The piano-hammer model of the previous section can also be configured as a plectrum by making the mass and damping small or zero, and by releasing the string when the contact force exceeds some threshold . That is, to a first approximation, a plectrum can be modeled as a spring (linear or nonlinear) that disengages when either it is far from the string or a maximum spring-force is exceeded. To avoid discontinuities when the plectrum and string engage/disengage, it is good to taper both the damping and spring-constant to zero at the point of contact (as shown below).
If the spring damping is much greater than twice the string wave impedance (), then the plectrum looks like a rigid termination to the string (force reflectance ), which makes physical sense.
If the damping is set to zero, i.e., if the plectrum is to be modeled as a simple linear spring, then the impedance becomes , and the force-wave reflectance becomes 
Digital Waveguide Plucked-String Model
When plucking a string, it is necessary to detect ``collisions'' between the plectrum and string. Also, more complete plucked-string models will allow the string to ``buzz'' on the frets and ``slap'' against obstacles such as the fingerboard. For these reasons, it is convenient to choose displacement waves for the waveguide string model. The reflection and transmission filters for displacement waves are the same as for velocity, namely, and .
As in the mass-string collision case, we obtain the one-filter scattering-junction implementation shown in Fig.9.23. The filter may now be digitized using the bilinear transform as previously (§9.3.1).
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
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
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
where, as first noted at Eq.(9.24) above,
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.
As discussed above, in a simple 1D plucking model, the plectrum comes up and engages the string when , and above some maximum force the plectrum releases the string. At this point, it is ``above'' the string. To pluck again in the same direction, the collision-detection must be disabled until we again have , requiring one bit of state to keep track of that.10.16 The harpsichord jack plucks the string only in the ``up'' direction due to its asymmetric behavior in the two directions . If only ``up picks'' are supported, then engagement can be suppressed after a release until comes back down below the envelope of string vibration (e.g., ). Note that intermittent disengagements as a plucking cycle begins are normal; there is often an audible ``buzzing'' or ``chattering'' when plucking an already vibrating string.
When plucking up and down in alternation, as in the tremolo technique (common on mandolins), the collision detection alternates between and , and again a bit of state is needed to keep track of which comparison to use.
The transmittance filter is again , and there is a one-filter form for the scattering junction as usual.
Since the pluck model is linear, the parameters are not signal-dependent. As a result, when the string and spring separate, there is a discontinuous change in the reflection and transmission coefficients. In practice, it is useful to ``feather'' the switch-over from one model to the next . In this instance, one appealing choice is to introduce a nonlinear spring, as is commonly used for piano-hammer models (see §9.3.2 for details).
Let the nonlinear spring model take the form
The foregoing suggests a nonlinear tapering of the damping in addition to the tapering the stiffness as the spring compression approaches zero. One natural choice would be
In summary, the engagement and disengagement of the plucking system can be ``feathered'' by a nonlinear spring and damper in the plectrum model.
Stiff Piano Strings
Piano Hammer Modeling