Nonlinear Piano-String Synthesis

When one-way transverse-to-longitudinal coupling is sufficiently accurate, we can model its effects separately based on observations of transverse waves. For example [30,28],

• longitudinal modes can be implemented as second-order resonators (``modal synthesis''), with driving coefficients given by

• orthogonally projecting [454] the spatial derivative of the squared string slope onto the longitudinal mode shapes (both functions of position $x$ ).

• If tension variations along the string are neglected (reasonable since longitudinal waves are so much faster than transverse waves), then the longitudinal force on the bridge can be derived from the estimated instantaneous tension in the string, and efficient methods for this have been developed for guitar-string synthesis, particularly by Tolonen (§9.1.6).

An excellent review of nonlinear piano-string synthesis (emphasizing the modal synthesis approach) is given in [30].