- Initial striking force determines the starting regime (1, 2, or 3 above).
- The string model is simplified as it decays.
Because the energy stored in a piano string decays monotonically after a hammer strike (neglecting coupling from other strings), we may switch to progressively simpler models as the energy of vibration falls below corresponding thresholds. Since the purpose of model-switching is merely to save computation, it need not happen immediately, so it may be triggered by a string-energy estimate based on observing the string at a single point over the past period or so of vibration. Perhaps most simply, the model-regime classifier can be based on the maximum magnitude of the bridge force over at least one period. If the regime 2 model includes an instantaneous string-length (tension) estimate, one may simply compare that to a threshold to determine when the simpler model can be used. If the longitudinal components and/or phantom partials are not completely inaudible when the model switches, then standard cross-fading techniques should be applied so that inharmonic partials are faded out rather than abruptly and audibly cut off.
To obtain a realistic initial shock noise in the tone for regime 1 (and for any model that does not compute it automatically), the appropriate shock pulse, computed as a function of hammer striking force (or velocity, displacement, etc.), can be summed into the longitudinal waveguide during the hammer contact period.
The longitudinal bridge force may be generated from the estimated string length (§9.1.6). This force should be exerted on the bridge in the direction coaxial with the string at the bridge (a direction available from the two transverse displacements one sample away from the bridge).
Phantom partials may be generated in the longitudinal waveguide as explicit intermodulation products based on the transverse-wave overtones known to be most contributing; for example, the Goetzel algorithm  could be used to track relevant partial amplitudes for this purpose. Such explicit synthesis of phantom partials, however, makes modal synthesis more compelling for the longitudinal component ; in a modal synthesis model, the longitudinal attack pulse can be replaced by a one-shot (per hammer strike) table playback, scaled and perhaps filtered as a function of the hammer striking velocity.
On the high end for regime 2 modeling, a full nonlinear coupling may be implemented along the three waveguides (two transverse and one longitudinal). At this level of complexity, a wide variety of finite-difference schemes should also be considered (§7.3.1) [53,555].
Checking the Approximations
Regimes of Piano-String Vibration