As an application of the theory, 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 the strings (see Fig. 3). An in-depth treatment of coupled strings can be found in [132].

To a first approximation, the bridge can be modeled as a lumped mass-spring-damper system, while for the strings, a distributed waveguide representation 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. Since the matrices and of (5) can be considered to be diagonal in this case, we could alternatively describe 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^{7} 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 (55) and (54) give the continuous-time load impedance

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

(56) |

We obtain

The factor in the impedance formulation of the scattering matrix (48) is given by

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

which can be implemented using a

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