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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:

$\displaystyle Ky''= \epsilon {\ddot y}+ \mu{\dot y} \protect$ (7.14)

Here, $ \mu>0$ 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 exponentially decaying traveling waves. When $ \mu=0$, we get non-decaying traveling waves as before. As discussed in §2.2.2, propagation losses may be introduced by the substitution

$\displaystyle 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.

Computational Savings

To illustrate how significant the computational savings can be, consider the simulation of a ``damped guitar string'' model in Fig.6.11. For simplicity, the length $ L$ string is rigidly terminated on both ends. Let the string be ``plucked'' by initial conditions so that we need not couple an input mechanism to the string. Also, let the output be simply the signal passing through a particular delay element rather than the more realistic summation of opposite elements in the bidirectional delay line. (A comb filter corresponding to pluck position can be added in series later.)

Figure 6.11: Discrete simulation of the rigidly terminated string with distributed resistive losses. The $ N$ loss factors $ g$ are embedded between the delay-line elements.
\includegraphics[width=\twidth]{eps/fstring}

In this string simulator, there is a loop of delay containing $ N = 2L/X= f_s/f_1$ samples where $ f_1$ is the desired pitch of the string. Because there is no input/output coupling, we may lump all of the losses at a single point in the delay loop. Furthermore, the two reflecting terminations (gain factors of $ -1$) may be commuted so as to cancel them. Finally, the right-going delay may be combined with the left-going delay to give a single, length $ N$, delay line. The result of these inaudible simplifications is shown in Fig. 6.12.

Figure 6.12: Discrete simulation of the rigidly terminated string with consolidated losses (frequency-independent). All $ N$ loss factors $ g$ have been ``pushed'' through delay elements and combined at a single point.
\includegraphics[width=\twidth]{eps/fsstring}

If the sampling rate is $ f_s=50$ kHz and the desired pitch is $ f_1=100$ Hz, the loop delay equals $ N=500$ samples. Since delay lines are efficiently implemented as circular buffers, the cost of implementation is normally dominated by the loss factors, each one requiring a multiply every sample, in general. (Losses of the form $ 1-2^{-k}$, $ 1-2^{-k}-2^{-l}$, etc., can be efficiently implemented using shifts and adds.) Thus, the consolidation of loss factors has reduced computational complexity by three orders of magnitude, i.e., by a factor of $ 500$ in this case. However, the physical accuracy of the simulation has not been compromised. In fact, the accuracy is improved because the $ N$ round-off errors per period arising from repeated multiplication by $ g$ have been replaced by a single round-off error per period in the multiplication by $ g^N$.

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Frequency-Dependent Damping
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The Ideal Struck String