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
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]