Book contents
- Frontmatter
- Contents
- Preface to the first edition
- Preface to the second edition
- Preface to the third edition
- Notes for reference
- 1 Free vibrations
- 2 Free vibrations in physics
- 3 Damping
- 4 Damping in physics
- 5 Forced vibrations
- 6 Forced vibrations in physics
- 7 Anharmonic vibrations
- 8 Two-coordinate vibrations
- 9 Non-dispersive waves
- 10 Non-dispersive waves in physics
- 11 Fourier theory
- 12 Dispersion
- 13 Water waves
- 14 Electromagnetic waves
- 15 De Broglie waves
- 16 Solitary waves
- 17 Plane waves at boundaries
- 18 Diffraction
- Answers to problems and hints for solution
- Constants and units
- Index
11 - Fourier theory
Published online by Cambridge University Press: 05 June 2012
- Frontmatter
- Contents
- Preface to the first edition
- Preface to the second edition
- Preface to the third edition
- Notes for reference
- 1 Free vibrations
- 2 Free vibrations in physics
- 3 Damping
- 4 Damping in physics
- 5 Forced vibrations
- 6 Forced vibrations in physics
- 7 Anharmonic vibrations
- 8 Two-coordinate vibrations
- 9 Non-dispersive waves
- 10 Non-dispersive waves in physics
- 11 Fourier theory
- 12 Dispersion
- 13 Water waves
- 14 Electromagnetic waves
- 15 De Broglie waves
- 16 Solitary waves
- 17 Plane waves at boundaries
- 18 Diffraction
- Answers to problems and hints for solution
- Constants and units
- Index
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.
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- Vibrations and Waves in Physics , pp. 192 - 212Publisher: Cambridge University PressPrint publication year: 1993