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Electromagnetic Geometric
Relationships
When
wavelengths and frequencies are used to define the
elements of a right triangle, over 2,000 years of Western mathematics
and science are merged into a pair of geometric relationships that
permit the basic units that describe the velocity of electromagnetic
waves to be defined mathematically.
There are no
mathematical reasons why wavelengths and frequencies cannot be used as
dimensions that define the elements of right triangles, but they cannot
be mixed among the elements in the same triangle. Within the right triangle
relationships one of the units that defines the wavelength frequency
constant of proportionality becomes a function of the angle.
When
Maxwell developed
the equations that define the characteristics of EM
waves, he could not
have envisioned how they
could have been expressed using geometric relationships. No one in the
Western world would have known until 1951 that the
precession emission of neutral hydrogen would be one of the critical
elements of the wavelength
frequency geometric relationships. Additionally, although the numerical
value of 2π can be
used as a frequency, it is not taught that this value is a mathematical
coincident point between frequency and angular frequency. This duality of 2π can be exploited in the mathematical relationships of paired right triangles.
The
two frequencies, 1420.405MHz and 628.315MHz are noted on the
elements of the triangle pair on
the right in the web page titled
Wavelength Frequency Triangle Pair - SI Units .
The hydrogen emission is one of the most widely known
frequencies in physics, and the numeric value of the second is
known quite precisely. The
text section titled "Angular Frequency" provides an explanation as to
why the second frequency, which is multiple of 2π, is given the
position of the vertical element of the right triangle labelled frequency.
Wavelength
Frequency Algebraic Relationships
It has been
known for some 400 years that a
wavelength
and frequency
have an inverse proportional relationship, and it was found just over
200 years ago that light waves and their associated frequencies follow
the same rules. The inverse proportional relationship between
wavelength and frequency is the velocity of the wave, the velocity
being determined by the characteristics of the medium in which a wave
can propagate. For EM waves, which includes light, the velocity is
typically
referred as the "speed of light" (SOL). The SOL is determined by the
permittivity of the material in which EM waves are permitted to
propagate, but the value is usually equated to the numeric value
assigned to this velocity by the System International (SI). It is
important that the scientific community has a reference value for the
SOL to provide a basis for comparative research. The
accuracy of the SI value for the SOL is based upon the numeric value
assigned for the duration of the second, which is currently limited
to about 11 significant figures.
The
equation set that describes the inverse proportional relationship
has three forms,
f = c / λ
λ = c / f
c = f *
λ (1)
where
f is frequency, λ
is wavelength, and c is the SOL. Frequency is
dimensioned in "cycles" per "unit time", wavelength in "length", and
the
SOL in "length" per "unit time". The relationships shown in the
equation (1) set are purely algebraic, no
angular elements are involved. The constant of
proportionality is the value of "c".
Angular
Frequency
Angular
frequency is expressed in radians per unit
time. Typically it
is presented in the equation form ω = 2πf, where f
is the number of repetitions (cycles) of the event. Usually the value
for f
is equated to the same meaning as regular
frequency, but this poses a conflict when the one and only numeric
value wherein the equation (1) set will return the same values for
angular frequency as that for regular frequency. When frequency has the
value of 6.28315... cycles per second, it
will return the correct value for the wavelength when angular frequency
has the value of 2π radians per second, the number of angular
cycles being one. The wavelength frequency relationships will be valid
for any tens multiple or division of the respective 2π
values. Using
SI units, the wavelength associated with any
tens
multiple or division of 2π will have a numeric value of
47713...,
differing only in decimal point placement.
For
angular frequency, the base frequency of the mathematical coincident
point represents a unit wavelength. One wavelength, a "unit wavelength" (λ) is representative of a complete wave cycle, which can be represented as a circle when its linear length is formed into a circle with a circumference with the length of one.
The
duality of 2π and λ establishes the angular synchronization between the triangle pair.
When constructing the triangle pairs, it is necessary to
designate one set of corresponding positions as reference values to provide angular synchronicity.
Regardless of the angle presented by the triangle pairs, the constant
pair assures that the changes in the other elements will always be
relative to the same angular reference. The vertical legs of the triangle
pairs were chosen to be the reference.
The
methodology used in developing the wavelength frequency geometric
relationships is straight forward once it was determined that the basic
value for frequency had to be the numeric value that coincided with the
convergence point for angular frequency and regular frequency.
The simplest starting point is drawing two 45 degree right triangles,
with the first representing wavelength having two legs equal to one
and the hypotenuse being the square root of 2, and the second
representing frequency having the two legs equal to 2π and the
hypotenuse
being 2π times
the square root of 2. The basis for the intrinsic
wavelength and frequency forms and the development of the
triangles are presented in the article titled, A Mathematical Process to Derive the Velocity of Light Using Mutually Defined Units.
As
noted in the above article, once the intrinsic values are scaled to
real-world values they have significance well beyond anything the
scientific community currently accepts.
It was found that expressing the
process using a trigonometric function was mathematically the most
efficient. The process that uses the trigonometric function exclusively
is presented in the article, The
Application of Geometric Relationships To Wavelengths and Frequencies.
“The
most universal standard of length which we could assume would be the
wavelength of a particular kind of light... Such a standard
would be independent of any changes in the dimensions of the earth, and
should be adopted by those who expect their writings to be more
permanent than that body.” James Clerk
Maxwell, 1873
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