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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 .

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 matrices and of (7) can be considered to be diagonal in this case, thus allowing a description of the system as four separate scalar waveguides. The pair of strings is described by the -variable impedance matrix (67)

The lumped elements forming the bridge are connected in series, so that the driving-point velocity  is the same for the spring, mass, and damper: (68)

Also, the forces provided by the spring, mass, and damper, add: (69)

We can derive an expression for the bridge impedances using the following relations in the Laplace-transform domain: (70)

Equations (68) and (67) give the continuous-time load impedance (71)

In order to move to the discrete-time domain, we may apply the bilinear transform (72)

to (69). The factor is used to control the compression of the frequency axis. It may be set to 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       The factor in the impedance formulation of the scattering matrix (62) is given by (73)

which is a rational function of the complex variable . The scattering matrix is given by (74)

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

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