Time-domain simulation of acoustical waveguides with arbitrarily spaced discontinuities

Abstract

The multiconvolution algorithm [Martı́nez et al., J. Acoust. Soc. Am. 84, 1620–1627 (1988)] to calculate the impulse response or reflection function of a musical instrument air column has proved to be useful, but it has the limitation that the spacing between discontinuites is constrained to be some multiple of cΔt (for phase velocity c and time step Δt). This paper presents an improved method, the continuous-time interpolated multiconvolution (CTIM), where such a limitation has been removed. The response of an air column, modeled as an arbitrary one-dimensional acoustic waveguide constructed using cylindrical or conical bore segments with viscothermal damping and tone-hole discontinuities, is obtained through continuous-time convolutions between analytical reflection and transmission functions and discrete-time pressure signals. The arbitrary spacing between discontinuites is accounted for by interpolation of the discrete-time pressure signals. Many musical instrument air columns possess tone holes that are opened or closed so that tones of different pitches are produced. A time-domain calculation is presented of the acoustic responses of tone-hole discontinuities that may be open or closed. The resulting reflection and transmission functions are well suited for use in the CTIM.