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
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