The 8-Segment String: Newton's Law
Consider first a light string divided into n (8 in this case) equal
segments of length L 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:
Once we have Newton's law, we can use the Feynman algorithm to plot the motion
for a variety of initial conditions. The program sets T/ML = 1, and offers
a selection of initial displacements: the
pluck and pulse configurations described previously, and seven
displacements of the form
known as normal modes. The motion is plotted frame by frame on the left,
and the motion of the mass one-quarter of the way along the string (j = 2) is
plotted on the right.
The pluck displacement generates a motion that is somewhat like that for
a continuous string, but the pulse displacement does not, and neither sets up
a periodic motion. The normal mode displacements, on the other hand, generate
periodic motion. The Feynman algorithm can be used to find the frequencies of
the periodic oscillations, and this is done next.
As the mode number k increases, the frequency increases along a sine curve.
The curve in the figure plots the analytic result obtained in the next
section.