Sampled Travelling Waves

Sampling is carried out by the change of variables

$\displaystyle x \rightarrow x_m$	$\displaystyle =$	$\displaystyle mX$
$\displaystyle t \rightarrow t_n$	$\displaystyle =$	$\displaystyle nT,$

where

is the temporal sampling interval, and $X \triangleq cT$ , is the spatial sampling interval

Substituting into the traveling-wave solution yields

$\displaystyle y(t_n,x_m)$	$\displaystyle =$	$\displaystyle y_r(t_n- x_m/c) + y_l(t_n+ x_m/c)$
	$\displaystyle =$	$\displaystyle y_r(nT- mX/c) + y_l(nT+ mX/c)$
	$\displaystyle =$	$\displaystyle y_r\left[(n-m)T\right]+ y_l\left[(n+m)T\right]$

Since multiplies all arguments it is typically suppressed.

This notation is also used to model pressure in acoustic tubes.

``Music 206: Digital Waveguides'' by Tamara Smyth, Department of Music, University of California, San Diego (UCSD).