Consider a light string divided into n equal segments of length d with a point mass m attached at the junction between each segment. If the tension in the string is T, and we use subscript j to identify the point masses in order from left to right, we can write Newton's law:
This is the numerical form of the classical wave equation for a string. It can be solved directly using the Feynman algorithm once the initial conditions are specified. The numerical equation takes on the analytic form when n tends to infinity:
The constant in the analytic equation, the tension divided by the mass per unit length, is usually written as v2, where v is the wave velocity. In this form, the classical wave equation applies to a variety of wave phenomena (including electromagnetic waves). Functions of the form f(x-vt) and f(x+vt) clearly satisfy the equation, and in the previous section the superposition of two such functions was used to generate the motion of an ideal continuous string.
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.
In the first applet, the string has 8 segments. There are three choices of initial displacement: the Pluck and Pulse displacements described previously, and the set of 7 normal modes. For the Pluck and Pulse displacements, the motion of the string is plotted frame by frame on the left, and the motion of a particle one-quarter of the way along the string (j=2) is plotted on the right. The normal mode oscillations are displayed in sequence frame by frame on the left, and frequency is plotted versus mode number on the right.
For n=8, the Pluck displacement generates a motion that is somewhat like the motion of the continuous string in the preceding section, but the Pulse displacement does not, and neither initial displacement sets up a periodic motion.
For the normal modes, as the mode number k increases, the frequency values increase, and the sine curve drawn through them is the analytic result obtained below. For an n-segment string clamped at both ends any displacement of the (n-1) point masses can be written as a superposition of the (n-1) normal modes. This superposition is known as the 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.
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 next applet shows the individual terms and their sum when we use a FFSS and a Fourier Sine Series (FSS) with the same number of terms to represent the Pluck and Pulse displacements of our 8-segment string.
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.
If the particles are at rest in the initial configuration (a "standing" wave), 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 for values of n from 8 to 384. You have the option of choosing an FFSS solution in place of the direct numerical solution (both give the same result).
As n increases, the motion begins to look more like the motion of a continuous string. The difference between the initial and final displacements is known as dispersion, and decreases as n increases. More will be said about dispersion in the next example.
The previous example had initial configurations with the point masses at rest, but the method can be extended to deal with a moving pulse. Feynman algorithm needs only to have the initial transverse velocities specified. The superposition requires a second FFSS with a sine rather than cosine time dependence and coefficients fitted to the initial transverse velocities. For a continuous string, a wave travelling in the +x direction, f(x-vt), has a transverse velocity at each point that is negative of the slope multiplied by v. In the next example there are two choices for the initial displacement: Pulse A which is a rounded version of the pulse used previously (1/4 of the length of the string in width), and Pulse B which has a number of oscillations within Pulse A as envelope. The string has 48, 96, 192 or 384 segments
Pulse A exhibits less dispersion than Pulse B, and both exhibit less dispersion as n increases. For Pulse B with n=48 dispersion obliterates the pulse, but in all other cases dispersion shows up in three ways. (1) It slows down the pulse - the pulse does not make it back to its initial position. (2) The pulse broadens and picks up trailing oscillations. (3) In pulse B, the crests within the envelope move faster than the envelope itself - leading crests decrease in amplitude and trailing crests increase in amplitude.
In terms of the dispersion relation, the velocity of the envelope of a series of wave crests, the group velocity, is the slope of the dispersion curve, dw / dk, and the velocity of the individual crests, the phase velocity, is w / k. The higher the value of n, the closer the group velocity is to the phase velocity.
The next applet plots the square of FFSS coefficient versus mode number for Pulse A and Pulse B. Dispersion curves for the selected n-value are superimposed on the plot. The mode scale is plotted up to a maximum of 64.
The FFSS coefficient distribution for Pulse B is centered about mode 32, the mode that modulates the pulse envelope. With n=48, the peak in the distribution is 2/3 of the way along the dispersion curve, so it is not surprising that the initial displacement 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.
If the string is composed of segments of differing mass density, waves will be partially reflected at the boundaries between them. The only modification required in the Feynman algorithm is the change in mass at each boundary. In the example, Pulse B is incident on a variable thickness segment where the mass density increases by a factor of 1.44 followed by a region where the mass density increases again by a factor of 1.44. This is a mechanical analog of the type of optical system studied in Matrix Wave Optics. The difference here is that we are dealing with wave pulses rather than plane waves. The "wavelength" (to the extent that the term is applicable) of Pulse B is 1/4 of the pulse width. The successive changes in mass density simulate an optical anti-reflection coating. The initial pulse width is 96 pixels, and this reduces to 80 pixels in the "film", so the film-thickness that gives 1/4 wave is 5 pixels.
The reflected pulse is inverted, and is strongly attenuated when the film thickness is 5 pixels. The transmitted pulse is narrower than the incident pulse and travels more slowly. Pulse B appears again in the quantum analog of this applet where it forms the real initial part of a complex pulse. The two applets illustrate the difference between classical and quantum pulse propagation, and act as a link to Matrix Wave Optics.