Wave Digital Mass-Spring Oscillator

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Wave Digital Mass-Spring Oscillator

Let's look again at the mass-spring oscillator of §Q.3.4, but this time without the driving force (which effectively decouples the mass and spring into separate first-order systems). The physical diagram and equivalent circuit are shown in Fig.Q.31 and Fig.Q.32, respectively.

**Figure Q.31:** Elementary mass-spring oscillator.
$\begin{figure}\input fig/tank.pstex_t \end{figure}$

**Figure Q.32:** Equivalent circuit for the mass-spring oscillator.
$\begin{figure}\input fig/tankec.pstex_t \end{figure}$

Note that the mass and spring can be regarded as being in parallel or in series. Under the parallel interpretation, we have the WDF shown in Fig.Q.33 and Fig.Q.34.^Q.5 The reflection coefficient $\rho$ can be computed, as usual, from the first alpha parameter:

$\displaystyle \rho = \alpha_1 - 1 = \frac{2\Gamma _1}{\Gamma _1+\Gamma _2} - 1 ... ..._1-\Gamma _2}{\Gamma _1+\Gamma _2} = \frac{R_2-R_1}{R_2+R_1} = \frac{m-k}{m+k}$

This result, $\rho=(m-k)/(m+k)$ , is just the ``impedance step over impedance sum'', so no calculation was really necessary.

**Figure Q.33:** Wave digital mass-spring oscillator.
$\begin{figure}\input fig/tankjunc.pstex_t \end{figure}$

**Figure Q.34:** Detailed wave-flow diagram for the wave digital mass-spring oscillator.

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Physical Audio Signal Processing
    Wave Digital Filters
       Wave Digital Modeling Examples
          Wave Digital Mass-Spring Oscillator