``Piano hammer in flight''

Suppose we wish to model a situation in which a mass of size kilograms is traveling with a constant velocity. This is an appropriate model for a piano hammer after its key has been pressed and before the hammer has reached the string.

Figure F.2 shows the ``wave digital mass'' derived previously.
The derivation consisted of inserting an infinitesimal
waveguide^{F.3} having (real) impedance
, solving for the force-wave reflectance of the mass as seen from
the waveguide, and then mapping it to the discrete time domain using
the bilinear transform.

We now need to attach the other end of the transmission line to a ``force source'' which applies a force of zero newtons to the mass. In other words, we need to terminate the line in a way that corresponds to zero force.

Let the force-wave components entering and leaving the mass
be denoted
and
, respectively (*i.e.*, we are dropping
the subscript `d' in Fig.F.2).
The physical force associated with the mass is

The zero-force case is therefore obtained when . This is illustrated in Fig.F.8.

Figure F.8a (left portion) illustrates what we derived
by physical reasoning, and as such, it is most appropriate as a
physical model of the constant-velocity mass. However, for actual
*implementation*, Fig.F.8b would be more typical in
practice. This is because we can always negate the state variable
if needed to convert it from
to
. It is
very common to see final simplifications like this to maximize
efficiency.

Note that Fig.F.8b can be interpreted physically as a wave
digital *spring* displaced by a constant force
.

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University