In reality, distributed linear propagation is never lossless. There is
always some attenuation and dispersion per unit distance traveled by a wave
in the medium. In principle, implementing loss and dispersion calls for a
digital filter to be inserted at the output of every delay element
to precisely simulate one sample of wave propagation. In practice this is
generally unnecessary. Instead, we may normally implement digital filters
*sparsely* along each digital waveguide [97,99].
Each filter
provides the attenuation and dispersion corresponding to the propagation
distance over the section it covers. In other words, lossy, dispersive
media are approximated by *piecewise lossless* media having the same
average
attenuation and dispersion over long distances. Loss and dispersion are
thus *lumped* at sparse points along the waveguide model
rather than being lumped at each spatial sampling point.

In the 1D case, it is possible to obtain exact results using sparsely
lumped losses and dispersion, thanks to the *commutativity* of linear,
time-invariant elements [99]. For example,
consider an acoustic cylinder which is samples long. If the per-sample
traveling-wave filter is given by , and if we only care what
happens at the endpoints of the tube, then we need only implement the
filter at each end of the -sample tube, prior to any
junctions. Since the per-sample filtering is typically very weak,
the stronger filter is normally also quite weak and readily
approximated to within audio specifications by a low-order digital filter.

It is easy to show that the per-sample traveling-wave filter for
any *passive* propagation medium such as a string or acoustic cylinder
must satisfy
for all
.
That is, wave propagation in a fixed wave impedance
cannot increase traveling-wave amplitude at any frequency.
This is a convenient stability criterion which is straightforward
to ensure in practice.

In summary, wave propagation in a general linear time-invariant medium is thus simulated using a (possibly interpolated) delay line to handle the time-delay associated with the propagation, and one or more digital filters to sparsely implement losses and dispersion where needed, such as prior to a bow-string contact model [90].

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