Physical modeling of the piano: An investigation into the effect of string stiffness on the hammer string interaction.
Charalampos Saitis, Sarah Orr, Maarten van Walstijn · The Journal of the Acoustical Society of America · 2009
The stiff string wave equation has four solutions, two of which are fast-decaying waves introduced by the string stiffness. In the case of digital waveguide modeling of piano strings these are normally neglected. Some recent reports have suggested that all four traveling waves should be considered, at least at the neighborhood of interaction points (i.e., the hammer and the boundaries). This paper investigates the effect of omitting string stiffness in the context of sound synthesis of the piano by physical modeling. A stiff, lossy string with a spatially distributed hammer force excitation is implemented using both a finite-difference time-domain scheme and a digital waveguide model. The two models are designed so as to have the exact same features but for the two stiffness-related solutions. Numerical experiments are employed to study the contact force and string velocity signals for different initial hammer velocity values. The results generally confirm that the two fast-decaying waves have only a marginal effect on the overall string motion. However, small audible differences result for bass strings struck with high initial hammer velocities.