If the (*n*-1) point masses on an *n*-segment string are given
initial displacements of the form

where the integer *k* is known as the mode number, periodic motion
results when the string is released from rest. The next example uses the
Feynman algorithm to show the oscillation and find the frequency for each
of the normal modes for the case when *n*=8.

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 FFSS.The FFSS is very useful for dealing with systems 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:

The expression relating frequency to mode number is known as the dispersion relation, and is the curve plotted in the previous example. A dispersion relation contains the "physics" of a problem dealt with by Fourier analysis. We will say more about this in the what follows.At this point we would like to show how the FFSS can be used to represent our pluck and pulse displacements. The next applet does this and shows the connection between the FFSS and the more familiar Fourier Sine Series (FSS).

The FFSS fits a finite number of points
exactly, whereas the FSS fits a continuous function over a specified range, but
requires an infinite number of terms to do so exactly. As *n* increases,
the FFSS approaches the FSS. The applet 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.

If the particles are at rest in the initial configuration (a "standing" pulse), the time dependence is generated by multiplying each term in the series by a cosine time dependence with the appropriate normal-mode frequency. The main advantage of this approach over the Feynman algorithm is that the displacement at any future time can be displayed directly without stepping time in small increments (this advantage is lost in simulations that require small time steps). The next applet treats the motion of the n-segment string with pluck and pulse initial displacements using the FFSS. It offers the option of replacing the sinusoidal dispersion relation with a linear "harmonic" dispersion relation with the initial slope of the sine curve.

With the true dispersion relation the FFSS results agree with those from
the Feynman algorithm. With the harmonic dispersion relation the FFSS results
preserve the linear segments observed in the applet in the introductory
section. It is, therefore, not necessary to use a large value for *n*
to model a continuous string: it is only necessary to modify the dispersion
relation

The FFSS can also be used to describe motion when the initial configuration is not at rest ("travelling" pulses). A second FFSS with a sine time-dependence is added with coefficients chosen to fit the initial velocities. The next applet uses this approach to treat the two pulses whose motion was studied previously using the Feynman algorithm.

In terms of the dispersion relation, the velocity of the envelope of a
series of wave crests is the slope of the dispersion curve at the mode
number of the wave crests, and the higher the value of *n*, closer
this is to the initial (harmonic) slope. The final applet plots the square
of FFSS coefficient versus mode number for the pulses
used in the two previous applets (Pluck, Pulse, Pulse A, and Pulse B), and
for the quantum analog of Pulse B. (Quantum wave pulses are
dealt with in the next section.) The calculation uses *n*=384, and the
mode scale is plotted only up to 64. The vertical scale is adjusted to make the
area shown in red the same for all five pulses. (In the case of Pluck and
Pulse, the first coefficient goes off scale by an amount that can be
estimated using this fact.) The two standing pulses, Pluck and Pulse, have
only odd non-zero coefficients. The travelling classical pulses, Pulse A and
Pulse B, have the initial velocity shown in green along with the displacement
in blue. The quantum pulse is complex. Its real part is identical to Pulse B,
and its imaginary part is what is plotted in green. Both Pulse B and the
quantum pulse have FFSS coefficient distributions centered about mode 32, the
mode that modulates the pulse envelope.

With *n*=48, the peak in the distribution for Pulse B is 2/3 of the
way along the dispersion curve, so it is not surprising that the pulse is
quickly obliterated. When *n*=384, the high-order FFSS coefficients
become vanishingly small, and the series need not extend over the full
spectrum. The applet prints out upper limit needed for the sum of the
squares of the coefficients to reach 99.99% of the total.