Numerical simulations of xylophones. II. Time-domain modeling of the resonator and of the radiated sound pressure

This paper presents a time-domain modeling for the sound pressure radiated by a xylophone and, more generally, by mallet percussion instruments such as the marimba and vibraphone, using finite difference methods. The time-domain model used for the one-dimensional (1-D) flexural vibrations of a nonuniform bar has been described in a previous paper by Chaigne and Doutaut [J. Acoust. Soc. Am. 101, 539–557 (1997)] and is now extended to the modeling of the sound-pressure field radiated by the bar coupled with a 1-D tubular resonator. The bar is viewed as a linear array of equivalent oscillating spheres. A fraction of the bar field excites the tubular resonator which, in turn, radiates sound with a certain delay. In the present model, the open end of the resonator is represented by an equivalent pulsating sphere. The total sound field is obtained by summing the respective contributions of the bar and tube. Particular care is given for defining a valid approximation of the radiation impedance, both in continuous and discrete time domain, on the basis of Kreiss’s theory. The model is successful in reproducing the main features of real instruments: sharp attack, tuning of the bar, directivity, tone color, and aftersound due to the bar-resonator coupling.