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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 $ T$ seconds, corresponding to a sampling rate $ f_s
\isdeftext 1/T$ samples per second. For CD-quality audio, we have $ f_s= 44.1$ kHz. The natural choice of spatial sampling interval $ X$ is the distance sound propagates in one temporal sampling interval $ T$ , or $ X\isdeftext cT$ 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

x &\to& x_m&=& mX, \nonumber \\
t &\to& t_n&=& nT. \nonumber

Substituting into the traveling-wave solution of the wave equation gives
$\displaystyle y(t_n,x_m)$ $\displaystyle \,\mathrel{\mathop=}\,$ $\displaystyle y_r(t_n- x_m/c) + y_l(t_n+ x_m/c) \protect$  
  $\displaystyle \,\mathrel{\mathop=}\,$ $\displaystyle y_r(nT- mX/c) + y_l(nT+ mX/c) \nonumber$  
  $\displaystyle \,\mathrel{\mathop=}\,$ $\displaystyle y_r\left[(n-m)T\right]+ y_l\left[(n+m)T\right].

Since $ T$ multiplies all arguments, let's suppress it by defining

$\displaystyle y^{+}(n) \isdefs y_r(nT), \qquad\qquad y^{-}(n) \isdefs y_l(nT). \protect$ (C.16)

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].

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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4
Copyright © 2023-08-20 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University