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 [

470]. 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

where

corresponds to a linear spring. The spring constant
linearized about zero

displacement is

which, for

, approaches zero as

. In other words, the
spring-constant itself goes to zero with its displacement, instead of
remaining a constant. This behavior serves to ``feather'' contact and
release with the string. We see from Eq.

(

9.23) above
that, as displacement goes to zero, the

reflectance approaches a
frequency-independent

reflection coefficient
,
resulting from the damping

that remains in the spring model. As
a result, there is still a discontinuity when the spring disengages
from the string.
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

so that

approaches zero at the same rate as

. It
would be interesting to estimate

for the spring and

damper from
measured data. In the absence of such data,

is easy to compute
(requiring a single multiplication). More generally, an interpolated
lookup of

values can be used.
In summary, the engagement and disengagement of the plucking system
can be ``feathered'' by a nonlinear spring and damper in the plectrum
model.

**Next Section:** Piano String
Wave Equation**Previous Section:** Digitization of the Damped-Spring Plectrum