### Terminated String Impedance

Note that the impedance of the*terminated*string, seen from one of its endpoints, is not the same thing as the wave impedance of the string itself. If the string is infinitely long, they are the same. However, when there are

*reflections*, they must be included in the impedance calculation, giving it an imaginary part. We may say that the impedance has a ``reactive'' component. The driving-point impedance of a rigidly terminated string is ``purely reactive,'' and may be called a

*reactance*(§7.1). If denotes the force at the driving-point of the string and denotes its velocity, then the driving-point impedance is given by (§7.1)

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*accuracy is improved*because the round-off errors per period arising from repeated multiplication by have been replaced by a single round-off error per period in the multiplication by .

**Next Section:**

Stiff String Synthesis Models

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Animation of Moving String Termination and Digital Waveguide Models