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).
Starting with the piano-hammer impedance of Eq.(9.19) and setting the mass to infinity (the plectrum holder is immovable), we define the plectrum impedance as
The force-wave reflectance of impedance
in Eq.(9.22), as
seen from the string, may be computed exactly as in
§9.3.1:
Again following §9.3.1, the transmittance for force waves is given by
and for velocity and displacement waves, the reflectance and transmittance are respectively and .
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 [129]