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Chapters

Physical Audio Signal Processing
    Wave Digital Filters
       Wave Digital Modeling Examples
          Mass and Dashpot in Series

Chapter Contents:

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Mass and Dashpot in Series

This is our first example illustrating a series connection of wave digital elements. Figure F.26 gives the physical scenario of a simple mass-dashpot system, and Fig.F.27 shows the equivalent circuit. Replacing element voltages and currents in the equivalent circuit by wave variables in an infinitesimal waveguides produces Fig.F.28.

**Figure F.26:** External force driving a mass which in turn drives a dashpot terminated on the other end by a rigid wall.
$\includegraphics{eps/massdash}$

**Figure:** Electrical equivalent circuit of the mass and dashpot system of Fig.F.26.
$\includegraphics{eps/massdashec}$

**Figure:** Intermediate wave-variable model of the mass and dashpot of Fig.F.27.
$\includegraphics{eps/massdashdt}$

**Figure F.29:** Wave digital filter for an ideal force source in parallel with the series combination of a mass and dashpot $\mu$ . The parallel and series adaptors are joined at an impedance which is calculated to suppress reflection from port 1 of the series adaptor.
$\includegraphics{eps/massdashjunc}$

The system can be described as an ideal force source connected in parallel with the series connection of mass and dashpot $\mu$ . Figure F.29 illustrates the resulting wave digital filter. Note that the ports are now numbered for reference. Two more symbols are introduced in this figure: (1) the horizontal line with a dot in the middle indicating a series adaptor, and (2) the indication of a reflection-free port on input 1 of the series adaptor (signal $f^{{+}}_1(n)$ ). Recall that a reflection-free port is always necessary when connecting two adaptors together, to avoid creating a delay-free loop.

Let's first calculate the impedance necessary to make input 1 of the series adaptor reflection free. From Eq. $\,$ (F.37), we require

$\displaystyle R = m + \mu$

That is, the impedance of the reflection-free port must equal the series combination of all other port impedances meeting at the junction.

The parallel adaptor, viewed alone, is equivalent to a force source driving impedance $R=m+\mu$ . It is therefore realizable as in Fig.F.20 with the wave digital spring replaced by the mass-dashpot assembly in Fig.F.29. However, we can also carry out a quick analysis to verify this: The alpha parameters are

$\begin{eqnarray*} \alpha_1 &\isdef & \frac{2\Gamma _1}{\Gamma _1+\Gamma _2} = \... ...\mu}\right)}{\infty+\left(\frac{1}{m}+\frac{1}{\mu}\right)} = 0 \end{eqnarray*}$

Therefore, the reflection coefficient seen at port 1 of the parallel adaptor is $\rho = \alpha_1 - 1 = 1$ , and the Kelly-Lochbaum scattering junction depicted in Fig.F.20 is verified.

Let's now calculate the internals of the series adaptor in Fig.F.29. From Eq. $\,$ (F.26), the beta parameters are

$\begin{eqnarray*} \beta_1 &\isdef & \frac{2R_1}{R_1+R_2+R_3} = \frac{2(m+\mu)}{... ...R_3}{R_1+R_2+R_3} = \frac{2m}{(m+\mu)+m+\mu} = \frac{m}{m+\mu} \end{eqnarray*}$

Following Eq. $\,$ (F.30), the series adaptor computes

$\begin{eqnarray*} f^{{+}}_J(n) &=& f^{{+}}_1(n)+f^{{+}}_2(n)+f^{{+}}_3(n) = f(... ...(n)\\ &=& \frac{\mu}{m+\mu}f^{{+}}_3(n) - \frac{m}{m+\mu} f(n) \end{eqnarray*}$

We do not need to explicitly compute $f^{{-}}_2(n)$ because it goes into a purely resistive impedance $\mu$ and produces no return wave. For the same reason, $f^{{+}}_2(n)\equiv\message{CHANGE eqv TO equiv IN SOURCE}0$ . Figure F.30 shows a wave flow diagram of the computations derived, together with the result of elementary simplifications.

**Figure F.30:** Wave flow diagram for the WDF modeling an ideal force source in parallel with the series combination of a mass and dashpot $\mu$ .
$\includegraphics{eps/massdashwdf}$

Because the difference of the two coefficients in Fig.F.30 is 1, we can easily derive the one-multiply form in Fig.F.31.

**Figure:** One-multiply form of the WDF in Fig.F.30.
$\includegraphics{eps/massdashwdfom}$

Subsections

Checking the WDF against the Analog Equivalent Circuit

Previous: Spring and Free Mass
Next: Checking the WDF against the Analog Equivalent Circuit

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

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