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\begin{document}

\title{Diametric Property of Spiral Wave Frequency Effect in Atrial
Fibrillation}



\author[1]{Donald Rubenstein}%
\affil[1]{Prisma Health}%


\vspace{-1em}



  
  \date{May 27, 2024}


\begingroup
\let\center\flushleft
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\maketitle
\endgroup





\selectlanguage{english}
\begin{abstract}
Introduction: Moving sources emitting spiral waves (SSp) such as cardiac
rotors, do not exhibit frequency changes typical of classic Doppler
effect (CDE) physics. Rotors exhibited fastest and slowest frequencies
at either side when migrating passed electrodes, not directly in front
and behind as in CDE. A new spiral wave frequency effect (SFE) equation
required derivation to accurately describe and predict WF frequency
changes observed near a moving SSp. Methods and Results: Rotational and
spiral math were developed to derive new rotational wave frequency
effect (RFE) and SFE equations in two dimensions. Wave front (WF)
strikes from SR occur when clock angle equaled the line of sight (LOS)
angle. WF strikes from spiral sources occurred when spiral summation
angle equaled LOS angle. SFE is analyzed by varying spiral size and
distance from SSp. New RFE and SFE equations predict diametric changes
in frequency that occur simultaneously on either side of a passing
rotor. Conclusions: WF frequency changes near a moving SSp exhibit 3
main differences compared to CDE: side-dependent frequency changes, a
strong-side unpaired WF strike, and a reversal of sequence of
activation. These differences, predicted by new RFE and SFE equations,
constitute the unique diametric property of the rotating waves. Moving
bodies that spin, or moving sources of WFs that rotate, result in
perceived frequency differences that are relative to side of
observation. Additionally, increasing and decreasing frequencies
observed, no longer always represent an approaching and receding SR and
SSp (respectively), especially when observed near the source.%
\end{abstract}%



\sloppy


\textbf{Methods and Results:}

Rotational and spiral math were developed to derive new rotational wave
frequency effect (RFE) and SFE equations in two dimensions. Wave front
(WF) strikes from S\textsubscript{R} occur when clock angle equaled the
line of sight (LOS) angle. WF strikes from spiral sources occurred when
spiral summation angle equaled LOS angle. SFE is analyzed by varying
spiral size and distance from S\textsubscript{Sp} . New RFE and SFE
equations predict diametric changes in frequency that occur
simultaneously on either side of a passing rotor.

\textbf{Conclusions:}

WF frequency changes near a moving S\textsubscript{Sp}exhibit 3 main
differences compared to CDE: side-dependent frequency changes, a
strong-side unpaired WF strike, and a reversal of sequence of
activation. These differences, predicted by new RFE and SFE equations,
constitute the unique diametric property of the rotating waves. Moving
bodies that spin, or moving sources of WFs that rotate, result in
perceived frequency differences that are relative to side of
observation. Additionally, increasing and decreasing frequencies
observed no longer always represent an approaching and receding
S\textsubscript{R} and S\textsubscript{Sp} (respectively), especially
when observed near the source.

\textbf{Keywords:} Doppler Effect, spiral, rotational, rotor, atrial
fibrillation, wave front, frequency effect, arrhythmia, diametric
property

\textbf{Non-standard Abbreviations and Acronyms}

CDE -- classic Doppler effect

CCW - counterclockwise

CW - clockwise

RFE - Rotational wave frequency effect

RDU -- rotational distance unit

SFE - Spiral wave frequency effect

S\textsubscript{C} - source of centrifugally propagating wave fronts

S\textsubscript{R} - rotational wave front source, a single
straight-line wave rotating about one end

S\textsubscript{Sp} - spiral wave front source, an Archimedean spiral
spinning around its center

T -- time period between wave front strikes

WF - wave front

tct -- transverse common tangent between 2 circles

? - wavelength

? -- line of sight angle as measured from the source's vector path being
the 0 angle

\selectlanguage{russian}\emph{ѱ}\textsubscript{\textbf{C}} -- clock angle

\selectlanguage{greek}ψ\selectlanguage{english}\textsubscript{h} -- hypotenuse angle

\selectlanguage{russian}ѱ\textsubscript{Sp} -- spiral angle

\selectlanguage{english}?\emph{\textsubscript{\textbf{S}}} -- angular velocity

\textbf{Introduction}

A Doppler effect or shift describes a change in the frequency that a
stationary observer senses when a source moves while emitting wave
fronts at a constant frequency. Christian Doppler in 1842, observed the
periodic back and forth shift in light frequency from infrared to
visible red that was being emitted from binary stars in rotation
(\emph{1} ). The higher frequency shift occurred when one star had its
maximal angular velocity directly towards the Earth. The Doppler
equation is derived on the assertion that periodic wave fronts (WFs)
propagate centrifugally outward in expanding circles with constant
propagation velocity.

\textbf{Three wave types, two different propagating paths}

A transverse or longitudinal WF is an oscillation of a field or
particles that propagates the energy through a mass or medium without
the transfer of mass itself (\emph{2} ). Both wave types propagate
outward centrifugally; the longitudinal disturbance is parallel, while
the transverse disturbance is perpendicular to the path of propagation.
The classic Doppler effect (CDE) and its predictive trigonometric-based
equation has been confirmed in the longitudinal sound WFs and all
transverse electromagnetic WFs. Fastest and slowest frequencies are
observed directly in front and behind the moving source of centrifugally
propagating WFs (S\textsubscript{C} ), with symmetric frequency changes
on either side of its motion. A recent addition to the classification of
wave types is the rotational wave (\emph{3} ). This form of WF
propagates around a central position. Physical examples of rotational WF
sources (S\textsubscript{R} ) ranging from micro to macro scales include
circular polarized light to hurricanes and spiral galaxies. Chemical
examples of spiral WFs were first realized with the Belousov-Zhabotinsky
reaction (\emph{4,5} ) and ultimately in electrophysiologic arrhythmias
(\emph{6} ). From here forward, it is important to further separate and
distinguish rotational WF sources based upon shape difference, a
rotating linear WF source (S\textsubscript{R} ) and that of a spiral WF
source (S\textsubscript{Sp} ). Both rotate about a central core, but
will have significantly different frequency change effects when
comparing S\textsubscript{R} and S\textsubscript{Sp} once these sources
are put into motion relative to the positions of observers.

Until now, it had been originally assumed (\emph{7} ), and then
perpetuated (\emph{8-11} ), that a moving source of spiral propagating
waves (S\textsubscript{Sp} ) would also exhibit the classical Doppler
effect (CDE). However, recent experimental evidence (\emph{12} ) showed
repeatedly that frequency changes near a migrating cardiac rotor were
too profoundly different to be explained by the CDE and its equation.
Asymmetric, increasing and decreasing frequencies were recorded
simultaneously on either side of the migrating rotor, not directly in
front and behind as would be expected from a CDE. New frequency effect
equations for rotational propagation, RFE, and spiral propagation, SFE,
required derivation to be provide accurate prediction of this side and
spin-dependent WF strike frequencies from either an S\textsubscript{R}
or S\textsubscript{Sp} . Clarity is needed at the outset; the name
Doppler effect and the Doppler equations are specific to WFs that have
an outward centrifugal path of propagation. Other rotational
Doppler-type equations have been proposed previously for
S\textsubscript{C} that only emit centrifugal propagating
electromagnetic WFs (\emph{13} ) and are not to be confused with the
frequency effect equations of WFs that have a rotational propagation
path.

\textbf{Cardiac electrophysiologic wave fronts -- sinus rhythm and rotor
classification}

A normal heart beat, is the result of an electrical WF called an action
potential, that propagates as a sudden depolarizing change of the
cellular membrane voltage. The electrical WF propagates from pacemaker
cells within the right atrium across the syncytium of cardiac cells as
it is transduced to a single forceful mechanical contraction.
Investigative research into this cardiac electrophysiology is vast and
the reader is directed to excellent reviews (\emph{14,15} ). The WF
propagates to neighboring cells via end-to-end and side-to-side cellular
intercellular gap junction connexin proteins. Although conduction of the
impulse during normal sinus rhythm may propagate anisotropically
(\emph{16} ), the action potential conduction is an aggregate of both
longitudinal and transverse waves that propagate roughly on a
centrifugal path outward from the sinus node.

During reentrant forms of cardiac arrhythmia, the WF propagation path is
altered to follow a circuit-like path. The upstroke of the action
potential propagates back around into cells recovering from their
refractory period, aptly named head-meets-tail propagation. More
disorganized arrhythmias, as in atrial fibrillation, a WF can pivot
around a region of inexcitable tissue (permanent or functional), and
break off into secondary propagating pathways. Single or multiple
changing and looping paths or circuits of WF conduction can last
indefinitely. The smallest of these circuits is the rotor (\emph{17} ).
There has been mounting evidence that cardiac rotors may be a sustaining
mechanism of fibrillatory activity (\emph{18-20} ). The rotor's spiral
shape may be stable for 1-2 wavelengths, where one \selectlanguage{english}? may range from
several hundred microns up to 2-3 cm (\emph{8,21-23} ). Within this
distance, the single WF of a cardiac rotor exhibits all 3 wave types.

\textbf{The SFE Deviations from the CDE}

The centrifugally outward propagation path from an S\textsubscript{C} to
an observer is always along the line of sight (LOS) to that observer.
Each subsequent WF is emitted as a distinctly separate oscillation of
energy. Frequency is measured by the number of WFs that pass an
observing site over time. In stark contrast, S\textsubscript{Sp} is a
single WF in the shape of a spiral with a propagation path, especially
within the first wavelength, \selectlanguage{english}?, that sweeps around a center of rotation
that gradually transitions to an outward path. Frequencies of WF strikes
generated from an S\textsubscript{Sp} are a result of complete
rotations. As S\textsubscript{Sp} moves along a vector path, the change
in the period observed between WF strikes is a result not only from the
change in the distance along LOS (as calculated for CDE), but also a
change in the arc length of rotation of the WF. Experimentally, compass
mapping of migrating rotors during human AF (\emph{12} ) identified
profound asymmetry of WF frequencies as S\textsubscript{Sp} passed
between electrodes (Figure 1A-C). One side recorded an increasing
frequency, while the other side recorded a simultaneous decreasing
frequency. The direction of frequency changes of were relative to the
side and spin orientation of the rotor source. The side-dependent
diametric frequency change was a highly consistent finding in 54 of 56
rotor transits passing the perimeter of electrodes. A 2-mm distance
separated highest frequency of WF strikes to lowest frequency.

Overall, we had identified three main reproducible observations for an
S\textsubscript{S} \textsubscript{p} that migrated between 2 stationary
electrodes: simultaneous diverging side-dependent frequency changes, an
unpaired WF strike at the point of closest approach of the
S\textsubscript{Sp} , and a reversal of sequence of WF strikes. The path
of S\textsubscript{S} \textsubscript{p} separated 2-D space into strong
and weak-sides of recordings. The strong-side label is attached to that
side which recorded increasing frequencies (during an approaching rotor)
and the unpaired WF strike. These 3 events can also be found by
examining other independent investigators recordings (Figure 1D,
(\emph{24} )). Importantly, CDE cannot predict these reproducible
observations. By standard Euclidean geometry and kinematics, it will be
shown below that the new RFE and SFE equations derived here predicts
these 3 components and ultimately constitute a diametric property of any
form of rotating sources of WFs. It will be shown that an
S\textsubscript{R} or S\textsubscript{Sp} may spin at a constant angular
velocity, but once that source moves relative to the position of an
observer, then the frequency shifts observed is relative to the spin
direction of the source and to which side of the source's motion that
the observer sits. Rotational and spiral sources of WFs at levels of
scale from atomic to galactic would be expected to exhibit this
diametric property of RFE and SFE. Once a spinning body moves or source
of rotating waves move, then frequencies changes are relative to side of
motion and spin direction of the source. This newly described phenomenon
of a diametric property unique to rotational WFs is easily visualized in
the human heart.

\textbf{METHODS}

To explore most simply the fundamental symmetry differences between a
moving S\textsubscript{C} and an S\textsubscript{R} or
S\textsubscript{Sp} , would be to compare the frequency of WF strikes at
recording sites placed equidistantly on either side a vector path of the
source of periodic WFs. This method will also allow the ability to
derive the new RFE and SFE equations.

The WF propagation path of a hypothetical S\textsubscript{R} passes
through all positions in 2-D space, ideally, during a single rotation.
This sets up a new unit, a rotational distance unit (RDU), that a moving
S\textsubscript{R} travelled during this one WF rotation, resulting in a
WF strike at any one some specific site in this 2-D space. Each
subsequent RDU provides the ability to calculate in time when the next
WF strike will occur at the same site. The second step reveals that the
vector path of S\textsubscript{R}splits 2-D space into 2 sides, strong
and weak, that depends upon the spin direction, clockwise (CW) or
counterclockwise (CCW). Determination of which side an observer sits
relative to the vector path of the S\textsubscript{R} determines whether
the directional change of frequency is either increasing or decreasing.
This diametric property, this asymmetry, unique to rotational waves,
identifies that frequency changes observed from sources that emit
rotational WFs is relative to the side that one observes. Lastly,
calculating the change in time periods between WF strikes from both
sides and at different distances from path of motion, unveils further
characteristics of this rotational relativity that differs from
symmetric CDE.

\textbf{Rotor rotation plane and the Rotational Distance Unit}

In two dimensions, the action potential WF of a cardiac rotor has been
shown to be spiral in shape. In three dimensions, the structure of the
rotor is described as a scroll wave (\emph{25} ). The unexcited core of
a scroll wave is extended perpendicularly to the rotational plane to
create an unexcited filament. Computer simulations and animal
experimental evidence (\emph{26-28} ) have shown scroll waves may have
different planes of rotation. A filament that extends from the
epicardial to endocardial surface is termed a transmural scroll wave.
Atrial chambers are thin, only a few mm in thickness. Most meandering
scroll waves were found to exist in the thinnest regions were transmural
(\emph{26} ). As such, the derivation of equations is in two-dimension
but could be expanded to three in later investigation.

A simplified S\textsubscript{R} is used as a foundational method to
calculate the period between WF strikes as the S\textsubscript{R} moves
relative to stationary observers (electrodes). The WF crest of
S\textsubscript{R} has the shape of a straight-line segment, or ray,
that pivots around a central core. The methods used here to derive the
RFE equation were essential to derive the SFE equation. The new
parameters developed here, methods, and graphical presentation provide
the ability to solve frequency related phenomena from rotational or
spiral WFs that appear in other forms of nature.

In Figure 2, an S\textsubscript{R} emits a clockwise rotational WF of
constant angular velocity (\(\mathbf{\omega}_{\mathbf{s}}\)) and moves with a constant
velocity (v\textsubscript{s} ). A single long clock hand simulates the
rotation of the WF crest. At greater radial distances from the center of
the source clock there is a proportional increase in the tangential
velocity (\selectlanguage{english}? \textsubscript{t} ) of the WF,\selectlanguage{english}
\begin{longtable}[]{@{}ll@{}}
\toprule
\(\ \mathbf{\omega}_{\mathbf{s}}\mathbf{=}\frac{\mathbf{\nu}_{\mathbf{t}}}{\mathbf{r}}\mathbf{\text{\ \ .}}\) & (1)\tabularnewline
\bottomrule
\end{longtable}

In biologic and physical systems, the tangential velocities of WF
propagation, including action potentials, are limited by physical and
biophysical restrictions (see Discussion). This figure shows how a WF
crest changes its position with both source motion and spin.

The RDU discussed above (Figure 2A) is defined as the distance travelled
at a constant source velocity (v\textsubscript{s} ) as the WF makes one
complete revolution around its center. One realizes that
?\textsubscript{s} and ?\textsubscript{s} are inversely proportional for
an RDU. An S\textsubscript{R} moving twice as fast, spinning twice as
fast result in the same RDU,\selectlanguage{english}
\begin{longtable}[]{@{}ll@{}}
\toprule
\(\mathbf{RDU=}\frac{\mathbf{\omega}_{\mathbf{s}}}{\mathbf{v}_{\mathbf{s}}}\) \textbf{,} & (2)\tabularnewline
\bottomrule
\end{longtable}

One needs to convert this to a single unit of rotation by substitution
of the angular velocity with ?\textsubscript{s} = 2??,\selectlanguage{english}
\begin{longtable}[]{@{}ll@{}}
\toprule
\(\mathbf{RDU=}\frac{\mathbf{2}\mathbf{\pi}}{\mathbf{v}_{\mathbf{s}}}\mathbf{f}\) & (3)\tabularnewline
\bottomrule
\end{longtable}

or\selectlanguage{english}
\begin{longtable}[]{@{}ll@{}}
\toprule
\(\mathbf{RDU=}\frac{\mathbf{2}\mathbf{\pi}}{\mathbf{v}_{\mathbf{s}}\mathbf{T}}\) & (4)\tabularnewline
\bottomrule
\end{longtable}

and replacing \textbf{?\textsubscript{s}T} with D, the distance
travelled, then the denominator simplifies to\selectlanguage{english}
\begin{longtable}[]{@{}ll@{}}
\toprule
\(\mathbf{RDU=}\frac{\mathbf{2}\mathbf{\pi}}{\mathbf{D}}\mathbf{\text{\ \ .}}\) & (5)\tabularnewline
\bottomrule
\end{longtable}

The side dependence of the WF crest position over time is clearly
evident. Tangential velocities are added to ?\textsubscript{s} on one
side, while on the other side it subtracted from ?\textsubscript{s}.

During the time that S\textsubscript{R} travelled distance D, the wave
front crest would have struck an electrode placed in any region of 2D
space during that single revolution. The RDU allows construct of other
units, parameters, and graphical presentations that will allow the
accurate computation of WF strikes, and frequency calculations at any
position of electrodes (or observers). The standard definition of period
(T ) is used as the time between WF strikes at a specific electrode.

In Figure 2B-D, graphical presentation of this simple rotational wave
between 2 electrodes simulates the experimental results previously
observed. By using this clock hand method, the clock angle
(\selectlanguage{russian}\emph{ѱ}\textsubscript{\textbf{c}} ) rotates at a constant
\selectlanguage{english}?\textsubscript{s}. The clock angle is measured by setting the vector
path as the zero angle, along a 3:00 direction. The sequence of WF
strikes at e\textsubscript{1} and e\textsubscript{2} progressively
separate in time (Figure 2B and 2C). WF strikes at
e\textsubscript{1}becomes earlier (arc length shortens on
S\textsubscript{R}'s clock, increasing frequency) and e\textsubscript{2}
become later (arc length lengthens on S\textsubscript{R} 's clock,
decreasing frequency). Passed the inflection point (\textbf{p} ), the
sequence of activation reverses as the WF strikes at e\textsubscript{2}
and then e\textsubscript{1} progressively approach the 9:00 direction.
As the S\textsubscript{R} moves between the electrodes, the direct line
of sight (LOS, Figure 2B and 2 D), angle \selectlanguage{english}? to e\textsubscript{1} and
e\textsubscript{2} changes. The LOS angle, \selectlanguage{english}?, has its zero angle also
aligned the 3:00 position. Therefore, acute angles are present with an
approaching source, but once past the inflection point p (where \selectlanguage{english}? =
90\textsuperscript{o}), angle \selectlanguage{english}? becomes obtuse. It is clear that during
each single rotation, a WF strike only can occur when the clock
angle,\emph{ps}\textsubscript{\textbf{c}} is equal to the LOS angle, \selectlanguage{english}?,
to that specific electrode (Figure 2B-D).\selectlanguage{english}
\begin{longtable}[]{@{}ll@{}}
\toprule
\(\mathbf{WF\ strike:\ \ \ }\mathbf{\psi}_{\mathbf{c}}\mathbf{=\theta.}\) & (6)\tabularnewline
\bottomrule
\end{longtable}

The change in arc length (S) between WF strikes is a change in the angle
of the clock governed by the equation\selectlanguage{english}
\begin{longtable}[]{@{}ll@{}}
\toprule
\(S=r\theta\ .\) & (7)\tabularnewline
\bottomrule
\end{longtable}

Importantly, had an observing electrode been placed directly in front of
the moving source (at inflection point \textbf{p} ), WF strikes at that
position would have occurred at 3:00 with each RDU as it approached,
then at 9:00 after passing \textbf{p} . Thus, no change in period, no
change in frequency with approaching or receding from point \textbf{p} ,
just an instantaneous phase shift in ? by 180\textsuperscript{o} or by ?
radians. Thus, the vector path of the S\textsubscript{R}separates 2-D
space into two regions, defined as a strong and weak-side. Observers, or
electrodes that sit on the strong-side sense the increasing frequency,
while those on the weak-side sense simultaneously, the diametric
opposite change, a decreasing frequency.

To quantitatively understand the asymmetry and diametric property
observed, a hypothetical timing experiment is presented (Figure 2C, D)
where an S\textsubscript{R} passes between 2 equidistant electrodes
(e\textsubscript{1} and e2), or observers, all represented as
synchronized clocks. The vector path of S\textsubscript{R} is a
perpendicular bisector, keeping e\textsubscript{1} and
e\textsubscript{2} equidistant to S\textsubscript{R} along its entire
vector path. Both clock angle\selectlanguage{russian}\emph{ѱ}\textsubscript{\textbf{c}} and LOS
angle \selectlanguage{english}?, have their 0\textsuperscript{o} angle aligned to the vector
path of S\textsubscript{R} that moves along the 3:00 line. The sweeping
clock hand represents the WF crest that extends far out beyond the
perimeter of the clock and moves as a single wave. The center of each
clock represents the exact location of S, e\textsubscript{1}, and
e\textsubscript{2}. A CCW rotation would reverse the designation of (+)
and (--) angles from the 0\textsuperscript{o} position.

Imaginary lines extending between 3:00 to 9:00 for each of the clocks
are parallel lines. It follows then that a clock hand line that extends
from the center of S\textsubscript{R} to one of the electrodes at the
moment of a WF strike, that line is the line of sight (LOS). The clock
times are exactly the same (congruent angles) by the theorem of
corresponding angles. The clocks at each electrode are large, such that
the clock perimeters hit at the inflection point p between the 2
electrodes. The radius of the e\textsubscript{1} clock to its perimeter
is the same for the e\textsubscript{2} clock (r\textsubscript{1} =
r\textsubscript{2}). The vector path of the S\textsubscript{R} is the
tangent line to both clocks at e\textsubscript{1} and e\textsubscript{2}
and intersect each clock's edge at the same point (\textbf{p} ) and
represents the point of the transverse common tangent (tct) of both
electrode clocks in 2D space. With each rotation of the WF crest as
S\textsubscript{R} approaches, then recedes from point p , the change in
\selectlanguage{english}?, or \selectlanguage{english}?\selectlanguage{english}? between WF strikes corresponds directly to a proportional
change in the period. Whether one adds or subtracts \selectlanguage{english}?\selectlanguage{english}? to the previous
position depends upon which side (strong or weak) the electrode sits.

\textbf{Determining Strong and Weak Sides: 3-point sequence rule}

A sequence-based approach (Figure 2D, right inset) is used to determine
whether the observing electrode (e\textsubscript{1}) sits on the strong
or weak-side of an approaching S\textsubscript{R} . The
S\textsubscript{R} vector path divides the plane into 2 sides. Two
points are added to the figure. One is placed on vector
path,\emph{forward} of the source's current position (arrow points
to\textbf{e\textsubscript{v}} , as a reminder). The second point is an
electrode (e\textsubscript{2}) placed on the opposite side of
S\textsubscript{R}'s vector path, then the sequence is analyzed as the
WF sweeps passed e\textsubscript{v}. The spin direction (CW or CCW) is
determined. An electrode that records a WF strikes that precedes that at
e\textsubscript{v}, sits on the strong-side. While an electrode that
records WF strikes after the e\textsubscript{v}, sits on the weak-side.
Strong-side electrodes record decreasing time periods, T between WF
strikes for an approaching S\textsubscript{R} , while the weak-side
electrodes record an increasing T . One now knows to subtract \selectlanguage{english}?\selectlanguage{english}? from
the previous period (faster frequency) using a strong-side electrode
position for an approaching S\textsubscript{R} , or add \selectlanguage{english}?\selectlanguage{english}? to the
previous period (slower frequency) using a weak-side electrode. The
3-point sequence rule remains useful whether the spin direction is CW or
counterclockwise (CCW) or if the source is approaching or receding from
the inflection point. Therefore, subscript labels (st) and (wk) will be
used for all subsequent calculations in this paper designating which
side the observing electrode sits relative to a moving rotational
source. The change in time period \selectlanguage{english}?T can be easily calculated knowing
that \selectlanguage{english}?\emph{\textsubscript{\textbf{s}}} is frequency measurement and the
inverse is equal to the period of rotation of the source,\selectlanguage{english}
\begin{longtable}[]{@{}ll@{}}
\toprule
\(\mathbf{T}_{\mathbf{s}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{\omega}_{\mathbf{s}}}\text{\ \ .}\) & (8)\tabularnewline
\bottomrule
\end{longtable}

The angular change in the LOS angle ? between WF strikes results in\selectlanguage{english}
\begin{longtable}[]{@{}ll@{}}
\toprule
\({\Delta T}_{\text{st}}=\frac{2\pi-\Delta\theta_{\text{st}}}{\omega_{s}}\) & (9)\tabularnewline
\bottomrule
\end{longtable}

and\selectlanguage{english}
\begin{longtable}[]{@{}ll@{}}
\toprule
\(T_{\text{wk}}=\frac{2\pi-\Delta\theta_{\text{wk}}}{\omega_{s}}\ \ ,\) & (10)\tabularnewline
\bottomrule
\end{longtable}

where ?? is a positive value at a strong-side electrode, and negative
value at a weak-side electrode.

\textbf{Rotational math and creating family of RDU lines}

The alignment of 0\textsuperscript{o} angle of the clock
angle\selectlanguage{russian}\emph{ѱ}\textsubscript{\textbf{c}} and LOS angle \selectlanguage{english}? (Figure 2C and
D) allows a direct calculation of the change of \selectlanguage{english}?, and therefore the
same angular change in \emph{ps}\textsubscript{\textbf{c}} . The change
in the clock angle \emph{ps}\textsubscript{\textbf{c}} is a direct
proportional change in the time period between WF strikes, translating
accurately as frequency changes.

In the example presented in Figure 2, the center of the emitting source
moves at a constant velocity, v\textsubscript{s} = 10 cm/s, at a
constant \selectlanguage{english}\emph{?\textsubscript{\textbf{s}}} =10 rotations/s =20\selectlanguage{english}?/s (in
radians). S\textsubscript{R} moves along the perpendicular bisector path
that approaches and intersects point\textbf{p} . Because
v\textsubscript{s} and \selectlanguage{english}?\emph{\textsubscript{\textbf{s}}} are both
constant, the RDU can be calculated. Note that time factors out, as one
simplifies the RDU such that the numerator is 2\selectlanguage{greek}π, \selectlanguage{english}as in Equation 2
above,

\begin{equation}
RDU=\frac{\left(\frac{20\pi}{s}\right)}{\left(\frac{10cm}{s}\right)}=\frac{2\pi}{1cm}\mathbf{\text{\ .\ }}\nonumber \\
 \end{equation}

Therefore, as the clock angle rotates 1 complete revolution, from ? to
-? (2?), S\textsubscript{R} will have travelled D distance (D = 1 cm).
The RDU can be used as the slope of the line that describes how the
angle \selectlanguage{greek}ψ\selectlanguage{english}\textsubscript{c} changes (radians) as a function of x (the
distance from p ). If one knows the source's clock hand position at a
specific point along the vector path of S\textsubscript{R} , then one
could plot the line on a graph (Figures 3A and B). Each complete
revolution 2? is plotted, showing the clock hand rotation angle from -\selectlanguage{greek}π
\selectlanguage{english}[?] ps\selectlanguage{english}\textsubscript{c} [?]+\selectlanguage{greek}π \selectlanguage{english}as a function of x distance travelled by
S\textsubscript{R} . Using the slope-intercept form, for
\selectlanguage{greek}ψ\selectlanguage{english}\textsubscript{c} = f(x), where a represents the y-intercept, the clock
hand angle is calculated by\selectlanguage{english}
\begin{longtable}[]{@{}ll@{}}
\toprule
\(\psi_{c}=RDUx+a\ .\) & (11)\tabularnewline
\bottomrule
\end{longtable}

At the end of one rotation, when \selectlanguage{greek}ψ\selectlanguage{english}\textsubscript{c}reaches 9:00 or -\selectlanguage{greek}π,
\selectlanguage{english}then the subsequent rotation would restart at +\selectlanguage{greek}π \selectlanguage{english}again as the clock hand
continues to rotate at a constant ?\emph{\textsubscript{\textbf{s}}} .
This next RDU line would be a parallel, having the same RDU slope from
+? to -\selectlanguage{greek}π, \selectlanguage{english}but the line would be one RDU closer (y-intercept, would
advance by 2?). This would result in subsequent y-intercepts being
advance by 2\selectlanguage{greek}π. \selectlanguage{english}A family or series of parallel RDU lines shows exactly
the clock hand angle for any S\textsubscript{R} position along its
entire vector path (Figure 4B green lines) by incrementing the
y-intercept by a factor 2\selectlanguage{greek}π \selectlanguage{english}for each subsequent rotation, where n
represents the integer of the rotation number\selectlanguage{english}
\begin{longtable}[]{@{}ll@{}}
\toprule
\(\psi_{c}=RDUx+\left(a+n2\pi\right)\mathbf{.}\) & (12)\tabularnewline
\bottomrule
\end{longtable}

\textbf{RESULTS}

\textbf{Superimposing plots of LOS angle ? and clock
angle\emph{ψ\textsubscript{c}}}\selectlanguage{english}

Figure 3A and 3B provides a graphical representation of how the clock
angle \selectlanguage{greek}ψ\selectlanguage{english}\textsubscript{c} (green lines) and the LOS angle ? to
equidistant electrode e\textsubscript{1} and e\textsubscript{2}(blue and
red curves) as a function of the distance S\textsubscript{R} is from the
inflection point. The intersection of these plots, where
\selectlanguage{greek}ψ\selectlanguage{english}\textsubscript{c} = ?, shows exactly when along the vector path that a
WF strikes e\textsubscript{1} and e\textsubscript{2}. The family of RDU
lines have a negative slope with CW rotation and
S\textsubscript{R}moving right to left (CCW rotation have a positive
slope). Because the source's vector path is a perpendicular bisector of
the line between e\textsubscript{1} and e\textsubscript{2,} right
triangles can be created between the electrode e\textsubscript{1} or
e\textsubscript{2}, point \textbf{p} , and the source location. The LOS
angle ? plotted as a function of distance x from point \textbf{p} is
superimposed on this graph as arccotangent curves. The arccotangent
function must be used instead of tangent since the LOS angle ? has a
range 0 [?] ? [?] \selectlanguage{greek}π \selectlanguage{english}for e\textsubscript{1}, and a range of 0 [?] ? [?] -\selectlanguage{greek}π \selectlanguage{english}for
e\textsubscript{2}, as the source moves passed these electrodes. The LOS
angle ? to e\textsubscript{1} is given a positive magnitude (uses + r to
e\textsubscript{1}, blue curve), while ? to e\textsubscript{2} is given
a negative magnitude (uses - r to e\textsubscript{2}, red curve).\selectlanguage{english}
\begin{longtable}[]{@{}ll@{}}
\toprule
\(\theta_{\text{st}}=\operatorname{arccot}\left(\frac{-x}{r}\right)\) & (13)\tabularnewline
\bottomrule
\end{longtable}

and\selectlanguage{english}
\begin{longtable}[]{@{}ll@{}}
\toprule
\(\theta_{\text{wk}}=arccot\left(\ \frac{-x}{-r}\ \right)\text{\ .}\) & (14)\tabularnewline
\bottomrule
\end{longtable}

Each new rotation advances n by 1 in equation 12 creating a new RDU line
that intersects with the arccotangent curve. In this example, an
arbitrary starting point for the first RDU, begins with the source clock
hand at 9:00 (or \selectlanguage{greek}π \selectlanguage{english}radians) with the source 3 cm away from
point\textbf{p} . Using the 3-point sequence rule above,
e\textsubscript{1}sits on the strong-side of the source's vector path
and e\textsubscript{2} sits on the weak-side. The value of x is made
negative to reverse the axis in keeping with a right to left motion of
the S\textsubscript{R} towards point p . There is symmetry of the arccot
curves about the x-axis. When one plots both \selectlanguage{greek}ψ\selectlanguage{english}\textsubscript{c} and ?
each as a function of x (Figure 4B), the points of intersection (white
circles) identifies when the WF strikes at e\textsubscript{1} and
e\textsubscript{2,} where the clock angle \selectlanguage{greek}ψ\selectlanguage{english}\textsubscript{c} = ? during
each RDU.

As S\textsubscript{R} approaches, there is one WF strike per rotation.
Initially, the timing of a WF strike at e\textsubscript{1}immediately
precedes a WF strike at e\textsubscript{2}. The time between WF strikes
at e\textsubscript{1} and e\textsubscript{2} increases with each RDU as
it approaches \textbf{p} (square box, ? = \selectlanguage{greek}π \selectlanguage{english}or -\selectlanguage{greek}π\selectlanguage{english}). Once
S\textsubscript{R} has passed \textbf{p} , the WF strike at
e\textsubscript{2} is delayed to such great extent that the WF strike at
e\textsubscript{2} now precedes the WF strike at e\textsubscript{1}. The
WF strike at e\textsubscript{1} was unpaired, signaling a reversal in
sequence. If one placed e\textsubscript{2} at positions closer to
inflection point (shorter r\textsubscript{2}), then the WF strikes would
exhibit less change in time periods at distance far from \textbf{p} ,
but quite more abrupt change with RDUs closest to p (black curved
lines). The change becomes more sudden for closer positioned electrodes
to the vector path of the rotational source.

The diametric property of rotational wave frequency effect occurs, but
slightly altered when S\textsubscript{R} approaches the electrodes from
an angle that is not a perpendicular bisector between electrodes (Fig
4(d-f)). The vector path is constructed to still have its closest
approach to e\textsubscript{1} be at 1 cm (\textbf{r\textsubscript{1}}
), but is tilted 30\textsuperscript{o}compared to 4A. A new radius is
calculated to its point of closest approach to e\textsubscript{2}
(\textbf{r\textsubscript{2}} with different red arccotan curve). The
distance between the points of closest approach to e\textsubscript{1}
and e\textsubscript{2} is length of the line segment of the transverse
common tangent (tct).

\textbf{Rotational Wave Frequency Effect equation}

The diametric property of rotational waves requires the identification
of the strong-side and weak-side of the source's movement to determine
time periods between subsequent WF strikes. Each of the WF strikes from
a moving rotational WF source is directly related to subsequent WF angle
change between strikes (??). Strong-side changes in the angle of the WF
strike results in ?? having a positive value termed now as
??\textsubscript{st}, while the weak-side having a negative value,
??\textsubscript{wk}. This allows for the derivation of the rotational
wave frequency effect formula.

The change in the clock hand angle \selectlanguage{greek}ψ\selectlanguage{english}\textsubscript{c} at
e\textsubscript{1}, between rotation 1 and rotation 2 is
??\textsubscript{st}. Similarly, ??\textsubscript{wk} represents the
clock hand angle change for the WF strikes occurring during rotation 1
and rotation 2 at e\textsubscript{2}. If one knows the constant angular
velocity of source's rotational wave, then the angles
?\textbf{?\textsubscript{st}} and ??\textsubscript{\textbf{wk}} equates
to the exact time period (T ) change between subsequent WF strikes.

The period (T ) for a rotational wave is defined as the time required to
make 1 complete revolution, and is inversely proportional to the angular
velocity of the source's WF,\selectlanguage{english}
\begin{longtable}[]{@{}ll@{}}
\toprule
\(T=\frac{2\pi}{\omega_{s}}\text{\ .}\) & (15)\tabularnewline
\bottomrule
\end{longtable}

Thus, the angular velocity is frequency (?) measurement is\selectlanguage{english}
\begin{longtable}[]{@{}ll@{}}
\toprule
\(\omega_{s}=2\pi f.\ \ \) & (16)\tabularnewline
\bottomrule
\end{longtable}

On the strong-side of the source's movement, the subsequent WF strike
occurs with less of a complete rotation (less arc length) or
2?-??\textsubscript{st} . The subsequent rotation results in a new
period T\textsubscript{st} that is reduced by ??\textsubscript{st} for
e\textsubscript{1}, while T\textsubscript{wk} is increased by
??\textsubscript{\textbf{wk}} or as shown above, such that\selectlanguage{english}
\begin{longtable}[]{@{}ll@{}}
\toprule
\(T_{\text{stNew}}=\ \left(\frac{2\pi-\Delta\theta_{\text{st}}}{\omega_{s}}\right)\text{T\ \ and\ \ }T_{\text{wkNew}}=\left(\frac{2\pi-\Delta\theta_{\text{wk}}}{\omega_{s}}\right)T\) & (17)\tabularnewline
\bottomrule
\end{longtable}

for e\textsubscript{1} and e\textsubscript{2} respectively. One can see
that subtracting ??\textsubscript{st} or
??\textsubscript{wk}(??\textsubscript{wk} has a negative value) result
in a shorter or longer period between WF strikes respectively. Note that
period change with subsequent RDUs, T\textsubscript{st} and
T\textsubscript{wk} for e\textsubscript{1} and e\textsubscript{2}
respectively, are not equal and opposite in time since the WF strikes
are not occurring simultaneously at e\textsubscript{1} and
e\textsubscript{2}. The source had moved closer to e\textsubscript{2} at
the time of the WF strike that is recorded there. One can convert the
change in period to a change in frequency observed at each electrode
since period and frequency are inversely proportional,\selectlanguage{english}
\begin{longtable}[]{@{}ll@{}}
\toprule
\(f=\frac{\omega_{s}}{2\pi-\Delta\theta}\text{\ .}\) & (18)\tabularnewline
\bottomrule
\end{longtable}

Importantly, ?? is either ??\textsubscript{st} or ??\textsubscript{wk},
they are not equal, and they are opposite in sign. The new frequency f
\textsuperscript{'} can be compared to the original frequency f
\selectlanguage{ngerman}\textsuperscript{º} , where the derived\textbf{Rotational Wave Frequency
Effect Equation} (RFE) is now\selectlanguage{english}
\begin{longtable}[]{@{}ll@{}}
\toprule
\(f^{{}^{\prime}}=\left(\frac{2\pi}{2\pi-\Delta\theta_{\text{St}}}\right)f^{o}\text{\ \ \ \ \ \ \ or\ \ \ \ \ \ \ }f^{{}^{\prime}}=\left(\frac{2\pi}{2\pi-\Delta\theta_{\text{Wk}}}\right)f^{o}\text{\ \ .\ }\) & (19)\tabularnewline
\bottomrule
\end{longtable}

And since ? = \selectlanguage{greek}ψ \selectlanguage{english}\textsubscript{c} at WF strikes, the change in the clock
hand angle results in\selectlanguage{english}
\begin{longtable}[]{@{}ll@{}}
\toprule
\(f^{{}^{\prime}}=\left(\frac{2\pi}{2\pi-\Delta\psi_{c,St}}\right)f^{o}\text{\ \ \ \ \ \ \ \ or\ \ \ \ \ \ \ \ \ }f^{{}^{\prime}}=\left(\frac{2\pi}{2\pi-\Delta\psi_{c,Wk}}\right)f^{o}.\) & (20)\tabularnewline
\bottomrule
\end{longtable}

The change in frequency for rotational waves is markedly different
compared to a classic doppler effect that has linear propagation paths
and unchanging propagation velocities,\selectlanguage{english}
\begin{longtable}[]{@{}ll@{}}
\toprule
\(f^{{}^{\prime}}=\ \left(\selectlanguage{greek}\frac{v_{\text{sound}}}{v_{\text{sound}}-v_{\text{source}}\text{cosθ}}\selectlanguage{english}\right)f^{o}.\) & (21)\tabularnewline
\bottomrule
\end{longtable}

A detailed comparison is made in Fig 4 for different electrode
separation from point p , keeping e\textsubscript{1} and
e\textsubscript{2} equidistant to S\textsubscript{R}that revolves with a
?\textsubscript{s} at 5 Hz. Such analysis allows further important
general characteristics of the diametric property of a rotational source
and can be compared to the diametric property of a spiral source
(S\textsubscript{Sp} ) described later.

Three main characteristics of the diametric property stand out for an
approaching, then receding source of rotational WFs.

\begin{enumerate}
\tightlist
\item
  The strong-side position of electrodes will sense an increased
  frequency while simultaneously, weak-side positions will sense a
  decreasing frequency during the approach, with opposite changes
  occurring when S recedes from p .
\item
  The strong-side will have one greater WF strike than the weak-side.
\item
  Reversal of sequence activation at the inflection point.
\end{enumerate}

Note that the RFE shows that directly in front and behind
S\textsubscript{R} , (either end of the graph), the frequency of WF
strikes approaches that of ?\textsubscript{s}. The fastest and slowest
frequencies observed are not directly in front as in CDE, it is an
asymmetric side-dependent phenomenon.

\textbf{THE DERIVATION OF A SPIRAL WAVE FREQUENCY EFFECT}

The cardiac rotor is not a simple rotational wave with a linear WF
crest, it is spiral. The creation of the RDU in the previous rotational
WF (S\textsubscript{R} ) example was the essential step to derive the
SFE equation. The RDU allows the accurate time calculation of a WF
strike at electrodes with each single rotation from a moving source of a
spiral wave (S\textsubscript{Sp} ). The rotor's spiral shape is
described to most nearly follow that of an Archimedean spiral (\emph{21,
29} ) with wave curvature and refractory tissue governing the ultimate
period of rotation. A ray drawn from the center of the Archimedean
spiral intersects each subsequent arm of the spiral at a constant
distance of separation. In Figure 5, an ideal Archimedean spiral is
shown. Similar to the clock hand spin, the spiral spin will also exhibit
a spin and side dependence to observed frequencies. The diametric
property of frequency changes exists here as well with movement of an
S\textsubscript{Sp} .

Determining frequency changes of WF strikes from an S\textsubscript{Sp}
have additional challenges compared to that of S\textsubscript{R} of
Figures 3 and 4. First, the spiral wave, even without spinning, has a
changing WF crest angle as one moves away from the emitting source's
center. Second, one needs to measure the spiral wave's specific size or
wavelength (?). Similar to S\textsubscript{R} , the position of
S\textsubscript{Sp} in 2-D described not only by the specific position
of its core, but also by its rotation angle as it moves. For simplicity,
it will be assumed that S\textsubscript{Sp} will maintain its spiral
shape for at least 2 wavelengths out from its center, or 2 complete
turns, as it moves along its vector path v\textsubscript{s} . In the
cardiac rotor, WF head (crest of WF) meets tail (absolute refractory
period) the distance ? (between 2 arms of the spiral) should not be able
to be compressed beyond this physical limit (see Discussion). The
calculations and plots below of ideal spirals with ?s at similar spiral
sizes, provide an accurate approximation for frequency effects as they
would be recorded from electrodes near rotors during atrial
fibrillation.

\textbf{Spiral math}

Similar to deriving RFE above, one must first develop a spiral math. The
general spiral equation is\selectlanguage{english}
\begin{longtable}[]{@{}ll@{}}
\toprule
\(r=a+\selectlanguage{greek}\text{bψ}\selectlanguage{english}_{\text{Sp}}^{1/c}\ \ ,\) & (22)\tabularnewline
\bottomrule
\end{longtable}

where radius r and \selectlanguage{greek}ψ\selectlanguage{english}\textsubscript{Sp} identify the distance from the
center and the angle of rotation to a specific point on the spiral. The
value a turns the spiral, b is the spiral size. Figure 5A is an
Archimedean spiral (c=1), and the polar wave equation is\selectlanguage{english}
\begin{longtable}[]{@{}ll@{}}
\toprule
\(r_{\text{Sp}}=a+\ \selectlanguage{greek}\text{bψ}\selectlanguage{english}_{\text{Sp}}\text{\ .}\) & (23)\tabularnewline
\bottomrule
\end{longtable}

Creating a spiral clock, \selectlanguage{greek}ψ \selectlanguage{english}\textsubscript{c}

To best visualize a clock hand on a spiral, the clock hand extends from
the spiral's center to the point of one compete revolution (2? radians).
The length of that clock hand is the spiral's size, b. This distance at
that one revolution is the wavelength, ?, and is the distance between
all subsequent spiral arms (Figure 5). The size b is derived from
equation 23 by setting a = 0, and r = ?, and at one revolution, \selectlanguage{greek}ψ
\selectlanguage{english}\textsubscript{Sp} = 2? radians,\selectlanguage{english}
\begin{longtable}[]{@{}ll@{}}
\toprule
\(b=\frac{\lambda}{2\pi}\text{\ \ .}\) & (24)\tabularnewline
\bottomrule
\end{longtable}

Substitution into equation 23 identifies a specific spiral angle \selectlanguage{greek}ψ
\selectlanguage{english}\textsubscript{Sp} ,\selectlanguage{english}
\begin{longtable}[]{@{}ll@{}}
\toprule
\(\psi_{\text{Sp}}=\ \frac{2\pi}{\lambda}r_{\text{Sp}}\ \ ,\) & (25)\tabularnewline
\bottomrule
\end{longtable}

and shows that the spiral angle \selectlanguage{greek}ψ \selectlanguage{english}\textsubscript{Sp} is inversely
proportional to ? of spirals of different size. If spiral A is twice the
size of spiral B, a line segment equal to spiral A's wavelength (2?)
would hit the spiral B's curve at 4? or at two full revolutions.

From here forward, S\textsubscript{Sp} will identify a source that emits
a spinning Archimedean spiral wave governed by the Equation
\textbf{25\emph{.}} Movement of the spiral will be defined by 2
velocities, the velocity of the center of the spiral along the vector
path (v\textsubscript{s} ), and the angular velocity of its spin
(?\textsubscript{\textbf{s)}} . Two angles, the clock angle \selectlanguage{greek}ψ
\selectlanguage{english}\textsubscript{c} and the spiral angle \selectlanguage{greek}ψ\selectlanguage{english}\textsubscript{Sp} , must be
determined to make frequency calculations of WF strikes from a spiral
wave. The clock angle \selectlanguage{greek}ψ\selectlanguage{english}\textsubscript{c} identifies the specific
position in its rotation that is referenced to a zero angle in 2-D
space, while the spiral angle \selectlanguage{greek}ψ \selectlanguage{english}\textsubscript{Sp} identifies the angle
measured from the clock angle \selectlanguage{greek}ψ \selectlanguage{english}\textsubscript{c} to a specific position
on the spiral wave. As seen in Figure 5, a spiral that spins in a
clockwise direction, its spiral angle \selectlanguage{greek}ψ\selectlanguage{english}\textsubscript{Sp} increases in
the opposite, or counterclockwise direction as one measures
progressively further from the spiral center along the spiral arm.
Conversely, a counterclockwise spinning spiral will have a spiral angle
\selectlanguage{greek}ψ \selectlanguage{english}\textsubscript{Sp} that increases in a clockwise direction.

This Archimedean spiral clock replaces the clock example of Figure 2 and
3 above. An imaginary clock hand is glued along the line
segment\textsubscript{0-2?}, the first ? (grey triangle, Figure 5B and
C). As in the prior rotational example (Fig 2A), the center of
S\textsubscript{Sp} moves along a vector path, v\textsubscript{s} ,
toward the 3:00 position, and is used as a referenced zero angle.
Therefore, the spiral's clock hand angle \selectlanguage{greek}ψ \selectlanguage{english}\textsubscript{c} is defined
as the angle between the clock hand and the vector path
v\textsubscript{s} .

\selectlanguage{greek}\textbf{b. Spiral summation angle, \emph{ψ\textsubscript{SA}}}\selectlanguage{english}

Any point on the spiral wave can now be described as a spiral summation
angle (\selectlanguage{greek}ψ\selectlanguage{english}\textsubscript{SA} ) of \selectlanguage{greek}ψ\selectlanguage{english}\textsubscript{c} and
\selectlanguage{greek}ψ\selectlanguage{english}\textsubscript{Sp} ,\selectlanguage{english}
\begin{longtable}[]{@{}ll@{}}
\toprule
\(\psi_{\text{SA}}=\psi_{c}+\psi_{\text{Sp}}\text{\ \ .}\) & (26)\tabularnewline
\bottomrule
\end{longtable}

In this example, spiral S\textsubscript{Sp} spins at a constant angular
velocity, ?\textsubscript{s}, that turns the clock hand angle \selectlanguage{greek}ψ
\selectlanguage{english}\textsubscript{c} . Any stationary observer that is placed on this 2-D
plane will have its clock perfectly synchronized to that of
S\textsubscript{Sp} . The change in the spiral's clock angle \selectlanguage{greek}ψ
\selectlanguage{english}\textsubscript{c} between any two sequential WF strikes of the spiral
arm at the observer is a measure of the time period (\textbf{T} )
between WF strikes and can be converted to a frequency measurement. The
T between WF strikes is measured far easier by the change in the
spiral's clock angle \selectlanguage{greek}ψ \selectlanguage{english}\textsubscript{c} , since the WF propagation path
and velocity are always changing. If the spiral spins, but its center is
stationary (v\textsubscript{s} = 0), then the WF from
S\textsubscript{Sp} will strike the stationary observer with a constant
T , no matter where it is positioned with each subsequent rotation. With
each one revolution, the WF crest moves one ? away in distance. The
frequency of WF strikes would be constant and equal to
?\textsubscript{s}. Setting the spiral in motion at constant
?\textsubscript{s} , the rotation distance unit, RDU, can be utilized.
The spiral will have completed one revolution after travelling D
distance (D= \emph{?\textsubscript{s \textbf{*}}} T of the RDU, Equation
5). Either the change in the spiral's clock hand angle or the change in
distance the spiral had moved between each WF strike provides the change
in T or frequency.

The \emph{?\textsubscript{s}} vector path of S\textsubscript{Sp} in
Figure 5D separates 2-D space in half. The 3-point sequence rule still
applies here to designate specific sides. If 2 observers were placed one
on either side of\emph{?\textsubscript{s}} , then the strong-side and
weak-side nomenclature holds for an approaching S\textsubscript{Sp} .
WFs from the spiral move faster through the strong side and slower on
the weak-side. Similar to S\textsubscript{R} , at certain radial
distances from the spiral center, shear zones and stall zones may exist
on strong-side and weak-sides respectively. These repeating regions of
shear and stall are staggered along the S\textsubscript{Sp} 's vector
path separated by the distance D of the RDU. Examples of likely
phenomena of shear and stall zones will be discussed below.

Similar to RFE, and to prove the diametric property exists for spiral
waves as well, stationary electrodes (e\textsubscript{1} and
e\textsubscript{2}) are placed on either side of the vector path
?\textsubscript{s} equidistantly from S\textsubscript{Sp} (Figure 6).
S\textsubscript{Sp} spins with a constant angular velocity
?\textsubscript{s}. The activation sequence of WF strikes at electrodes
e\textsubscript{1} and e\textsubscript{2} that straddle either side of
\emph{v\textsubscript{s}} , must reverse their sequence as the spinning
S\textsubscript{Sp} passes the inflection point (p ) of the transverse
common tangent. The LOS angle, ?, ranging from 0\textsuperscript{o} to
180\textsuperscript{o}, remains the angle between the line of the
forward vector path v\textsubscript{s} set at 0\textsuperscript{o} and
the line from the center of S\textsubscript{Sp} to e\textsubscript{1} or
e\textsubscript{2}. LOS angle ? is positive if the electrode is above
the vector line \emph{v\textsubscript{s}} , and negative if the
electrode sits below v\textsubscript{s} . Each electrode sits at the
center of imaginary perfectly circular clocks created in the same 2-D
plane (blue and red circles). The clock perimeters are of specific
radial size, such that\emph{v\textsubscript{s}} intersects the
transverse common tangent to both clock's perimeter at point p . Point p
is therefore a common inflection point (p ) where the moving
S\textsubscript{Sp}changes from being an approaching S\textsubscript{Sp}
to a receding S\textsubscript{Sp} . The measured distance h along the
LOS angle ? to either e\textsubscript{1} or e\textsubscript{2}, is
simply the hypotenuse of a right triangle that is created from
S\textsubscript{Sp} , p , and either e\textsubscript{1} or
e\textsubscript{2},\selectlanguage{english}
\begin{longtable}[]{@{}ll@{}}
\toprule
\(\mathbf{h=}\sqrt{\mathbf{x}^{\mathbf{2}}\mathbf{+}\mathbf{r}^{\mathbf{2}}}\mathbf{\ \ ,}\) & (27)\tabularnewline
\bottomrule
\end{longtable}

where r is the radius of the imaginary clock perimeter of
e\textsubscript{1} or e\textsubscript{2}, and x is the distance from the
S\textsubscript{Sp} to the inflection point \textbf{p} . A positive
value of x is used as the S\textsubscript{Sp} approaching p and negative
if receding from p . Although the value of x is squared here, it's
positive and negative values will need to be remembered and used later
with frequency calculations. Since e\textsubscript{1} and
e\textsubscript{2} are equidistant to the spiral S all along its path,
the LOS angle ?, is also equal all along its path. Any diametric changes
(increasing vs decreasing) in frequency detected by the electrodes will
be a result of strong and weak-side of spin direction of the spiral.

Hypotenuse used as a spiral angle, \selectlanguage{greek}ψ\selectlanguage{english}\textsubscript{h}

To determine when a WF strike occurs for an Archimedean spiral at an
electrode, the size of the spiral, clock angle \selectlanguage{greek}ψ\selectlanguage{english}\textsubscript{c} , and
position along its vector path must be known. Similar to
S\textsubscript{R} , the S\textsubscript{Sp} orientation angle changes
all along its path, governed by ?\textsubscript{s} of the clock angle
\selectlanguage{greek}ψ\selectlanguage{english}\textsubscript{c} . Also, along this path of motion, the distance h to
the electrode changes by equation 27. At each position of
S\textsubscript{Sp} , that new distance h to the electrode can be
inserted into equation 25, substituting h for r in equation 25,\selectlanguage{english}
\begin{longtable}[]{@{}ll@{}}
\toprule
\(\psi_{h}=\frac{2\pi}{\lambda_{\text{Sp}}}\text{h\ .\ \ \ \ }\) & (28)\tabularnewline
\bottomrule
\end{longtable}

The calculated angle on the spiral, \selectlanguage{greek}ψ\selectlanguage{english}\textsubscript{h} , and the h value
can be oriented to the current spiral clock angle position by summing
\selectlanguage{greek}ψ\selectlanguage{english}\textsubscript{h} + \selectlanguage{greek}ψ\selectlanguage{english}\textsubscript{c} , creating a new summation angle
\selectlanguage{greek}ψ\selectlanguage{english}\textsubscript{SA} that replaces equation 26,\selectlanguage{english}
\begin{longtable}[]{@{}ll@{}}
\toprule
\(\psi_{\text{SA}}\left(x\right)=\psi_{c\left(x\right)}+\psi_{h\left(x\right)}\text{\ .}\) & (29)\tabularnewline
\bottomrule
\end{longtable}

As the S\textsubscript{Sp} moves, the \selectlanguage{greek}ψ\selectlanguage{english}\textsubscript{SA} can be
calculated all along this path of motion. Importantly, it is only when
the \selectlanguage{greek}ψ\selectlanguage{english}\textsubscript{SA} is equal to the LOS angle ? does a WF strike
occur (Figure 6C and D). Either the change in the clock angle
\selectlanguage{greek}ψ\selectlanguage{english}\textsubscript{c} between two subsequent WF strikes (first position,
S\textsubscript{Sp1} and the second position, S\textsubscript{Sp2} ), or
a measure of the distance x travelled by S\textsubscript{Sp} can be
computed as a time period or instantaneous frequency. As seen in Figure
6C, there is a sequence reversal of WF strikes at e\textsubscript{1} and
e\textsubscript{2} as S\textsubscript{Sp} moves from position
S\textsubscript{Sp1} (approaching p ) to position
S\textsubscript{Sp2}(recedes from point p ). A sequence reversal
indicates that the frequency changes at e\textsubscript{1} must be
opposite to frequency changes at e\textsubscript{2} (as will be shown
below) and therefore the diametric property of strong and weak-sides
must also exist for spiral waves.

\textbf{Movement of spinning spiral wave and plotting spiral summation
curves}

To analyze the diametric property changes in WF strike frequency in
detail, the snapshot examples of Figure 6 are put into motion as the
spiral spins while travelling in a straight line. In contrast to
S\textsubscript{R} , WF strikes from S\textsubscript{Sp} can be
determined accurately at the intersection of the family of
\selectlanguage{greek}ψ\selectlanguage{english}\textsubscript{SA} curves for each rotation (similar to family of RDU
lines in Figure 3) with the arccotan curves of the LOS angle ? as a
function of x (Figure 7). Here for S\textsubscript{Sp} , WF strikes
occur when \selectlanguage{greek}ψ\selectlanguage{english}\textsubscript{SA} = ? for each rotation. Each clock angle
\selectlanguage{greek}ψ\selectlanguage{english}\textsubscript{c} as a function of x is the same calculation of RDU
lines as Equation 12. The plot is again limited from -\selectlanguage{greek}π \selectlanguage{english}[?]
ps\selectlanguage{english}\textsubscript{c} [?]+\selectlanguage{greek}π \selectlanguage{english}such that at the end of one rotation, when
\selectlanguage{greek}ψ\selectlanguage{english}\textsubscript{c} reaches 9:00 or -\selectlanguage{greek}π, \selectlanguage{english}the y-intercept and is
incremented by 2?. The hypotenuse angle
\selectlanguage{greek}\emph{ψ}\selectlanguage{english}\textsubscript{\textbf{h}} , converted from the h distance of
S\textsubscript{Sp,} is added to the clock angle \selectlanguage{greek}ψ\selectlanguage{english}\textsubscript{c} for
all values of x . Substitution of the component parameters
of\selectlanguage{greek}\emph{ψ}\selectlanguage{english}\textsubscript{\textbf{c}} and
\selectlanguage{greek}\emph{ψ}\selectlanguage{english}\textbf{\textsubscript{h}} into Equation 29 yields\selectlanguage{english}
\begin{longtable}[]{@{}ll@{}}
\toprule
\(\mathbf{\psi}_{\mathbf{\text{SA}}}\left(\mathbf{x}\right)\mathbf{=\ -}\frac{\mathbf{2}\mathbf{\pi}}{\mathbf{D}}\mathbf{x+}\left(\mathbf{a+n}\mathbf{2}\mathbf{\pi}\right)\mathbf{+}\frac{\mathbf{2}\mathbf{\pi}}{\mathbf{\lambda}_{\mathbf{\text{Sp}}}}\sqrt{\mathbf{x}^{\mathbf{2}}\mathbf{+}\mathbf{r}^{\mathbf{2}}}\mathbf{\ \ ,}\) & (30)\tabularnewline
\bottomrule
\end{longtable}

which can accurately identify where on the spiral WF crest a distance h
is measured from the spiral wave's center at any specific x-position and
spin orientation. The y-intercept component essentially sets the clock
time at a specific portion of an RDU. Every multiple n distance D from
that position on the x-axis will have the same \selectlanguage{greek}ψ\selectlanguage{english}\textsubscript{c} . The
value a is within the range 0[?] \emph{a} [?] 2?, and n is an integer from
-[?] \textless{} n \textless{} +[?]. For one specific \selectlanguage{greek}ψ\selectlanguage{english}\textsubscript{SA}
curve, one could replace that y-intercept with the variable K for a
where K = a + n2?, and simplifies equation 30 to\selectlanguage{english}
\begin{longtable}[]{@{}ll@{}}
\toprule
\(\mathbf{\psi}_{\mathbf{\text{SA}}}\left(\mathbf{x}\right)\mathbf{=-}\frac{\mathbf{2}\mathbf{\pi}}{\mathbf{D}}\mathbf{x+}\frac{\mathbf{2}\mathbf{\pi}}{\mathbf{\lambda}}\sqrt{\mathbf{x}^{\mathbf{2}}\mathbf{+}\mathbf{r}^{\mathbf{2}}}\mathbf{+K\ \ .}\) & (31)\tabularnewline
\bottomrule
\end{longtable}

Thus, each spiral summation curve is made up of two components for its
slope, a clock spin orientation RDU and spiral size-specific distance.

\textbf{Spiral wave frequency effect equation}

In the example provided in Figure 7, the parameters of
?\textsubscript{s} and ?\emph{\textsubscript{\textbf{s}}} were chosen to
allow easy visualization separating specific spiral positions at
sequential WF strikes along vector path of S\textsubscript{Sp} . Each
spiral curve plotted (Figure 7A) are at the moment of subsequent WF
strikes at equidistant electrodes, e\textsubscript{1} and
e\textsubscript{2} (bold dot-dash spirals, thin dash spirals
respectively). The points of intersection (white dots, Figure 7B)
between the family of \selectlanguage{greek}ψ \selectlanguage{english}\textsubscript{SA} lines (black lines) with the
arccotangent curves for the LOS angle ? to electrodes
e\textsubscript{1}and e\textsubscript{2} (blue and red lines
respectively) indicate exactly when a WF strike occurs
(\selectlanguage{greek}ψ\selectlanguage{english}\textsubscript{SA} = ?). The comparison of frequency relative to the
side of observation can now be made (Figure 7C). As the
S\textsubscript{Sp} approaches both electrodes of same distance of
separation, strong-side electrode e\textsubscript{1} records an
increasing frequency simultaneously to a weak-side decreasing frequency
at e\textsubscript{2}.

These initial motion parameters were chosen specifically to provide less
WF strikes than would typically occur and simplify the visualization of
the diametric property. WF strikes occurred with each rotation when \selectlanguage{greek}ψ
\selectlanguage{english}\textsubscript{SA} = ?. The clock angle component of \selectlanguage{greek}ψ
\selectlanguage{english}\textsubscript{SA} provides the solution to frequency and must be
differentiated from the clock angle of a straight arm rotational WF. The
clock angle \selectlanguage{greek}ψ\selectlanguage{english}\textsubscript{c} , for WF strike calculations of the RFE
is a direct function of ?,\selectlanguage{english}
\begin{longtable}[]{@{}ll@{}}
\toprule
\(\text{Rotational\ }\mathbf{\psi}_{\mathbf{c}}\left(\mathbf{x}\right)\mathbf{=\theta}\)\textbf{.} & (32)\tabularnewline
\bottomrule
\end{longtable}

To avoid confusion, by adding a superscript Sp to \selectlanguage{greek}ψ\selectlanguage{english}\textsubscript{c} ,
the distinction can be made where WF strikes calculation to assess the
spiral wave frequency effect, is a function of \selectlanguage{greek}ψ \selectlanguage{english}\textsubscript{SA} --
\selectlanguage{greek}ψ\selectlanguage{english}\textsubscript{h} ,\selectlanguage{english}
\begin{longtable}[]{@{}ll@{}}
\toprule
\(\text{Spiral\ }\mathbf{\psi}_{\mathbf{c}}^{\mathbf{\text{Sp}}}\left(\mathbf{x}\right)\mathbf{=}\mathbf{\psi}_{\mathbf{\text{SA}}\left(\mathbf{x}\right)}\mathbf{-}\mathbf{\psi}_{\mathbf{h}\left(\mathbf{x}\right)}\mathbf{\text{\ .}}\) & (33)\tabularnewline
\bottomrule
\end{longtable}

The RFE equation can now be modified to a spiral solution,\selectlanguage{english}
\begin{longtable}[]{@{}ll@{}}
\toprule
\(f^{{}^{\prime}}=\left(\frac{2\pi}{2\pi-\Delta\psi_{c-St}^{\text{Sp}}}\right)f^{o}\text{\ \ \ \ \ \ \ \ or\ \ \ \ \ \ \ \ \ }f^{{}^{\prime}}=\left(\frac{2\pi}{2\pi-\Delta\psi_{c-Wk}^{\text{Sp}}}\right)f^{o}\text{\ .\ }\) & (34)\tabularnewline
\bottomrule
\end{longtable}

More simply the \textbf{Spiral Wave Frequency Equation (SFE)} is,\selectlanguage{english}
\begin{longtable}[]{@{}ll@{}}
\toprule
\(f^{{}^{\prime}}=\left(\frac{\omega_{s}}{2\pi-\Delta\psi_{c-St}^{\text{Sp}}}\right)\text{\ \ \ \ \ \ \ \ or\ \ \ \ \ \ \ \ \ }f^{{}^{\prime}}=\left(\frac{\omega_{s}}{2\pi-\Delta\psi_{c-Wk}^{\text{Sp}}}\right)\text{\ .}\) & (35)\tabularnewline
\bottomrule
\end{longtable}

The frequency calculations from WF strikes can also be calculated most
easily by using the spiral ? distance-time, D as a scale of frequency,
and the change in x-values between WF strikes measured against D.
An\textbf{alternate SFE} is,\selectlanguage{english}
\begin{longtable}[]{@{}ll@{}}
\toprule
\(f=\frac{D}{\Delta x_{\text{St}}}\omega_{s}\ \ \ \ \ \ \ \ or\ \ \ \ \ \ f=\frac{D}{\Delta x_{\text{Wk}}}\omega_{s}\text{\ .}\) & (36)\tabularnewline
\bottomrule
\end{longtable}

\textbf{f. Special case of electrode sits directly on vector path
of\emph{S\textsubscript{Sp}}}

Two changes occur with frequency calculations when an electrode sits
directly on the ?\textsubscript{s} vector path of S\textsubscript{Sp} .
First, the WF strikes would occur with LOS ? = 0, no matter the distance
from inflection point p . The subsequent WF strikes, in this special
case, would occur only when \selectlanguage{greek}ψ\selectlanguage{english}\textsubscript{SA} = 0 radians.
\selectlanguage{greek}ψ\selectlanguage{english}\textsubscript{SA} is independent of the distance from p , and
therefore not on a changing ? governed by a side-specific arccotangent
curve. Second, r=0, within the spiral size-specific distance component
of the Equation 31. The \selectlanguage{greek}ψ\selectlanguage{english}\textsubscript{SA} equation simplifies to\selectlanguage{english}
\begin{longtable}[]{@{}ll@{}}
\toprule
\(\mathbf{\psi}_{\mathbf{\text{SA}}}\left(\mathbf{x}\right)\mathbf{=-}\frac{\mathbf{2}\mathbf{\pi}}{\mathbf{D}}\mathbf{x\pm}\frac{\mathbf{2}\mathbf{\pi}}{\mathbf{\lambda}}\mathbf{x+K\ ,}\) & (37)\tabularnewline
\bottomrule
\end{longtable}

where the (-) sign is used as S\textsubscript{Sp}approaches the
electrode, and (+) as S\textsubscript{Sp}recedes from it. The solution
results in 2 families of parallel straight lines, approaching and
receding. The slopes of the line or first derivative provide the ability
to make a calculation of frequency,\selectlanguage{english}
\begin{longtable}[]{@{}ll@{}}
\toprule
\(\mathbf{slope:\ \ }\mathbf{\psi}^{\mathbf{{}^{\prime}}}\left(\mathbf{x}\right)\mathbf{=}\left|\frac{\mathbf{2}\mathbf{\pi}}{\mathbf{D}}\right|\mathbf{\pm}\frac{\mathbf{2}\mathbf{\pi}}{\mathbf{\lambda}}\mathbf{\text{\ \ }}\text{.\ \ \ \ \ \ \ \ \ \ }\) & (38)\tabularnewline
\bottomrule
\end{longtable}

An absolute value of the D component of the RDU is now used to convert
the reversed x-axis that was originally needed to measure clock angles
\selectlanguage{greek}ψ\selectlanguage{english}\textsubscript{c} in Figures 3, and 7. The slope on the polar graph can
be converted to a frequency by multiplying the slope by
v\textsubscript{s} /2?,\selectlanguage{english}
\begin{longtable}[]{@{}ll@{}}
\toprule
\(\mathbf{f=}\left(\left|\frac{\mathbf{2}\mathbf{\pi}}{\mathbf{D}}\right|\mathbf{\pm}\frac{\mathbf{2}\mathbf{\pi}}{\mathbf{\lambda}}\right)\mathbf{\ }\frac{\mathbf{v}_{\mathbf{s}}}{\mathbf{2}\mathbf{\pi}}\mathbf{\ \ ,}\text{\ \ \ \ \ \ \ \ \ }\) & (39)\tabularnewline
\bottomrule
\end{longtable}

or\selectlanguage{english}
\begin{longtable}[]{@{}ll@{}}
\toprule
\(\mathbf{f=}\left|\frac{\mathbf{v}_{\mathbf{s}}}{\mathbf{D}}\right|\mathbf{\pm}\frac{\mathbf{v}_{\mathbf{s}}}{\mathbf{\lambda}}\text{\ \ .}\) & (40)\tabularnewline
\bottomrule
\end{longtable}

This equation also provides the same result in frequency for electrodes
not on the vector path \textbf{?\textsubscript{s}} when
S\textsubscript{Sp} is far away from the inflection point. The r value
becomes insignificant in the calculation of the hypotenuse angle \selectlanguage{greek}ψ
\selectlanguage{english}\textsubscript{h}of Equation 31. The WF strikes, intersection points on
the arccotangent curves approach asymptotically the x-axis (approach y =
0). It is only at relatively short distances of S\textsubscript{Sp} ,
within the first few ? to the inflection point, that the side-dependent
frequency changes, or the diametric property, would be observed. In the
case of cardiac rotors with ? being up to 2-3 cm, then the diametric
property would be obvious. These frequency changes are recordings from a
single site and are different than previous descriptions labelled as
phase shifts comparing activation sequence at multiple different
recording sites of a single revolution. The side difference is more than
a mere phase shift or slip, there is an actual loss of a WF strike in
number.

Both components of slope of Equation 38 have numerators of 2? which
allows direct summation of angles in a polar perspective. It will be
shown below that by constructing this equation in this manner, D and ?
become an important ratio with regard to frequency of WF strikes when
comparing approaching and receding spirals.

\textbf{Electrode separation, comparison of diametric property of SFE
and CDE}

Cardiac rotors during atrial fibrillation move about tenfold slower than
in the teaching example of Figure 7 above. Typical human cardiac rotors
that have angular frequencies ranging from 4-8 Hz (\emph{30} ), and can
drift at 10-20 mm/sec (\emph{8} ). The diametric property is more
apparent at these slower \textbf{?\textsubscript{s}} rates. Computed
comparisons of various electrode separation and spiral sizes are shown
respectively in Figures 8 and 9 below to help identify more specific
characteristics of the diametric property of the SFE. In both figures, a
spiral WF source, S\textsubscript{Sp} with ?\textsubscript{s} at 5 Hz,
moves 1 cm/sec between 2 equidistant electrodes. If an electrode sat
directly on the v\textsubscript{s} vector path, the approaching and
receding WF strike frequency would be at a constant 5.32Hz (T=188 msec)
and 4.68 Hz (T= 213 msec), respectively (using Equation 40). However,
just 2 mm off to the side of the vector path (Figure 8, dark blue and
dark red curves), the diametric property of the SFE shows that the
strong-side has a peak WF frequency of 5.78 Hz (T= 173 msec) while
simultaneously, the weak-side has a nadir frequency of 4.24 Hz (T= 238
msec). Electrodes positioned closest to the side of the vector path of
S\textsubscript{Sp} have the greatest magnitude changes in frequency
compared to electrodes placed further to the sides, or directly in front
or back. Contrast this for a moving source, S\textsubscript{c} , that
emitting periodic centrifugal propagating WFs. Electrodes positioned
equidistantly on either side of\textbf{?\textsubscript{s}} , would
experience WF strikes simultaneously and the change in frequency would
be governed by the classical Doppler Equation 21 as seen in Figure 8B.
Peak and nadir frequency are seen directly in front or behind, not to
the sides as that observed from an S\textsubscript{Sp} .

\textbf{h. Spiral wavelength effect on diametric property of SFE, ?/D
ratio, and surfing}

The diametric property of the SFE is more apparent with closer
equidistant electrodes to S\textsubscript{Sp} . Utilizing that
observation, further assessment of the diametric property was made by
varying the spiral size ? as the S\textsubscript{Sp}moved between
electrodes spaced only 2 mm on either side of ?\textsubscript{s} (Figure
9). All spiral sizes exhibited the diametric property with strong and
weak-sides showing their greatest difference in frequencies at the
inflection point. The peak frequency of the strong-side occurred earlier
than the nadir frequency on weak-side, resulting in a skewed
diamond-shaped separation at center. The later weak-side nadir is a
result of the unpaired strong-side WF signaling a sequence reversal. The
largest spiral sizes (appearing more like a radial arm of an
S\textsubscript{R}) approached frequency changes most similar to RFE
seen in Figure 4, where approaching and receding frequencies were equal
at relatively long distances from p with a frequency of
?\emph{\textsubscript{\textbf{s}}} . At the other extreme, decreasing ?,
resulted in a larger spiral-size component in the slope of the summation
angle, \selectlanguage{greek}ψ\selectlanguage{english}\textsubscript{SA} of Equation 31 and Equation 38, causing
greater variation between the approaching and receding
S\textsubscript{Sp} . In the special situation where ? = D, the
approaching frequency is twice that of
?\emph{\textsubscript{\textbf{s}}} , and the receding frequency is zero.
At this surfing ratio, there are no WF strikes because the WF crests
continuously straddles the electrodes. At smaller ratio, where ?/D
\textless{} 1, then WF sequence does not reverse when
S\textsubscript{Sp} recedes from p (not shown).

\textbf{DISCUSSION}

WFs emitted from rotational and spiral sources do not follow classic
Doppler effect phenomena. In a previous investigation of rotors
migrating passed a perimeter of electrodes in human AF (\emph{12} ),
three new events were observed repeatedly that were unable to be
predicted by the CDE equation. This is corroborated by same events that
can be found when analyzing recordings of independent investigators
(Figure 1D, (\emph{24} )). These events, diametric side-dependent
frequency changes, an unpaired strong-side WF strike, and activation
reversal, constitute the diametric property of a spiral or rotational
wave frequency effect. The newly derived RFE and SFE equations here,
predict all 3 events of this diametric property that were seen
experimentally. The Doppler effect and its equation are specific to the
application of S\textsubscript{c} , but not from a source that emits a
rotating wave by an S\textsubscript{R} or S\textsubscript{Sp} . A first
attempt to describe frequency changes from a migrating spiral wave in a
Belousov -- Zabotinsky reaction were ``interpreted'' as a phase shift of
the Doppler effect (\emph{7)} . The technical limitations could not have
identified strong and weak sides. A simple phase shift or phase slip
does not accurately describe loss of a weak-side WF strike, nor
simultaneous increasing and decreasing side and spin-dependent frequency
changes. The importance of the asymmetry of WF strike number near a
moving rotational source will be made more obvious in the examples
provided below. Interestingly, the SFE results in frequency changes that
asymptotically approach that calculated for a CDE after several ? of the
spiral center (Figure 9).

Rotational and spiral math were developed from a new unit, the RDU, that
allows accurate calculation of WF strikes with each rotation. WF strikes
from a hypothetical spinning straight radial WF crest occurred only when
the clock angle, \selectlanguage{russian}ѱ\textsubscript{c} , equaled the LOS angle \selectlanguage{english}?. WF
strikes from a spinning spiral WF crest occurred only when the spiral
summation angle, ps\textsubscript{SA} , equaled the LOS angle \selectlanguage{english}?. The
vector line of motion of either S\textsubscript{R} or
S\textsubscript{Sp} separates 2-D space into a strong-side (increasing
frequencies with approaching source) and a weak-side (decreasing
frequencies with approaching source). Reversing the spin direction
reverses the sides. The side-dependent diametric frequency changes of
both S\textsubscript{R} and S\textsubscript{Sp} result from a
strong-side unpaired WF strike at the inflection point when sequence
reversal occurs (Figure 5 and 8). In both RFE and SFE, observation
positions closer to the vector path \selectlanguage{english}\emph{?\textsubscript{s}} of either
S\textsubscript{R} or S\textsubscript{Sp} exhibited more profound
frequency differences between strong and weak-sides. Observing WF
strikes from a position directly on the vector path as the source passes
by, results in an instantaneous phase shift with S\textsubscript{R} or
an instantaneous frequency shift with S\textsubscript{Sp} . Frequency
changes observed on the vector path of motion for either
S\textsubscript{c} or S\textsubscript{Sp} are equal. A most important
conclusion from these new equations is that 2 mm may separate recordings
of highest and lowest frequency of action potential activation, and may
have just signal that a rotor had just passing between then. A bipolar
electrode spanning both sides would be expected to record transient
double potentials (12).

Geometrically, both S\textsubscript{R} and S\textsubscript{Sp} showed
staggered potential shear and stall zones on strong and weak-sides
respectively which confirm likely regions of the biophysical constraints
of rotor migration (\emph{11} ). Stall zones clearly represent regions
where the electrical action potential restitution, or refractoriness,
would prevent WF propagation and would be a site of a functional block
(\emph{11} ). Shear zones, where tangential velocity propagation
velocity hits upper biophysical limits might result in regions where the
spiral shape is transformed to more centrifugal outward propagation.
Additionally, the migration or sliding of the spiral core tip is limited
by the electrical restitution of the atrial tissue, the critical radius
of curvature, and the propagation conduction velocity restitution
(\emph{11,21} ). Taken together, these factors that steer/curve a
rotor's path, and show that a single region of atrial tissue may have
multiple roles as being both active and passive contributors to the
arrhythmia. Can normal atrial tissue be considered as a true source of
arrhythmia if it is just the stability of the rotor's shape and its
ability to recruit neighboring cells to behave abnormally transiently
just as it passes through? Slight adjustment of the AF model lead to new
insights into why current treatment strategies may be ineffective
(\emph{32} ).

\textbf{Atrial fibrillation: The Slide-Pivot-Slide/Triggered Model,
Maximizing the Slide}

The more detailed electrical recordings of migrating rotors and the
identification of the diametric property of SFE allows one to slightly
modify the description of atrial fibrillation. The majority of migrating
rotors found in our previous study and others (\emph{12,23,24} ), were
not sustained more than a few seconds. Our methods had selected 5
rotations or more (typically lasting at least 1 second). Even if a WF
pivoted with just 1 or 2 rotations (consider naming this a swirl), great
variation in the time of tissue activation, either from frequency
changes, and/or phase shifts of adjacent tissue, would be manifest.

A WF crest propagates or slides between lateral boundaries, and then
pivots around the end of each boundary. Continuously changing barriers
from time-dependent refractory periods, constantly change the
propagation path. The WF emerges beyond a barrier with a bend, a pivot,
then slide between new barriers, or may even shed new
S\textsubscript{Sp} (\emph{23,29,32} ). Focal ectopic source of APs,
either from spontaneous or from triggered automaticity would add to
additional drivers or functional barriers of conduction to sustain the
duration of AF (\emph{33} ). New pivot points, swirls, or rotors sustain
WF propagation. Just the ability to pivot to unexcited heart tissue
faster than propagation of WFs from the sinus node pacemaker cells may
be just as important as high dominant frequency (HDF) of transient
rotors.

Observing the migration of rotors, one can conclude that any one
specific atrial tissue site may change its component function of
arrhythmia. Within moments, atrial cells act either as a slide, a
barrier, a pivot, swirl, rotor, or a triggered site. The diametric
property of SFE shows that the actual path of a rotor is a line
separating highest frequency and lowest frequency. If the spin
orientation was reversed, then the side of fastest and slowest frequency
would also be reversed. Ablation of these moving targets of rotor
migration based only upon sites of HDF might still fail. Should atrial
tissue that is recruited momentarily to function abnormally be
considered a true ``source'' of arrhythmia and ablated? Possibly, a more
reasonable approach might be to shift from a moving target of normal
cells behaving badly, to one that ablates regions of atrial tissue
regions that have a higher ability to spawn new rotors, or regions that
exhibit frequent pivots in the propagation path. New studies are being
planned. Ablation may become more customized in the process to maximize
the slide and minimize the pivots with a goal to limit tissue
destruction.

\textbf{Side-Relative Frequency Perception and The Unpaired WF Strike}

When the center of an S\textsubscript{R} or S\textsubscript{Sp} is
stationary but emits a rotating WF at a constant \selectlanguage{english}?\textsubscript{s}, all
stationary observation sites record the same WF strike frequency, just
at different phases of the rotation. However, once the center of an
S\textsubscript{R} or S\textsubscript{Sp} moves relative to stationary
observers, symmetry breaks down and frequencies that are recorded are
relative to the side and spin direction that one observes the
S\textsubscript{R} or S\textsubscript{Sp} . The vector path of motion
splits 2-D space into 2 sides, a strong and weak side. Which of the two
sides that an observer sits, is determined by the 3-point sequence rule
described above. As an S\textsubscript{R} or S\textsubscript{Sp} moves
past 2 stationary observers that are positioned on either side of the
vector path, the strong-side will record one more WF strike than the
weak-side. This side difference in WF strike count is independent of
\selectlanguage{english}?\textsubscript{s} and \selectlanguage{english}?\textsubscript{s}. Importantly had an observer
moved together with the S\textsubscript{R} or S\textsubscript{Sp} ,
parallel along the same vector path, keeping the same distance, then the
frequency observed would be the same as \selectlanguage{english}?\textsubscript{s} of
S\textsubscript{R} or S\textsubscript{Sp} , as if neither moved.

Figure 10 provides a simple illustration where Person R is x meters from
2 closely positioned stationary observers S (strong-side) and W
(weak-side). Person S and W turn their heads to keep their eyes focused
on Person R. Person R spins at constant
\selectlanguage{english}?\emph{\textsubscript{\textbf{s}}} while moving at constant velocity
directly between S and W to a position x meters away. R completes 1/2
rotation. Person R faced observer W along its entire path and saw no
turns of R, while person S saw essentially one full turn. Now, in a
second pass, person R makes 1\selectlanguage{ngerman}½ turns, spinning again in the same
direction, moving that same 2x meter distance. W observed 1 full turn,
while S observed 2 full turns. The number of turns observed is relative
to which side of the vector path that the observer sits. Independent of
\selectlanguage{english}?\textsubscript{s} and \selectlanguage{english}?\textsubscript{s}, S observes one more turn than
W. Most fascinating, is that the opposite relative changes in frequency
between strong and weak-sides are because of just one greater number of
rotations at the inflection point between the 2 sides. The side-relative
diametric frequency changes are apparent and becomes increasingly
important as one investigates interactions closer to a moving
S\textsubscript{R} or S\textsubscript{Sp} with its local environment or
media.

This side-dependent frequency differences (diametric property of RFE or
SFE) found in WFs from cardiac rotors, also apply directly to any
rotating mass that moves through space relative to other objects (Figure
10C). One might ask if this diametric property affects the original
Doppler conclusions? The short answer is yes, in a small way, but not in
our lifetime. A hypothetical condition answers this question a bit more
fully. A binary star system moves along a vector path directly between
two imaginary Christian Dopplers, named S and W. If the rotation plane
of the binary star and the 2 Doppler observers are coplanar, then the
diametric property of RFE of an S\textsubscript{R} applies. The star
rotation position that results with greatest angular velocity towards
each Doppler is simply when the clock angle \selectlanguage{russian}\emph{ѱ} \textsubscript{c} =
LOS angle \selectlanguage{english}?. The moment that a peak frequency is observed would be
identical to that of a WF strike from an S\textsubscript{R} . Both
Dopplers would record an oscillation in frequencies, but with marked
differences in the timing of when those peaks occurred. For an
approaching binary star, strong-side Doppler would record a shortening
period between the peaks of frequency, as well as one greater peak count
than what the weak-side Doppler records. Although the 2 men would have
identified peak frequencies to occur when the tangential velocities of
the star aligned with his LOS, they would have developed opposite
conclusions regarding the timing of peak frequency and their theories
would not have completely matched. Importantly, two other Doppler
observers, D and D', that travel in the same inertial plane with the
binary star, remaining the same exact distance from the binary would not
observe any changes. Their plane of reference would not result in the
development of strong and weak sides. Both would observe a constant
frequency of the timing of the peaks, their 2D space would not have been
split into strong and weak sides.

The proof of the existence of diametric property of rotational
propagating thereby limits the definition of the Doppler effect to be a
specific symmetric phenomenon only of centrifugal propagating WFs of
S\textsubscript{C} . In addition, to the loss of symmetry along the
sides of the path of motion of S\textsubscript{R} or S\textsubscript{Sp}
, is the loss of a sharp demarcation at the inflection point of
perceived blue-shifted frequency (or higher frequency observed than
emitted) and perceived red-shifted frequency (or lower frequency
observed than emitted). When an S\textsubscript{R} or
S\textsubscript{Sp} moves relative to observer positions, the diametric
property of RFE and SFE expands the zones of higher and lower
frequencies that can be observed. Therefore, a weak-side observer can
perceive slower then emitted frequencies even though S\textsubscript{R}
or S\textsubscript{Sp} is approaching. The converse being true also
(shaded red and blue regions, Figures 4 and 8). It would be inaccurate
to assume that higher frequencies and lower frequencies observed,
represent always an approaching and receding S\textsubscript{R} and
S\textsubscript{Sp} respectively, especially when observers are near to
the source.

The asymmetric labelling of strong-side and weak-side of hurricanes
(\emph{34} ) describing greater tangential wind speeds and storm surges
(Ekman transport) may no longer be analogous or coincidental phenomena
to cardiac rotors but are likely consequential of the diametric property
of rotational relativity. The energy of rotating WFs subject to the
diametric property of RFE and SFE may have direct influence on mass
within its media to result in the spiral lattice structure of crystal
formation (\emph{35} ) at one level of nature to the accretions of
matter into the spiral stellar arms within the galaxies at another
level. The creation of one's own spiral wave and surfing between its
crests for a means of propulsion would require computations of specific
surfing ratio of D and \selectlanguage{english}? as seen in Figure 9. The diametric property of
RFE and SFE appears to be the critical link between many spinning
phenomena at all sizes of nature. Future investigations of the diametric
property of RFE, and SFE might improve the understanding of chemical
reactions, enantiomer formation, air turbulence, water eddies, Reynold's
number, wing lift, weather models, and plotting paths of space travel.

\textbf{Acknowledgements}

I gratefully acknowledge the support, multiple discussions and
identification of additional references by Mason Rubenstein.

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Permission Letter for Figure 4B of Ghoraani that is used in Figure 1D

\textbf{Supplemental Materials}

Figure 3 Data and weblink

Figure 4 Data and weblink

Figure 7 Data and weblink

Figure 8A Data and weblink

Figure 8B Data

Figure 9 Data and weblink

\textbf{TABLES}\selectlanguage{english}
\begin{longtable}[]{@{}llllllllllll@{}}
\toprule
Blue & e\textsubscript{1} & & & Red & e\textsubscript{2} & & & Skew &
e\textsubscript{2} & &\tabularnewline
\midrule
\endhead
x & ?x & ? & Deg & x & ?x & ? & Deg & x & ?x & ? & Deg\tabularnewline
3.54 & & & 15.7 & 3.46 & & & -16.2 & 3.47 & & & -9.3\tabularnewline
2.56 & 0.98 & 10.2 & 21.3 & 2.44 & 1.02 & 9.8 & -22.3 & 2.47 & 1.01 & 10
& -11.9\tabularnewline
1.59 & 0.97 & 10.3 & 32.2 & 1.40 & 1.04 & 9.6 & -35.5 & 1.45 & 1.01 &
9.8 & -16.6\tabularnewline
0.66 & 0.93 & 10.7 & 56.6 & 0.30 & 1.11 & 9.1 & -73.5 & 0.42 & 1.03 &
9.7 & -27.2\tabularnewline
-0.22 & 0.88 & 11.5 & 192.2 & -0.86 & 1.16 & 8.6 & -130.8 & -0.68 & 1.11
& 9.1 & -66.7\tabularnewline
-1.12 & 0.9 & 11.1 & 138.1 & -1.92 & 1.06 & 9.4 & -152.5 & -1.89 & 1.20
& 8.3 & -140.6\tabularnewline
-2.07 & 0.95 & 10.5 & 154.2 & -2.95 & 1.02 & 9.8 & -161.3 & -2.94 & 1.05
& 9.5 & -159.3\tabularnewline
-3.05 & 0.98 & 10.2 & 161.9 & -3.96 & 1.01 & 9.9 & -165.8 & -3.96 & 1.02
& 9.8 & -166.1\tabularnewline
\bottomrule
\end{longtable}

Table I. WF strikes positions at e\textsubscript{1},
e\textsubscript{2}and skewed e\textsubscript{2} along the
v\textsubscript{s} vector path.

Values of x represent the distance S\textsubscript{R} is from point p in
Figure 3. Note that as S\textsubscript{R} approaches the inflection
point, there is an increasing frequency (?) on strong-side (blue), while
simultaneous decreasing ? on weak-side (red).

\textbf{FIGURE LEGENDS}

Figure 1. Three components of electrode recording pattern of a migrating
rotor breaching perimeter of electrodes.

\begin{enumerate}
\tightlist
\item
  Unipolar recordings of a clockwise spinning rotor as it migrates from
  within ring of 16 electrodes. (Methods in (\emph{12} )). A
  head-meets-tail activation sequence (slanted left dotted yellow oval)
  is observed as CW rotor spins within a circular perimeter of
  electrodes. As rotor migrates between electrodes
  e\textsubscript{10}and e\textsubscript{11}, three events occur:
  simultaneous diametric frequency shifts (strong-side (s)
  e\textsubscript{10} increases in frequency while weak-side (w)
  decreases in frequency), an unpaired WF strike occurs on strong-side
  e\textsubscript{10}, and a reversal of activation sequence
  e\textsubscript{8}-e\textsubscript{13} (yellow arrows).
\item
  Schematic of rotor positions of A above, as core migrations breaches
  perimeter of electrodes. Actual recordings (12) included unipolar,
  adjacent bipolar and cross-circle bipolar confirming the breach.
\item
  Schematic timing of WF strikes at
  e\textsubscript{8}-e\textsubscript{13}. Note the 3-component pattern
  of WF strikes that occurs when a rotor breaches the perimeter;
  simultaneous strong-side increasing frequency with a weak-side
  decreasing frequency, an unpaired strong-side WF strike, and a
  reversal of activation sequence. This pattern was observed repeatedly
  54 of 56 transits (\emph{12} ).
\item
  Independent confirmation of the 3-component pattern of diametric
  property is seen in recordings of a counterclockwise spinning rotor
  meandering across a circular array of electrodes. Strong-side (s) and
  weak-side (w) have been labelled here, showing an unpaired strong-side
  WF strike, and reversal of activation sequence (green dashed oval with
  arrow). Reprinted and adapted Fig 4B from Ghoraani et al (\emph{24} )
  with permission from Elsevier Publishing Copyright 2013, Heart Rhythm.
\end{enumerate}

Figure 2. Rotational Distance Unit (RDU), clock
angle\selectlanguage{russian}\emph{ѱ}\textsubscript{\textbf{c}} and LOS angle \selectlanguage{english}?.

A. The S\textsubscript{R} is theoretical clock that slides on a surface
at a constant velocity. The clock hand represents a WF crest that
rotates about its center at a constant \selectlanguage{english}?\textsubscript{s}. The
rotational distance unit (RDU) is defined as a single
revolution/distance travelled 2\selectlanguage{english}?/D. Regions of WF that exhibit greatest
velocity, shear zones, and regions of slowest velocity, stall zones are
separated by consecutive RDUs.

B. \emph{Top} : The rotational WF of an S\textsubscript{R} is
represented as a green clock hand that spins clockwise at constant
\selectlanguage{english}?\textsubscript{s}. Recording electrodes e\textsubscript{1} and
e\textsubscript{2} are on the tissue surface. S\textsubscript{R} moves
at a constant\selectlanguage{english}\textbf{?\textsubscript{s}} along its vector path (green
arrow) that is a perpendicular bisector between e\textsubscript{1} and
e\textsubscript{2}, such that the rotor is always equidistant to
e\textsubscript{1} and e\textsubscript{2}. The distance travelled
between each complete revolution is the RDU.

\emph{Bottom:} Each RDU (vertical dotted lines) is shown as the clock
approaches inflection point (\textbf{p} ). The colored dots correspond
to the time at which the wave front strikes the electrodes. The
clockwise rotation in the plan of atrial surface results in sequential
activation of e\textsubscript{1} followed by e\textsubscript{2}. With
each RDU, the time period between WF strikes at e1 shortens, while time
between WF strikes at e\textsubscript{2} lengthens. When rotor passes
inflection point \textbf{p} , the sequence reverses. Changing the clock
rotation to counterclockwise would reverse the sequence changes.

C. (a). Clock Angle \selectlanguage{greek}ψ\selectlanguage{english}\textsubscript{c} . Upper inset shows angle
0\textsuperscript{o} is set at forward vector path of motion at the 3:00
position. During CW spin, the clock hand (thin yellow triangle) of the
source (\textbf{S} ), is initially positioned at 9:00 (\selectlanguage{greek}π\selectlanguage{english}). The source's
clock hand rotates with constant\textbf{?\textsubscript{s}} at 10 Hz
(20\selectlanguage{greek}π/\selectlanguage{english}s). The clock hand is a positive angle at \selectlanguage{greek}π \selectlanguage{english}and decreases as
\selectlanguage{greek}ψ\selectlanguage{english}\textsubscript{c}rotates towards 3:00 passing +\selectlanguage{greek}π/\selectlanguage{english}2 set at 12:00. When
clock hand passes 12:00, fastest forward tangential velocity is parallel
to\textbf{?\textsubscript{s}} that may create a potential WF shear zone
at greater radii. Once the clock hand passes 3:00 (0
radians),\selectlanguage{greek}\emph{ψ\textsubscript{c}}\selectlanguage{english} is a negative angle as it rotates
back towards 9:00 (-\selectlanguage{greek}π\selectlanguage{english}), passing -\selectlanguage{greek}π/\selectlanguage{english}2 set at 6:00. As the clock hand
passes 6:00, the slowest forward velocity is parallel to
?\textsubscript{s} that may create a potential WF stall zone at greater
radii. The RDU (small clocks along vector ?\textsubscript{\textbf{s}} )
is shown as the clock hand makes one complete revolution. Shear and
stall regions would be expected to occur with each revolution at same
respective \selectlanguage{greek}ψ\selectlanguage{english}\textsubscript{c} , separated by distance D of the RDU.
Electrodes e\textsubscript{1} and e\textsubscript{2}, depicted at the
center of synchronized clocks, are positioned 1 cm on either side of the
vector path that is a perpendicular bisector. Four subsequent WF strikes
are shown at each electrode as \textbf{S} moves passed the inflection
point \textbf{p} .

D. Line of Sight (LOS) angle ?. Note that a WF strike occurs when
\selectlanguage{greek}ψ\selectlanguage{english}\textsubscript{c} = LOS angle ? as it sweeps past clock with each
rotation. Upper inset shows that angle 0\textsuperscript{o} is set
directly along the forward vector path of the source. The positive and
negative angle ?s are set to align with the positive and negative angle
\selectlanguage{greek}ψ\selectlanguage{english}\textsubscript{c} as seen in 3C. Note that once\textbf{S} passes the
inflection point \textbf{p} , LOS angle ? is obtuse. The change in angle
? (??) between subsequent WF strikes is a positive value at a
strong-side electrode, and negative value at a weak-side electrode.
Right inset shows sequence of WF strikes is
e\textsubscript{1}-e\textsubscript{v}-e\textsubscript{2}. Since
e\textsubscript{1} precedes e\textsubscript{v} (dashed white circle),
then e\textsubscript{1} sits on the strong-side.

Figure 3. Electrodes on either side of S\textsubscript{R} 's vector path
reveals the diametric property of RFE.

A. Source's clock, and electrode clocks are synchronized, aligned, and
coplanar. Clockwise S\textsubscript{R} with constant angular velocity
20\selectlanguage{greek}π/\selectlanguage{english}sec, moves at constant velocity 10 cm/sec (RDU= 2\selectlanguage{greek}π/\selectlanguage{english}1cm) along a
perpendicular bisector vector path of line
e\textsubscript{1}e\textsubscript{2} (2.0 cm apart). Electrodes
e\textsubscript{1} and e\textsubscript{2} are equidistant to
S\textsubscript{R} along its entire path. Clocks at e\textsubscript{1}
and e\textsubscript{2} constructed with each radius 1.0 cm touch each
other at the common tangent point \textbf{p.} The source clock reads
9:00 when the source is 3mm away from point\textbf{p.} One complete
revolution of the clock occurs every 1 cm (RDU = 2\selectlanguage{greek}π/\selectlanguage{english}1cm) as the source
moves toward the electrodes. The clock hand (yellow triangle) extends
beyond the perimeter of the clock, rotates and strikes
e\textsubscript{1} before striking e\textsubscript{2}. The first 3 RDUs
result in WF strikes at e\textsubscript{1} at line a, b, and c, and at
e\textsubscript{2} at lines d, e, and f. The distance from\textbf{p} and
the angle of the WF strikes are listed in table 1. The clock hand
strikes the e\textsubscript{1} clock earlier in its rotation
(strong-side) while striking e\textsubscript{2} later in its rotation
(weak-side). See Supplementary Materials Figure 3 Data File. B.
Graphical presentation of both f(x) = \selectlanguage{greek}ψ\selectlanguage{english}\textsubscript{c} for clock hand
angle and f(x) = ? for LOS angle to both electrodes as
S\textsubscript{R} approaches and recedes from inflection point
\textbf{p,} the closest approach to electrodes. Note x-axis is reversed.
Angle ? from the source to e\textsubscript{1} and e\textsubscript{2} is
calculated by the arccotangent for strong-side (blue curve) and the
arccotan curve for weak-side (red curve) from Equation 13 and Equation
14 respectively. Starting point for first RDU (black dot with dashed
arrow), begins with the source clock hand at 9:00 (\selectlanguage{greek}π \selectlanguage{english}radians) at a
position 3 cm away from point \textbf{p}. The RDU lines of the clock
angle \selectlanguage{greek}ψ\selectlanguage{english}\textsubscript{c} (green lines) intersect the arccot curves for
the LOS ? at the moment of WF strike at e\textsubscript{1} and
e\textsubscript{2} (white circles). Intersection points at a-f
correspond to the distance from \textbf{p} and clock angles shown in 3A
above and listed in Table 1. RDUs \#1-3 show WF strikes at a-c at
e\textsubscript{1} and d-f at e\textsubscript{2}. Arccot curves are
shown for comparison if e\textsubscript{2} sits closer to the inflection
point at distances \selectlanguage{ngerman}½, ¼, and \selectlanguage{english} 1/8  of r\textsubscript{2} (black curves). C.
Similar to A, the S\textsubscript{R} clock, and electrode clocks are all
synchronized, aligned, and coplanar. The vector path of the source is at
a tilted (\selectlanguage{greek}\emph{ϕ} \selectlanguage{english}= 30\textsuperscript{o}) compared to A. The path is
created such that e\textsubscript{1} is still 1 cm from the vector's
path of closest approach. The inset of the enlarged electrodes shows the
trigonometry used to calculate the transverse common tangent line
segment length (tct) and the e\textsubscript{2} distance to the closest
approach of the source (r\textsubscript{2}). Electrodes
e\textsubscript{1} and e\textsubscript{2} remain at their same position
with L =2.0 cm of separation. The tct is 1.0 cm (tct = L sin \selectlanguage{greek}\emph{ϕ}\selectlanguage{english}).
The distance (r\textsubscript{2}) of e\textsubscript{2} to the point of
closest approach of the S\textsubscript{R}'s vector path is calculated
at 0.73 cm (r\textsubscript{2} = x\textsubscript{2} cos\selectlanguage{greek}\emph{ϕ}\selectlanguage{english}). D.
Similar to B, simultaneous plots of the clock hand angle \selectlanguage{greek}ψ
\selectlanguage{english}\textsubscript{c} and LOS angle ? as function of distance from both
electrodes. The plot of \selectlanguage{greek}ψ\selectlanguage{english}\textsubscript{c} and the arccot curve for ? of
the LOS to e\textsubscript{1} remains the same. The arccot curve for
e\textsubscript{2} is calculated for an r\textsubscript{2} = 0.73 cm and
skewed by the tct = 1.0 cm. Intersection of curves identify when and
where WF strikes occur. RDUs \#1-3 show WF strikes at a-c at
e\textsubscript{1} and d-f at e\textsubscript{2}.

Figure 4. The diametric property of an S\textsubscript{R} at different
distances of equidistant electrodes.

S\textsubscript{R} has a ?\emph{\textsubscript{\textbf{s}}} = 5Hz, that
moves with ?\textsubscript{s} = 1cm/s along a vector path is a
perpendicular bisector between e\textsubscript{1} and
e\textsubscript{2}. Electrode pairs sit at 1mm, 2mm, 5mm, 1 cm and 2 cm
from the inflection point p. Computed WF strikes are plotted as a
function of x from the p with the instantaneous frequency (hollow
circles) on the y-axis. RDU is 2?/0.2cm. Frequency effects on the
strong-side and weak-side of the S\textsubscript{R} are shown by
smoothed blue shaded curves and red/yellow curves respectively for a
particular electrode distance. The approaching S\textsubscript{R} shows
that the sequence of WF strikes occur at e\textsubscript{1} immediately
precedes e\textsubscript{2.} After S\textsubscript{R}passes inflection p
(at x=0), the sequence of WF strikes reverses, e\textsubscript{2}
strikes precede e\textsubscript{1} (hollow black ovals). At about 0.5
cm, marked separation of progressive WF strike frequency occurs
comparing strong and weak-sides (blue and red shaded ovals). At far left
and right of the plot, frequency approaches
?\emph{\textsubscript{\textbf{s}}} at 5 Hz. An unpaired WF strike on the
strong-side (black bracket) occurs at the inflection point for each
distance. Closer electrodes show more abrupt and greater frequency
changes than electrodes that straddle the vector path at further
distances. Equations and data from these plots are provide in the
Supplementary Materials Figure 4 Data File.

Figure 5. Archimedean spiral and time reference.

A source S\textsubscript{Sp} emits a clockwise Archimedean spiral WF
governed by equation\selectlanguage{greek}\emph{r = ψ} \selectlanguage{english}\textbf{\textsubscript{Sp}} for 2 full
rotations with a constant wavelength ? =2?.

Expanded view of first revolution of spiral with defining position of
clock hand (\selectlanguage{greek}ψ \selectlanguage{english}\textsubscript{c} grey triangle) along the first ?. The
length of line from 0 to 2? is the spiral wavelength ? that defines its
size. Radial spokes from center to the spiral arm show the spiral angles
\selectlanguage{greek}ψ \selectlanguage{english}\textsubscript{Sp} at increments of 30\textsuperscript{o}, or ?/6
radians, and increases in the CCW direction. The clock hand angle
\selectlanguage{greek}ψ\selectlanguage{english}\textsubscript{c} is measured by having the 0\textsuperscript{o}angle
aligned at 3:00, the vector path of \selectlanguage{greek}ψ\selectlanguage{english}\textsubscript{Sp} . An example
point of reference, the spiral angle\selectlanguage{greek}\emph{ψ} \selectlanguage{english}\textbf{\textsubscript{Sp}}
= 90\textsuperscript{o} is noted at the arrow.

C. As spiral source S\textsubscript{Sp} spins clockwise ?/4 radians, a
point on the spiral wave can still be computed by a spiral summation
angle \selectlanguage{greek}ψ \selectlanguage{english}\textsubscript{SA}. The \selectlanguage{greek}ψ \selectlanguage{english}\textsubscript{SA} is generated by
first identifying the clock hand angle \selectlanguage{greek}ψ \selectlanguage{english}\textsubscript{c} compared to
the zero angle and then following the wave back CCW by the spiral
angle\selectlanguage{greek}\emph{ψ} \selectlanguage{english}\textbf{\textsubscript{Sp}}. The clock angle has positive
values above the line v\textsubscript{s} and negative values below line
v\textsubscript{s}. The new position of spiral angle \selectlanguage{greek}\emph{ψ}
\selectlanguage{english}\textbf{\textsubscript{Sp}} = 90\textsuperscript{o} is noted at the
arrow. D. The rotational distance unit (RDU, 2?/D) is applied to spiral
wave. The distance D travelled is function a constant velocity
\textbf{?\textsubscript{s}} of the spiral as it makes one completes
revolution at a constant angular velocity, ?\textsubscript{s}. The
3-point sequence rule developed for rotational waves is used here to
label the strong-side and weak-sides of a spiral wave (blue and red
rectangles, respectively). Similar to Fig 3(a), separated by D
distances, WFs would move faster through strong-side shear zones and
slower through weak-side move slower (stall zones).

Figure 6. Spiral math components determine WF strikes.

A and B. A moment in time that a clockwise spinning S\textsubscript{Sp}
(green striped curve) approaches the inflection point (\textbf{p})
between 2 equidistant electrodes e\textsubscript{1} and
e\textsubscript{2} (r\textsubscript{1} = r\textsubscript{2}, center of
blue and red circles) on either side of the vector path
\textbf{v\textsubscript{s}} of S\textsubscript{Sp}. \textbf{Step 1}:
determine the clock hand angle (thin gray triangle aligned along ray
from 0 --\textgreater{} 2?, or spiral's 1\textsuperscript{st} ?) as
measured from the 0\textsuperscript{o} angle represented by the forward
vector path (\textbf{v\textsubscript{s}}) of S\textsubscript{Sp} (same
as in Fig 3A). \textbf{Step 2}: determine the distance from S to
e\textsubscript{1}, the hypotenuse h, of the right triangle
S-p-e\textsubscript{1}. The distance \emph{h} is converted to the angle
\selectlanguage{greek}\emph{ψ\textsubscript{h}}\selectlanguage{english} and is plotted on the spiral curve in a
counterclockwise direction from \selectlanguage{greek}ψ\selectlanguage{english}\textsubscript{c}. The resultant point
on the spiral, at the
\selectlanguage{greek}\emph{ψ\textsubscript{SA}}\selectlanguage{english}\textbf{\textsubscript{,}} in 2-D space
identifies that the spiral WF crest did not intersect with
e\textsubscript{1} at that position and time. C and D. Two examples show
when the spiral WF crest strikes e\textsubscript{1}. When
S\textsubscript{Sp} is at position \textbf{S\textsubscript{Sp1}}
(approaching the inflection point\textbf{p)} and at position
\textbf{S\textsubscript{Sp2}} (receding from point \textbf{p)}. At
\textbf{S\textsubscript{Sp1}}, the clock hand angle \selectlanguage{greek}ψ \selectlanguage{english}\textsubscript{c1}
is added to the hypotenuse angle \selectlanguage{greek}\emph{ψ \textsubscript{h1}}\selectlanguage{english} resulting
in an acute angle\selectlanguage{greek}\emph{ψ \textsubscript{SA}}\selectlanguage{english} that is equal the acute LOS
angle ?\textsubscript{\textbf{1}}. At \textbf{S\textsubscript{Sp2}}, the
clock hand angle \selectlanguage{greek}ψ \selectlanguage{english}\textsubscript{c2} is added to the hypotenuse angle
\selectlanguage{greek}\emph{ψ \textsubscript{h2}}\selectlanguage{english} resulting in an obtuse angle \selectlanguage{greek}\emph{ψ
\textsubscript{SA}}\selectlanguage{english} that is equal the obtuse LOS angle
?\textsubscript{\textbf{2}}. Close inspection
of\textbf{S\textsubscript{Sp1}} shows that the WF strike at
e\textsubscript{1} has occurred before a WF strike would occur at
e\textsubscript{2}, whereas the sequence is reversed when source is
passed \textbf{p} at position \textbf{S\textsubscript{Sp2}}. Here, the
WF strike at e\textsubscript{2} had already occurred just before the
strike at e\textsubscript{1}.

Figure 7. Equidistant electrodes on either side of source's vector path
reveals the diametric property of spiral WF wave frequency effects.

A. S\textsubscript{Sp} has a wavelength of ? radians, a spiral size ? =
?cm and spins with an angular velocity of 10 rotations/s (20?/s).
S\textsubscript{Sp} moves on a vector path at\emph{v\textsubscript{s}} =
15 cm/s that is a perpendicular bisector between electrodes
e\textsubscript{1} and e\textsubscript{2}, each 0.5 cm from the point of
closest approach (transverse common tangent to blue and red circles). A
reference position of S\textsubscript{Sp} at 3 cm (thick green spiral)
from the inflection point between electrodes is the spiral source's
clock angle \selectlanguage{russian}ѱ\textsubscript{c} is at 9:00 or \selectlanguage{english}? radians. The spiral wave
in its x position and spin orientation is shown at each WF strike as
calculated to occur at positions 1-13. WF strikes at e\textsubscript{1}
(bold dash-dot spirals) are compared to WF strikes at e\textsubscript{2}
(dashed spirals). Black and yellow dashed spirals show the sequence
reversal of WF strikes secondary to the diametric property by opposite
and progressive changes of T comparing strong and weak-sides (positions
7-9). Strong-side has one more WF strike than weak-side (unpaired WF
strike at position 9), identical to that seen during rotor migration
breaching perimeter of electrodes (\emph{2}) that we described as a \selectlanguage{ngerman}½
cycle drop-off. See Supplementary Materials Figure 7 Data File. B. The
LOS angle \selectlanguage{english}? is plotted as a function of distance x to inflection point
by the arccotangent curves for electrodes e\textsubscript{1} and
e\textsubscript{2} (blue and red curve, respectively). Superimposed on
graph is the spiral clock angle \selectlanguage{greek}ψ\selectlanguage{english}\textsubscript{c}(green lines), the
distance to electrodes in radians as the hypotenuse angle
\selectlanguage{russian}ѱ\textsubscript{h} (small portion of the pink parabolic curve is
illustrated), the spiral summation angle ѱ\textsubscript{SA} (black
lines). The points of intersection of ѱ\textsubscript{SA} with the
arccotangent curves (white circles) indicate when ѱ\textsubscript{SA} =
\selectlanguage{english}?, identifying a WF strike. Although both e\textsubscript{1}and
e\textsubscript{2} record faster frequencies during an approaching
S\textsubscript{Sp} compared to receding, the WF strikes at
e\textsubscript{1} during the approach accelerate (S\textsubscript{Sp}
positions at 1,3,5,7) while simultaneously the WF strikes at
e\textsubscript{2}(S\textsubscript{Sp} positions at 2,4,6,8) decelerate.
Example for frequency determination at WF strikes at
e\textsubscript{1}from position 5 to position 7 can be calculated by
either \selectlanguage{english}?ps \textsubscript{c} or \selectlanguage{english}?x . The \selectlanguage{english}?ps \textsubscript{c} method
would be via using ps\textsubscript{c} values at hollow stars (-2.941,
-0.779, respectively) and plugging into frequency equation. 35, or by
the \selectlanguage{english}?x method by equation 36. C. Side-Dependent Frequency Changes of
Spiral Wave. Plots of the instantaneous frequencies at each WF strike
are shown for both strong and weak electrodes (e\textsubscript{1} and
e\textsubscript{2}, respectively). The approaching spiral wave shows a
progressive increase in frequency of WF strikes at the strong-side
electrode e\textsubscript{1}. Simultaneously, there is a progressive
decrease in frequency of WF strikes at the weak-side electrode
e\textsubscript{2}. This diametric progression leads to an unpaired WF
at e\textsubscript{1}(position 9, near the inflection point, p)
resulting in the sequence reversal with WF strikes at e\textsubscript{2}
preceding e\textsubscript{1}. From relatively far distances
(x\textgreater{}\textgreater{}r) during the approach, strong and
weak-sides frequencies of WF strikes are nearly the same, at 14.78 WF
strikes /sec. When S\textsubscript{Sp} recedes from p, frequencies of
both strong and weak-sides then approach a final receding frequency of
spiral 5.22 WF strikes/sec as per Equation 40 or 10 \selectlanguage{ngerman}± 4.78 WF
strikes/sec.

Figure 8. Comparison of frequencies observed from a moving spinning
S\textsubscript{Sp} source and a moving

\emph{S\textsubscript{C}}\textbf{.}

A. Comparison of frequency changes with propagation of a spiral wave
between different distances of equidistant electrodes. The speed of
spiral source v\textsubscript{s} =1 cm/sec. Angular frequency is 5
rotations/sec, RDU = 0.2 cm\textsuperscript{-1}, with \selectlanguage{english}? = \selectlanguage{english}?cm.
Comparison of calculations were made with stationary electrodes placed
at 0.2 cm, 03 cm, 0.5 cm, and 2 cm from the inflection point p. Closer
electrodes to the S\textsubscript{Sp} center create a greater diametric
change in \selectlanguage{english}? between the strong and weak-side electrodes (gradation of
darker blue and darker red lines, respectively). Similar to CDE,
S\textsubscript{Sp} at larger distances from electrodes, the approaching
\selectlanguage{english}? is faster than \selectlanguage{english}?\emph{\textsubscript{\textbf{s}}}, then gradually
decreases, becoming less than \selectlanguage{english}?\emph{\textsubscript{\textbf{s}}} after
receding from p (light blue and light yellow lines, respectively). Using
Equation 40, the approaching \selectlanguage{english}? = 5.32 Hz and the receding \selectlanguage{english}?= 4.68 Hz.
See Supplementary Materials Figure 8A Data File. B. Classical Doppler
effect on \selectlanguage{english}? changes sensed by equidistant electrodes (same as in (a)
above) of a moving source S\textsubscript{c} (same
\selectlanguage{english}?\textsubscript{\emph{s}} = 1 cm/s) that emits waves propagating
centrifugally away at \selectlanguage{english}? cm/s (same as spinning spiral source above if
not moving in x-direction) utilizing Equation 21. For this to occur,
with same \selectlanguage{english}? at \selectlanguage{english}? cm and same frequency at 5 Hz, the wave propagation
velocity is 5\selectlanguage{english}? cm/s. If the electrode was directly on
the\selectlanguage{english}\emph{?\textsubscript{s}} vector path, the approaching \selectlanguage{english}? = 5.34 Hz,
and the receding \selectlanguage{english}? = 4.70 Hz). Note that from far away, greater
electrode distance, the WF strike \selectlanguage{english}? is almost identical to that of a
spiral wave at a distance (several \selectlanguage{english}? away). See Supplementary Materials
Figure 8B Data File.

Figure 9. Spiral size on diametric property of SFE.

Plots of different spiral sizes vs frequency are shown when electrodes
e\textsubscript{1} and e\textsubscript{2} (each 2 mm from \textbf{p})
are equidistant to S\textsubscript{Sp} along its vector path of motion.
S\textsubscript{Sp} moves along the vector with constant
\selectlanguage{english}?\textsubscript{s} = 1 cm/sec, while spinning at constant
\selectlanguage{english}?\emph{\textsubscript{\textbf{s}}} = 5 Hz, with RDU = 2\selectlanguage{english}?/0.2cm. Curves
are smoothed lines linking the subsequent WF strikes of
S\textsubscript{Sp} of same size \selectlanguage{english}?. Pairs of curves show frequency
differences between strong-side e\textsubscript{1}(light or dark green
dots) and weak-side e\textsubscript{2} (light or dark pink dots). At far
distances (x\textgreater{}\textgreater{}r), either approaching or
receding, the WF strikes at e\textsubscript{1} and e\textsubscript{2}
from the largest spiral size (200 \selectlanguage{english}? cm) have values near
\selectlanguage{english}?\emph{\textsubscript{\textbf{s}}}. As \selectlanguage{english}? gets smaller, the approaching \selectlanguage{english}?
is faster, and receding \selectlanguage{english}? is slower. The greatest side-dependent
difference occurs near the inflection point p. An unpaired WF strike on
the strong-side (3 dark blue arrows) with that on the weak-side (2 red
arrows) results in the final step before sequence reversal is complete
(dashed ovals). When \selectlanguage{english}? = D at 0.2 cm, the approaching WF \selectlanguage{english}? is 10Hz
(twice the value of \selectlanguage{english}?\emph{\textsubscript{\textbf{s}}}), but when
receding from p, no WF strides occur (electrodes would be surfing
between spiral WFs). See Supplementary Materials Figure 9 Data. Figure
10. The side-relative differences of frequency observed from
S\textsubscript{R} or S\textsubscript{Sp} are a result of the diametric
property of RFE and SFE seen at all levels of nature. A. Person R moves
along a path between persons S and W while spinning 1/2 turn. Person W
observed 0 turns since person R faced person W along its entire path.
Person S observed 1 turn. B. Person R moves along a path between persons
S and W with a faster spin, completing 1 and 1/2 turns over the same
distance. Person S observed 2 turns. Person W observed 1 turn. C. A
rotating binary star exhibits peak Doppler effect frequencies when the
tangential vector path of the small orange star is directed toward
observers and the location of the peak frequency can be considered as
the clock angle \selectlanguage{russian}ѱ\textsubscript{c} for an S\textsubscript{R}. When
binary star moves on path between S and W, the timing of peaks will
exhibit diverging trends of the diametric property of
S\textsubscript{R}. Yet, for observers D and D' that travel with the
binary star, stationary in the same inertial plane, then they perceive
no diverging frequency shifts and no difference in count of peak
frequencies. The side-dependent timing of peak frequencies exhibit the
diametric property of RFE that is relative to motion. On a grander, and
slower scale, if one were to replace the binary with a star within a
spinning and moving spiral galaxy, then the peak frequency count and
timing changes between peaks must similarly follow the diametric
property of SFE.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/JCEFig1A/JCEFig1A}

\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/JCEFig1BC/JCEFig1BC}

\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/JCEFig1D/JCEFig1D}

\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/JCEFig2AB/JCEFig2AB}

\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/JCEFig2C/JCEFig2C}

\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/JCEFig2D/JCEFig2D}

\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/JCEFig3ABCDeps/JCEFig3ABCDeps}

\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/JCEFig4/JCEFig4}

\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/JCEFig5/JCEFig5}

\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/JCEFig6/JCEFig6}

\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/JCEFig7AB/JCEFig7AB}

\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/JCEFig7C/JCEFig7C}

\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/JCEFig8A/JCEFig8A}

\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/JCEFig8B/JCEFig8B}

\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/JCEFig9/JCEFig9}

\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/JCEFig10ABC/JCEFig10ABC}

\end{center}
\end{figure}

\selectlanguage{english}
\FloatBarrier
\end{document}

