As discussed in [121], displacement waves
are often preferred over force or velocity waves for guitar-string
simulations, because such strings often hit obstacles such as frets or
the neck. To obtain displacement from velocity at a given
,
we may time-integrate
velocity as above to produce displacement at any spatial sample along the
string where a collision might be possible. However, all these
integrators can be eliminated by simply going to a displacement-wave
simulation, as has been done in nearly all papers to date on plucking
models for digital waveguide strings.
To convert our force-wave simulation to a displacement-wave
simulation, we may first convert force to velocity using the Ohm's law
relations
and
and then
conceptually integrate all signals with respect to time (in advance of
the simulation).
is the same on both sides of the finger-junction, which means we
can convert from force to velocity by simply negating all left-going
signals. (Conceptually, all signals are converted from force to
velocity by the Ohm's law relations and then divided by
, but the
common scaling by
can be omitted (or postponed) unless signal
values are desired in particular physical units.) An
all-velocity-wave simulation can be converted to displacement waves
even more easily by simply changing
to
everywhere, because
velocity and displacement waves scatter identically. In more general
situations, we can go to the Laplace domain and replace each
occurrence of
by
, each
by
,
divide all signals by
, push any leftover
around for maximum
simplification, perhaps absorbing it into a nearby filter. In an
all-velocity-wave simulation, each signal gets multiplied by
in
this procedure, which means it cancels out of all definable transfer
functions. All filters in the diagram (just
in this
example) can be left alone because their inputs and outputs are still
force-valued in principle. (We expressed each force wave in terms of
velocity and wave impedance without changing the signal flow diagram,
which remains a force-wave simulation until minus signs, scalings, and
operators are moved around and combined.) Of course, one can
absorb scalings and sign reversals into the filter(s) to change the
physical input/output units as desired. Since we routinely assume
zero initial conditions in an impedance description, the integration
constants obtained by time-integrating velocities to get displacements
are all defined to be zero. Additional considerations regarding the
choice of displacement waves over velocity (or force) waves are given
in §E.3.3. In particular, their initial conditions can be very
different, and traveling-wave components tend not to be as well
behaved for displacement waves.