To carry the traveling-wave solution into the ``digital domain,'' it
is necessary to *sample* the traveling-wave amplitudes at
intervals of
seconds, corresponding to a sampling rate
samples per second. For CD-quality audio, we have
kHz. The natural choice of *spatial sampling
interval*
is the distance sound propagates in one temporal
sampling interval
, or
meters. In a lossless
traveling-wave simulation, the whole wave moves left or right one
spatial sample each time sample; hence, lossless simulation requires
only digital delay lines. By lumping losses parsimoniously in a real
acoustic model, most of the traveling-wave simulation can in fact be
lossless even in a practical application.

Formally, sampling is carried out by the change of variables

Substituting into the traveling-wave solution of the wave equation gives

Since multiplies all arguments, let's suppress it by defining

This new notation also introduces a `` '' superscript to denote a traveling-wave component propagating to the right, and a `` '' superscript to denote propagation to the left. This notation is similar to that used for acoustic tubes [299].

[How to cite this work] [Order a printed hardcopy] [Comment on this page via email]

Copyright ©

Center for Computer Research in Music and Acoustics (CCRMA), Stanford University