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Wave Digital Dashpot

Starting with a dashpot with coefficient $ \mu $ , we have

$\displaystyle R(s) = \mu$   (Dashpot Impedance)

and reflectance

$\displaystyle \hat{\rho}_\mu(s) = \frac{\mu - R_0 }{\mu + R_0}

This time, choosing $ R_0$ equal to the element value gives

$\displaystyle \hat{\rho}_\mu(s) = 0

Conformally mapping the zero function yields the zero function so that

$\displaystyle \fbox{$\displaystyle \hat{\tilde{\rho}}_\mu(z) = 0$}
\qquad\mbox{(Wave Digital Dashpot)}

as well. Thus, the WDF of a dashpot is a ``wave sink,'' as diagrammed in Fig.F.4.

Figure F.4: Wave flow diagram for the Wave Digital Dashpot.

In the context of waveguide theory, a zero reflectance corresponds to a matched impedance, i.e., the terminating transmission-line impedance equals the characteristic impedance of the line.

The difference equation for the wave digital dashpot is simply $ f^{{-}}(n)=0$ . While this may appear overly degenerate at first, remember that the interface to the element is a port at impedance $ R_0=\mu$ . Thus, in this particular case only, the infinitesimal waveguide interface is the element itself.

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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