The 8-Segment String: Finite Fourier Sine Series (FFSS)
The Finite Fourier Transform (FFT) is defined by
The proof that the relations are valid involves only simple geometry in
the complex plane. If we are dealing with an odd sequence of real numbers, i.e.
we need be concerned only with the first half of the sequence, and the FFT
takes the form of the Finite Fourier Sine Series (FFSS)
The FFSS is very useful for dealing with systems such as our string with
"zero boundary conditions":
To show that each term in the
FFSS is a normal mode, we substitute into Newton's law, and use "twice
sine half-sum cosine half-difference" to add sines:
This confirms what we found using the Feynman algorithm: if the system is
released from rest in one of the sinusoidal normal mode displacements, all
of the masses execute periodic motion at a common frequency. The analytic
expression derived above is plotted along with the Feynman algorithm results
in the previous section. Analytic expressions of this type are known as
dispersion relations, and contain the "physics" of problems dealt with by
Fourier analysis. We will say more about this in the following sections.
At this point we would like to show the FFSS graphically for our pluck and
pulse displacements, and relate it to the more familiar Fourier Sine Series
(FSS). The FFSS is a finite series which "fits" a finite number of points
exactly; the FSS is an infinite series which "fits" a continuous function over
a specified range. As n increases, the FFSS approaches the FSS. The program
shows the individual terms and their sum
when we use a FFSS and a FSS with the same number of terms
to represent the pluck and pulse displacements of our 8-segment string.