Sampled Traveling Waves

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 [301].

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Subsections

• Digital Waveguide Model
• Digital Waveguide Interpolation
• Relation to the Finite Difference Recursion

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Order a Hardcopy of Physical Audio Signal Processing

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.