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Wave Digital Mass-Spring Oscillator

Let's look again at the mass-spring oscillator of §F.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.F.34 and Fig.F.35, respectively.

Figure F.34: Elementary mass-spring oscillator.

Figure F.35: Equivalent circuit for the mass-spring oscillator.

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.F.36 and Fig.F.37.F.4 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
= \frac{\Gamma _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 F.36: Wave digital mass-spring oscillator.

Figure F.37: Detailed wave-flow diagram for the wave digital mass-spring oscillator.

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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