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 .
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 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
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).10.16 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 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.