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 . 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 .