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Coupled Strings

We have just discussed the coupling between vertical and horizontal planes of vibration along a single string. There is also important coupling among different strings on the same instrument. For example, modern pianos are constructed having up to three physical strings associated with each key. These strings are slightly mistuned in order to sculpt the shape of the decay envelope, including its beating characteristics and two-stage decay. A two-stage decay is desired in piano strings in order to provide a strong initial attack followed by a long-sustaining ``aftersound'' [543], [18, Weinreich chapter].


A simple approximation to the effect of coupled strings is obtained by simply summing two or more slightly detuned strings. While this can provide a realistic beating effect in the amplitude envelope, it does not provide a true two-stage decay. A more realistic simulation of coupling requires signal to flow from each coupled string into all others. When the bridge moves in response to string vibrations, traveling waves are generated along all other strings attached to the bridge. In the simplest case of a bridge modeled as a rigid body, the generated wave is identical on all strings. In §C.13, an efficient scattering formulation of string coupling at a bridge is derived for this case [439]. It can be seen as a simplification of the general coupling matrix shown in Fig.6.20 for the two-string (or two-polarization) case. Additionally, an eigenanalysis of the coupling matrix is performed, thereby extending the analysis of §6.12.2 above.
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Longitudinal Waves
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Asymmetry of Horizontal/Vertical Terminations