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Frequency-Dependent Friction

With a further generalization, we can consider losses that are dependent on frequency, so that the friction coefficient $\Upsilon $ is replaced by $\Upsilon(\omega_s)$. In such case, all the formulas up to (49) will be recomputed with this new $\Upsilon(\omega_s)$.

Quite often, losses are deduced from experimental data which give the value $V_R(\omega_s)$. In these cases, it is useful to calculate the value of $\Upsilon(\omega_s)$ so that the wave admittance can be computed. From (33) we find

\Theta = \frac{\vert V_R\vert}{\sqrt{\frac{\omega_s^2}{c^2} + {V_R}^2}}
\end{displaymath} (51)

and, therefore, from (46) we get
\Upsilon(\omega_s) = \frac{2 \vert V_R(\omega_s)\vert}{\omega_s} {\sqrt{\frac{\omega_s^2}{c^2} + {V_R(\omega_s)}^2}} \; .
\end{displaymath} (52)

For instance, in a radius-$a$ cylindrical tube, the visco-thermal losses can be approximated by the formula [45]

\vert V_R(\omega_s)\vert = \frac{3.0\times 10^{-5}}{a} \sqrt{\frac{\omega_s}{2 \pi}} \; ,
\end{displaymath} (53)

which can be directly replaced into (53).

In vibrating strings, the viscous friction with air determines a damping that can be represented by the formula [45]

\vert V_R(\omega_s)\vert = a_1 \sqrt{\frac{\omega_s}{2 \pi}} + a_2 \; ,
\end{displaymath} (54)

where $a_1$ and $a_2$ are coefficients that depend on radius and density of the string.


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``Generalized Digital Waveguide Networks'', by Julius O. Smith III and Davide Rocchesso, preprint submitted for publication, Summer 2001.
Copyright © 2008-03-12 by Julius O. Smith III and Davide Rocchesso
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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