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 [350].

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

where denotes string vertical position at the plucking point . This may be called the

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

When the spring engages the string ( ) and begins to compress, the upward force on the string at the contact point is given by

where again . The force is applied given (spring is in contact with string) and given (the force at which the pluck releases in a simple max-force model).

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

where denotes the string wave impedance (p. ). Solving Eq.(9.25) for the velocity at the plucking point yields

or, for displacement waves,

Substituting and taking the Laplace transform yields

Solving for and recognizing the force reflectance gives

where, as first noted at Eq.(9.24) above,

We can thus formulate a one-filter scattering junction as follows:

This system is diagrammed in Fig.9.24. The manipulation of the
minus signs relative to Fig.9.23 makes it convenient for
restricting
to positive values only (as shown in the
figure), corresponding to the plectrum engaging the string going up.
This uses the approximation
,
which is exact when
, *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.

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University