The qualitative approach

Ninety percent of all physics is concerned with vibrations and waves of one sort or another. The same basic thread runs through most branches of physical science, from acoustics through engineering, fluid mechanics, optics, electromagnetic theory and X-rays to quantum mechanics and information theory. It is closely bound to the idea of a signal and its spectrum. To take a simple example: imagine an experiment in which a musician plays a steady note on a trumpet or a violin, and a microphone produces a voltage proportional to the instantaneous air pressure. An oscilloscope will display a graph of pressure against time, F(t), which is periodic. The reciprocal of the period is the frequency of the note, 440 Hz, say, for a well-tempered middle A – the tuning-up frequency for an orchestra.

The waveform is not a pure sinusoid, and it would be boring and colourless if it were. It contains ‘harmonics’ or ‘overtones’: multiples of the fundamental frequency, with various amplitudes and in various phases, depending on the timbre of the note, the type of instrument being played and on the player. The waveform can be analysed to find the amplitudes of the overtones, and a list can be made of the amplitudes and phases of the sinusoids which it comprises. Alternatively a graph, A(ν), can be plotted (the sound-spectrum) of the amplitudes against frequency (Fig. 1.1).