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Example in Acoustics

As an application of the theory developed herein, we outline the digital simulation of two pairs of piano strings. The strings are attached to a common bridge, which acts as a coupling element between them (see Fig. 7). An in-depth treatment of coupled strings can be found in [30].

To a first approximation, the bridge can be modeled as a lumped mass-spring-damper system, while for the strings, a distributed representation as waveguides is more appropriate. For the purpose of illustrating the theory in its general form, we represent each pair of strings as a single 2-variable waveguide. This approach is justified if we associate the pair with the same key in such a way that both the strings are subject to the same excitation. Actually, the $2 \times 2$ matrices $\mbox{\boldmath$M$}$ and ${\mbox{\boldmath$T$}}$ of (7) can be considered to be diagonal in this case, thus allowing a description of the system as four separate scalar waveguides.

Figure 7: Two pairs of strings coupled at a bridge.
\epsfysize =140pt \epsfbox{figure/bridge2.eps}\end{figure}

The $i^{th}$ pair of strings is described by the $2$-variable impedance matrix

{\mbox{\boldmath$R$}}_i = \left[ \begin{array}{cc} {R_{i,1}} &
0 \\ 0 & {R_{i,2}} \\
\end{array} \right] \,.
\end{displaymath} (67)

The lumped elements forming the bridge are connected in series, so that the driving-point velocity[*] $u$ is the same for the spring, mass, and damper:

u(t)=u_m(t)=u_k(t)=u_{\mu}(t) \,.
\end{displaymath} (68)

Also, the forces provided by the spring, mass, and damper, add:
p(t)=p_m(t)+p_k(t)+p_{\mu}(t) \,.
\end{displaymath} (69)

We can derive an expression for the bridge impedances using the following relations in the Laplace-transform domain:
P_k(s)=(k/s) U_k(s)\\
P_m(s)={m s}U_m(s)\\
P_{\mu}(s)=\mu U_{\mu}(s) \,.
\end{array}\end{displaymath} (70)

Equations (68) and (67) give the continuous-time load impedance
R_{\rm L}(s) = \frac{P(s)}{U(s)} = m\frac{s^2 + s \mu/m + k/m}{s} \,.
\end{displaymath} (71)

In order to move to the discrete-time domain, we may apply the bilinear transform
s \leftarrow \alpha \frac{1 - z^{-1}}{1 + z^{-1}}
\end{displaymath} (72)

to (69). The factor $\alpha $ is used to control the compression of the frequency axis. It may be set to $2/T$ so that the discrete-time filter corresponds to integrating the analog differential equation using the trapezoidal rule, or it may be chosen to preserve the resonance frequency.

We obtain

$\displaystyle R_{\rm L}(z)$ $\textstyle =$ $\displaystyle \left[ (\alpha^2 - \alpha \mu/m + k/m)z^{-2} \right.$  
  $\textstyle +$ $\displaystyle (-2 \alpha^2 + 2 k/m)z^{-1}$  
  $\textstyle +$ $\displaystyle \left. (\alpha^2 + \alpha \mu/m + k/m) \right] \left/ \left[ \alpha / m (1 - z^{-2}) \right] \right. \,.$  

The factor $S$ in the impedance formulation of the scattering matrix (62) is given by
S(z) = \left[ {\displaystyle \sum_{i,j=1}^{2}{R_{i,j}} } + R_{\rm L}(z) \right]^{-1} \,,
\end{displaymath} (73)

which is a rational function of the complex variable $z$. The scattering matrix is given by
{\mbox{\boldmath$A$}}= 2 S \left[ \begin{array}{llll} {R_{1,...
...& {R_{2,2}} \\
\end{array} \right] - {\mbox{\boldmath$I$}}\,,
\end{displaymath} (74)

which can be implemented using a single second-order filter having transfer function (71).

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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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