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.(4.1) 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 §H.5, solutions to this wave equation can be expressed as sums of left- and right-going exponentially decaying traveling waves. When , we get non-decaying traveling waves as before. As discussed in §1.2.2, propagation losses may be introduced by the substitution

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 lumping of propagation loss at one point along the waveguide serves to minimize both computational cost and round-off error. In general finite difference schemes, such a simplification is usually either not possible or nonobvious.

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Subsections

• • Computational Savings

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and au