Finite-difference modeling of the piano

Abstract

The construction of a complete computational model of the piano is described. This model treats the three major subsystems of the instrument: the hammers and strings, the soundboard, and the air (the sound radiation). Hammer nonlinearities and hysteresis are included, along with realistic stiff strings with frequency-dependent damping, following the work of Chaigne and Askenfelt. [Chaigne and A. Askenfelt, ‘‘Numerical simulations of piano strings. I. A physical model for a struck string using finite difference methods,’’ J. Acoust. Soc. Am. 95, 1112–118 (1994); A. Chaigne, J. Acoust. 5, 181 (1992)]. The strings are coupled to a soundboard model that includes the added thickness and stiffness of ribs and bridges. The vibrating soundboard drives the room air, with walls modeled through a complex acoustic impedance following Botteldooren. [D. Botteldooren, ‘‘Acoustical finite-difference time-domain simulation in a quasi-Cartesian grid,’’ J. Acoust. Soc. Am. 95 2313–2319 (1994); 98, 3302 (1995)]. All of the calculations take a finite difference approach, producing a time domain computation. Calculated tones are presented and future directions discussed. [Work supported in part by NSF grant PHY-9988562.]