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of the output. Higher spatial and temporal resolu-
tions can be used for a more accurate simulation
but this costs more in terms of processing cycles
and these techniques can become computationally
expensive for large objects. For this reason, they
are not often the preferred method for environ-
ments involving many objects or in cases where
efficiency is important and sound quality is not
the top priority.
In the mid-1980s, digital waveguide synthesis
emerged as a computationally cheap method for
modelling one-dimensional (1D) wave propaga-
tion in systems with no stiffness. As many stringed
and woodwind instruments approximately fit this
description the technique proved popular and has
been heralded as “the most successful physical
modelling technique to date” (Bilbao, 2009, p.
18). It is explained as follows.
The vibrational behaviour of a 1D system
with no stiffness can be modelled as two travel-
ling waves propagating through the structure,
bouncing back at the ends and not interacting
with each other. These waves originate from an
excitation signal and initially leave the excitation
point in opposite directions. Digital waveguide
synthesis involves employing two memory buf-
fers as a delay line for each direction. The size
of the buffers corresponds to the time taken for a
sound wave to travel the length of the structure
being modelled and is therefore calculated using
the speed of sound in the medium. To model at-
tenuation, the waves are passed through a digital
filter as they bounce at each end. The filters are
designed to apply the correct attenuation for the
structure being modelled and can apply frequency-
dependent attenuation, giving the output sound the
signature of different materials. As locations in
the memory buffers correspond to positions on a
real object it is easy to simulate the structure being
excited at different points by adding an excitation
signal to the corresponding memory location of
each buffer. Audio output can easily be extracted
from the model by summing the contents of the
memory location of each buffer corresponding to
the desired pickup position.
Compared to the techniques discussed so far,
which may require hundreds or thousands of
arithmetic operations per temporal interval, digital
waveguide synthesis is a very inexpensive tech-
nique. The movement of a signal through a buffer
requires no arithmetic operations and a second
order digital filter will often suffice for applying
attenuation at the ends. It should be stressed that
this technique does not gain computational ef-
ficiency over other methods through some crude
approximation or by sacrificing veracity. Instead,
it exploits the harmonic nature of waves in a 1D
system without stiffness and employs a very cheap
method of delaying a signal in time.
An obvious drawback with digital waveguides
is their limitation to 1D systems. When they have
been extended to more dimensions, creating a
“waveguide mesh”, the performance gains have
been lost and the number of arithmetic calcula-
tions involved invariably approaches that of a
finite difference scheme (Bilbao, 2006). Another
limitation is their inability to model the effect
that the vibration amplitude has on how a system
resonates, that is, they cannot easily be used to
model distributed nonlinearities. This effect is
apparent when one listens to how the sound of a
gong changes as it resonates.
To understand another drawback of digital
waveguide synthesis one must understand the
effect that bending stiffness has on how an ob-
ject vibrates. The composite frequencies of a 1D
sounding object with no bending stiffness will be
harmonically related, that is, the higher frequen-
cies will be integer multiples of the fundamental
frequency. Bending stiffness causes waves to
propagate more quickly through a medium and,
significantly, this effect is more pronounced in
higher frequencies. Therefore, the spectrum of a
1D sounding object with bending stiffness will
not be harmonic. Because many stringed and
woodwind instruments are not greatly affected by
stiffness, this disadvantage has not been enough
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