Sound Modelling
111
Karplus-Strong synthesis
waveguide junctions
digital waveguide networks
(t)
(t)
(t - nT)
(t + nT)
p
+
p
-
Wave Delay
Wave Delay
p
+
p
-
Figure 14: Wave propagation propagation in a ideal (i.e. linear, non-dissipative
and non-dispersive) medium can be represented, in the discrete-time domain,
by a couple of digital delay lines.
Let us consider deviations from ideal propagation due to losses and disper-
sion in the resonator. Usually, these linear effects are lumped and simulated
with a few filters which are cascaded with the delay lines. Losses due to ter-
minations, internal frictions, etc., give rise to gentle low pass filters, whose pa-
rameters can be identified from measurements. Wave dispersion, which is often
due to medium stiffness, is simulated by means of allpass filters whose effect
is to produce a frequency-dependent propagation velocity [83]. The reflecting
terminations of the resonator (e.g., a guitar bridge) can also modeled as filters.
In virtue of linearity and time invariance, all the filters can be condensed in a
single higher-order filtering block, and all the delays can be connected to form a
single longer delay line. As a result, we would get the recursive comb filter, de-
scribed in chapter 3, which forms the structure of the Karplus-Strong synthesis
algorithm [47].
One-dimensional waveguide models can be connected together by means of
waveguide junctions, thus forming digital waveguide networks, which are used
for simulation of multi-dimensional media (e.g., membranes [34]) or complex
acoustic systems (e.g., several strings attached to a bridge [17]). The general
treatment of waveguide networks is beyond the scope of this book [85].
tive with the difference (p(i + 1, n) - p(i, n))/X, and the first-order time derivative with the
difference (p(i, n + 1) - p(i, n))/T
7
The D'Alembert solution can be derived by inserting the exponential eigenfunction e
st+vx