## 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

*exponentially decaying*traveling waves. When , we get non-decaying traveling waves as before. As discussed in §2.2.2, propagation losses may be introduced by the substitution

*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 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.)*all*of the losses at a single point in the delay loop. Furthermore, the two reflecting terminations (gain factors of ) 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 , delay line. The result of these inaudible simplifications is shown in Fig. 6.12.

*three orders of magnitude,*

*i.e.*, by a factor of in this case. However, the physical accuracy of the simulation has not been compromised. In fact, the

*accuracy is improved*because the round-off errors per period arising from repeated multiplication by have been replaced by a single round-off error per period in the multiplication by .

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Frequency-Dependent Damping

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The Ideal Struck String