Coupled Oscillators and Normal Modes

The Plucked String: Traveling Wave Solution

When a first-year class is asked to describe the motion of a string released from rest in a "plucked" (isosceles triangle) configuration, there is usually a lot of discussion, but no general agreement even when the professor puts in his two cents worth. (The string is assumed to be ideal: perfectly flexible, small lateral displacements, etc.) If one "cheats" and describes the displacement as the superposition of two waves traveling in opposite directions, one can generate a plot of the motion fairly simply, and this is the approach taken in the first example.

The motion of the string is plotted frame by frame on the left, and the displacement of a point one-quarter of the way along the string is plotted versus time on the right. The program offers a choice between the "pluck" and a "pulse" in which only the central quarter of the string has an initial displacement. The program generates amusing pictures, but leaves the physics of the problem to be dealt with in the sections that follow.