Wave Motion

Introduction: The Plucked String

When a class is asked to describe the motion of a string released from rest in a "Pluck" (isosceles triangle) configuration, most students form their fingers into a tent, and move them up and down. The instructor usually has to work very hard to convince everyone that there is no net force on a straight section of string, so only the segment at the kink can have an acceleration. The folowing applet offers two initial configurations: one where the inital displacement extends across the entire string (Pluck), and one where it is confined to the central quarter (Pulse). (The usual assumptions are made: the string is perfectly flexible; the tension is high enough for gravity to be ignored; the transverse displacements, although plotted on an expanded scale, are much smaller than the length of the string.) The motion of the string is plotted frame by frame on the left, and the displacement of a point one-quarter of the way from one end is plotted versus time on the right. The fact that there is acceleration only at a kink shows up clearly for both initial displacements. The Pulse option shows the initial displacement breaking up into travelling pulses that invert on reflection from the fixed ends of the string.

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The algorithm that generates the motion makes use of the fact that the classical wave equation (Newton's Law) has travelling wave solutions that can be superimposed to describe the motion of a string with fixed endpoints. This generates amusing pictures, but says nothing about the physics of the problem.

In what follows, two alternate approaches are taken to generating the motion of the string. In the first, a numerical solution of the wave equation is generated directly. In second, the "normal modes" and their frequencies of oscillation are found first, then are superimposed to represent the initial displacement of the string. For a uniform string clamped at both ends, the normal modes make up what is known as the Fourier Sine Series (FSS), and the superposition can be made as accurate as one wishes by including sufficient terms. For a string composed of segments of differing linear density, the normal modes do not form a set of orthogonal functions, so their superposition can only approximate an initial displacement.

The quantum version of a string with fixed endpoints is called the infinite square well. The quantum wave equation is the Schrodinger Equation, and the infinite square well has the same set of normal modes (stationary states) as the classical clamped string. Initial "Pluck" and "Pulse" wavefunctions generate oscillations quite different from those on a classical string. In the direct solution this is due to the difference in the wave equations, and in the series solution it is due to the difference in the dispersion relations (the dependence of frequency on mode number) and the complex exponential time dependence in the quantum case. Unlike classical normal modes, the quantum stationary states always form orthogonal sets.

The examples in the following sections enable one to compare and contrast the behavior of waves on classical and quantum strings.

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