The 64-Segment String: Newton's Law and FFSS

The techniques developed in the previous sections can be applied directly to a 64-segment string. The first program uses the Feynman algorithm and compares the final displacement with the FFSS result. The motion of the string is plotted frame by frame in blue on the left, and the motion of the mass one-quarter of the way along the string (j = 16) is plotted on the right. The FFSS result is superimposed in red on the final displacement.

There are two points to note: (1) the difference between the blue and red final displacements is a measure of the Feynman algorithm cumulative error; (2) the difference between the initial and final displacements is a measure of the error when a 64-segment string is used to model a continuous string. As expected, the Feynman algorithm provides adequate accuracy, and the 64-segment model of a continuous string works better for a pluck displacement than a pulse displacement. There is little point in trying to improve the accuracy of the Feynman algorithm by reducing the time step, or the accuracy of the model by increasing the number of segments (both of which increase computation time) when the FFSS solution is available. The next program plots the motion using the FFSS, and offers an additional option: the choice between dispersive (D) and harmonic (H) motion.

As we stated previously, in Fourier analysis the physics of a problem is contained in the dispersion relation. The dispersion relation we derived for our n-segment string generates dispersive motion. The pulse displacement has greater contributions from the higher modes than has the pluck displacement, and its shape changes more with time. If we replace the sine dispersion relation by a linear relation with the same initial slope, we generate harmonic motion, and get the motion of a continuous string that we obtained using traveling waves at the beginning.