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
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: