11 - Fourier theory

Summary

At this point it is worth bringing together several things we have discovered about vibrating systems in earlier chapters.

(1) If a number of harmonic driving forces act simultaneously on a linear system, the resulting steady-state vibration ψ(t) is a superposition of harmonic vibrations whose frequencies are those of the driving forces: each harmonic force makes its own independent contribution to ψ(t). This is an example of the principle of superposition (section 5.2).

(2) The free vibration of a non-linear system is not harmonic, but something more complicated; we found it possible, however, to express an anharmonic vibration ψ(t) as a series of terms consisting of the fundamental vibration and a series of harmonics (sections 7.1 and 7.2).

(3) At small amplitudes, the application of a harmonic driving force to a non-linear system leads to a steady-state ψ(t) which contains the driving frequency and harmonics of that frequency (section 7.3).

(4) The standing waves that are possible on a non-dispersive string of finite length have frequencies in a sequence like v₁, 2v₁, 3v₁,…; when a number of standing waves are excited simultaneously, the vibration ψ(t) of any given point on the string must therefore consist of a series of superposed harmonics.

It is clear from these examples alone that the harmonic type of vibration on which we have spent so much time has a fundamental significance as the building block for more complicated motions.