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Sampled Traveling Waves in a String

For discrete-time simulation, we must sample the traveling waves

For a vibrating string with length $ L$ and fundamental frequency $ f_0$ ,

$\displaystyle c = f_0 \cdot 2L
\quad\quad
\left(\frac{\hbox{meters}}{\hbox{sec}}
=
\frac{\hbox{periods}}{\hbox{sec}}
\cdot \frac{\hbox{meters}}{\hbox{period}} \right)
$

so that

$\displaystyle X= cT = (f_0 2L)/f_s= L [ f_0 / (f_s/2) ]
$

Thus, the number of spatial samples along the string is

$\displaystyle \zbox{L/X= (f_s/2) / f_0}
$

or

$\displaystyle \zbox{\hbox{Number of spatial samples} = \hbox{Number of string harmonics}}
$

Examples:


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Download StringWaves.pdf
Download StringWaves_2up.pdf
Download StringWaves_4up.pdf

``Digitizing Strings Waves in Vibrating Strings'', by Julius O. Smith III, (From Lecture Overheads, Music 420).
Copyright © 2014-03-24 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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