Normal Modes and Fourier Series

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.

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As the mode number k increases, the frequency values increase, and the sine curve drawn through them is the analytic result obtained below. The normal modes, although interesting in their own right, owe their importance to the fact that any displacement of the (n-1) point masses can be written as a superposition of the (n-1) normal modes. For an n-segment string clamped at both ends the superposition is known as a 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 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).

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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.

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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.

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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.

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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.

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