Preface

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n 1636 the French mathematician Marin Mersenne published his observation that a vibrating string produces multiple pitches simultaneously. The most audible pitch corre- sponds to the lowest frequency of vibration, called the fundamental tone of the string. Mersenne also detected higher pitches, at integer multiples of the fundamental fre- quency. (The relationship between frequency and pitch is logarithmic; doubling the frequency raises the pitch by one octave.)

The higher multiples of the fundamental frequency are called overtones of the string. Figure 5.1 shows the frequency decomposition for a sound sample of a bowed violin string, with a fundamental frequency of 440 Hz. The overtones appear as peaks in the intensity plot at multiples of 440. >/p>

At the time of Mersenne’s observations, there was no theoretical model for string vibration that would explain the overtones. The wave equation that d’Alembert subsequently developed (a century later) gave the first theoretical justification. However, this connection is not apparent in the explicit solution formula developed below. To understand how the overtones are predicted by the wave equation, we need to organize the solutions in terms of frequency

 

Hyperbolic Differential Equations

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References