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:

$$Ky''= \epsilon {\ddot y}+ \mu{\dot y} \protect$$ (5.12)

Here, $\mu>0$ can be thought of as a very simple friction coefficient, or resistance. As derived in §G.5, solutions to this wave equation can be expressed as sums of left- and right-going exponentially decaying traveling waves. When $\mu=0$, we get non-decaying traveling waves as before. As discussed in §1.2.2, propagation losses may be introduced by the substitution

$$z^{-1}\leftarrow gz^{-1}, \quad \left\vert g\right\vert\leq 1$$

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 $N$ is equivalent to simply scaling the output of the delay line by $g^N$. 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.