each
electromagnetic wave mode has a mean thermal energy equal
to kT, where k is Boltzmann’s constant. Half of this mean
energy is electrical and the other half is magnetic. For N
different wave modes in the system, both the electrical and
magnetic fields carry a total energy equal to NkT/2.
It is easy to see that the assumption that, in a hypothetical
wave system, these wave energies are less than the above
given values violates not only the Energy Equipartition
Theorem but also the Principle of Detailed Balance [6] and the
Second Law of Thermodynamics: Coupling this hypothetical
wave system to a regular one would hit the Detailed Energy
Balance of equilibrium between wave modes in the
hypothetical and regular systems because it would yield an
energy flow toward the hypothetical one. This energy flow
could then be utilized for perpetual motion machines of the
second kind, i.e., violate the Second Law of Thermodynamics.
C. Violation of Planck’s Law and experimental facts for
blackbody radiation
Planck [7] deduced his law of thermal radiation, for
simplicity, from the properties of a box with black walls, i.e.,
internal walls with unity absorptivity and emissivity. In
thermal equilibrium, thermal radiation within an infinitely
large box with walls of arbitrary absorptivity, emissivity and
color has a power spectral intensity for each polarization given
by
I(f)=4
π
hf 3
c2
1
ehf /kT −1
, (15)
where h is Planck’s constant. The derivation of this formula is
based on counting existing wave modes in the frequency range
f > fmin , where the minimum frequency is obtained by the same
frequency quantization as we claim exists in a cable. A certain
misconception exists that, in closing the finite-size cable by
the wave impedance (resistance) Rw at its two ends, all of the
lower frequencies, at f < fmin , will also be available for wave
modes as a consequence of the unity absorptivity and
emissivity of the impedance match. However, one must realize
that a cable closed by the wave impedance (resistance) Rw at
its two ends is a one-dimensional realization of Planck’s box
with perfectly absorbing (black) walls. Allowing the f < fmin
frequency range for wave modes results in a non-quantized
continuum distribution of wave modes, which yields an
infinite number N of wave modes in any finite frequency band.
Such a situation in the finite-size cable results in an infinite
amount of thermal energy NkT in any finite frequency band
within the classical-physical frequency range f << kT/h. This
situation is not a problem in a cable or box with infinite size.
However, in a finite-size box, infinite thermal energy in the
sum of wave modes would naturally result in infinite intensity
of blackbody radiation.
Furthermore, the situation is similar when the cable ends are
terminated by RA and RB rather than by Rw. According to
Planck’s results, discussed above, the thermal radiation field
in the closed box and cable does not depend on the
absorptivity and emissivity of the walls—i.e., on the cable
termination resistances—since otherwise Planck’s Law would
be invalid and his formula violate the Second Law of
Thermodynamics. Therefore the above argumentation
regarding infinite energies and infinite thermal radiation holds
for arbitrary termination and wall color.
If there is loss in the cable and the cable is terminated by RA
and RB , the same situation holds provided RA, RB and the
cable have the same (noise) temperature, because the system is
still in thermal equilibrium.
In conclusion, assuming waves in the quasi-static limit (Rel.
2) violates Planck’s Law and experimental facts about
blackbody radiation.

V. CONCLUSIONS
We discussed how non-physical elements used for
calculation in engineering can lead to physical results, and
why the non-physical elements should not be considered
physical. We defined what physical waves are and showed
that low-frequency signal propagation in a short cable does not
entail waves but retarded potentials. Finally, we proved that
the assumption of waves in the quasi-static limit violates
several laws of physics, including (i) the Energy Equipartition
Theorem, (ii) the Second Law of Thermodynamics (i.e.,
allowing the construction of a perpetual motion machine of the
second kind, (iii) the Principle of Detailed Balance, and (iv)
Planck’s formula at the low-frequency end (which would
permit infinitely strong black body radiation there).
REFERENCES
[1] L.J. Gunn, A. Allison, D. Abbott, “A directional wave measurement
attack against the Kish key distribution system”, (Nature) Science
Reports, vol. 4, 2014, p. 6461.
[2] L.B. Kish, “Totally secure classical communication utilizing Johnson(-
like) noise and Kirchoff’s law”, Physics Letters A, vol. 352, 2006, pp.
178–182.
[3] L.B. Kish, C.G. Granqvist, “On the security of the Kirchhoff-law–
Johnson-noise (KLJN) communicator”, Quantum Information
Processing, vol. 13, 2014, pp. 2213–2219.
[4] H.P. Chen, L.B. Kish, C.G. Granqvist, G. Schmera, “On the ‘cracking’
scheme in the paper ‘A directional coupler attack against the Kish key
distribution system’ by Gunn, Allison and Abbott”, Metrology and
Measurement Systems, vol. 21, 2014, pp. 389–400.
[5] L.B. Kish, Z. Gingl, R. Mingesz, G. Vadai, J. Smulko, C.G. Granqvist,
“Analysis of an attenuator artifact in an experimental attack by Gunn–
Allison–Abbott against the Kirchhoff-law–Johnson-noise (KLJN) secure
key exchange system”, Fluctuation and Noise Letters, vol. 14, 2015, p.
1550011.
[6] L.B. Kish, C.G. Granqvist, “Elimination of a Second-Law-attack, and all
cable-resistance-based attacks, in the Kirchhoff-law–Johnson-noise
(KLJN) secure key exchange system”, Entropy, vol. 16, 2014, pp. 5223–
5231.
[7] H.P. Chen, L.B. Kish, C.G. Granqvist, G. Schmera, “Do electromagnetic
waves exist in a short cable at low frequencies? What does physics
say?”, Fluctuation and Noise Letters, vol. 13, 2014, p. 1450016.