#### Incorporating Control Motion

Let denote the vertical position of the*mass*in Fig.9.22. (We still assume .) We can think of as the position of the

*control point*on the plectrum,

*e.g.*, the position of the ``pinch-point'' holding the plectrum while plucking the string. In a harpsichord, can be considered the

*jack*position [347].

Also denote by the

*rest length*of the spring in Fig.9.22, and let denote the position of the ``end'' of the spring while not in contact with the string. Then the plectrum makes

*contact*with the string when

*collision detection*equation. Let the subscripts and each denote one side of the scattering system, as indicated in Fig.9.23. Then, for example, is the displacement of the string on the left (side ) of plucking point, and is on the right side of (but still located at point ). By continuity of the string, we have

^{10.15}For or the applied force is zero and the entire plucking system disappears to leave and , or equivalently, the force reflectance becomes and the transmittance becomes . During contact, force equilibrium at the plucking point requires (

*cf.*§9.3.1)

where as usual (§6.1), with denoting the string tension. Using Ohm's laws for traveling-wave components (p. ), we have

Substituting and taking the Laplace transform yields

*i.e.*, when the plectrum does not affect the string displacement at the current time. It is therefore exact at the time of collision and also applicable just after release. Similarly, can be used to trigger a release of the string from the plectrum.

**Next Section:**

Successive Pluck Collision Detection

**Previous Section:**

Digital Waveguide Plucked-String Model