The Damped Plucked String

Without damping, the ideal plucked string sounds more like a cheap electronic organ than a string because the sound is perfectly periodic and never decays. Static spectra are very boring to the ear. The discrete Fourier transform (DFT) of the initial ``string loop'' contents gives the Fourier series coefficients for the periodic tone produced.

The simplest change to the ideal wave equation of Eq. (6.1) that provides damping is to add a term proportional to velocity:

Here, can be thought of as a very simple friction coefficient, or resistance. As derived in §C.5, solutions to this wave equation can be expressed as sums of left- and right-going

in each delay element (or wherever one sample of delay models one spatial sample of wave propagation). By commutativity of LTI systems, making the above substitution in a delay line of length is equivalent to simply scaling the output of the delay line by . This

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University