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:
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.