Page 60 - Maxwell House

40 Chapter 1

is called the in-phase () and quadrature () component, respectively. We will show

bellow that Fourier transform of Maxwell’s equations, as we will prove late, makes them time
independent and much easier to solve.
In case of periodic signals on the domain [−, ] the Fourier transform (1.78) can be written as
a linear combination of phasors or exponents

() = ∑ ∞ 0 (1.81)

=−∞
Here = 2 is called the fundamental angular frequency, T is the fundamental period of
⁄
0
signal, and
1 − 0
= ∫ () (1.82)

2 −
is the component of the discrete complex spectrum. The Fourier transform (1.78) can be viewed
as the limit of Fourier series with fundamental period extended to (−∞, ∞) and can be applied
to a wider variety of functions. That is why we prefer to use it in our following analysis.
French mathematician Jean Baptiste Joseph Fourier in 1807 presented his great discovery, later
called by his name, at the meeting of the French Academy but it was rejected with a
12
recommendation to continue the work. Harmonic series representations similar to (1.81) had
a long history back to Old Babylonian times (2000-1600 B.C.) and was used to compute tables
of planetary positions. The connection between mathematics and music goes back at least as
far as the sixth century B.C. when it was discovered the relationship between numbers and
sound. All musical tuning systems are based on this discovery. Fourier analysis lets identify
naturally occurring harmonics (which are, simply put, the basis of all musical composition), to
model sound, and break up sound into the pieces that define it. It is quite sad to think that the
genius music of Mozart is the set of Fourier harmonics, but that is the fact. A little bit more
complicated Fourier series can “describe” any of Rembrandt’s or Michelangelo’s artwork,
Freedom Tower in New York or the marvelous Gothic style Cathedral of Saint Mary of the
Flower in Florence, Italy.
Before applying (1.78) or (1.81) to Maxwell’s House note that mainly invisible EM fields can
be detected through the measurements associated with the energy transfer from EM fields to
test instrument. However, the vast majority of such instruments are not designed to be fast
enough and be capable of measuring an instantaneous waveform of signals. For example, the
frequency of quite moderate 1 GHz signal must involve more than 1,000,000,000 measurements
per second! Only several companies in the world produce the instruments for such kind of
measurement, and they are very complicated, costly and not broadly available. The common
go-around is the measurement of the time-averaged or mean-square volumes. For example, the
averaged energy of the monochromatic or single frequency harmonic signal is defined as [16]

2
2
〈() 〉 ≝ 1 ∫ () (1.83)
2 −
Applying (1.83) to the in-phase and quadrature component from (1.80) we have

12 The first English translation of his works was published in 1878 only! More than 20 years later.