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 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

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:

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

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

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

(72) |

We obtain

The factor in the impedance formulation of the scattering matrix (62) is given by

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

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

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

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