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Introduction to the Physics of Waves by Tim Freegarde

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14

Fourier series and transforms

14.1 Fourier synthesis and analysis

In our discussion of linearity and dispersion in Chapter 13, we have seen how simple sinusoidal components may be combined to form more complex waveforms: two frequencies produce periodic beats, and further components allow a single wavepacket to be isolated. With sufficient components spanning a great enough range of wavenumbers, the single wavepacket may also be made arbitrarily narrow. If a single narrow wavepacket can be synthesized by adding sinusoidal components then, by the principle of superposition, so can an arbitrary waveform, for we may construct the desired form from a series of consecutive pulses whose individual magnitudes correspond to the amplitude of ...

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