Digitizing

via the centered second-order difference
[Eq.

(

7.5)]

we obtain the following

*explicit finite-difference scheme* (§

D.1):

where

is the external input vector for exciting the string
(driving three adjacent

masses in simulations to date).
Note that requiring three adjacent spatial string samples to be in
contact with the

piano hammer during the attack (which helps to
suppress

aliasing of spatial frequencies on the string during the
attack) implies a

sampling rate in the vicinity of 6 megahertz
[

265]. Thus, the model is expensive to compute!
However, results to date show a high degree of accuracy, as desired.
In particular, the stretching of the partial

overtones in the

stiff-string model of
Fig.

has
been measured to be highly accurate despite using only three

spring
attachment points on one side of each mass disk [

265].
See [

53] for alternative

finite-difference formulations
that better preserve physical energy and have other nice properties
worth considering.

**Next Section:** Force-Pulse Synthesis**Previous Section:** Nonlinear Piano-String Equations of Motion in State-Space Form