- longitudinal modes can be implemented as second-order
resonators (``modal synthesis''), with driving coefficients
- orthogonally projecting  the spatial
derivative of the squared string slope onto the longitudinal mode
shapes (both functions of position ).
- 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 .
Regimes of Piano-String Vibration