Time-domain simulation of acoustical waveguides with arbitrarily spaced discontinuities

Abstract

The multiconvolution algorithm ͓Martínez et al., J. Acoust. Soc. Am. 84, 1620-1627 ͑1988͔͒ 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 c⌬t ͑for phase velocity c and time step ⌬t͒. This paper presents an improved method, the continuous-time interpolated multiconvolution ͑CTIM͒, 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. INTRODUCTIONThe simulation of the acoustical behavior of wind instruments requires the dynamical description of the main two parts of such systems, the bore and the driver, and their mutual interaction. As the driver is essentially a nonlinear system, it is always described in the time domain through a differential equation. For that reason, even if the bore behaves, under playing conditions, approximately as a linear system, so that its description can be made either in time domain or in frequency domain, it is a time-domain response ͓usually its input impulse response h(t)͔ that is used when studying the bore-to-driver feedback.There are mainly two techniques to calculate h(t): either as the inverse Fourier transform ͑FT͒ of the corresponding frequency response ͓the input impedance Z()͔ or directly in the time domain. The first technique implies all the inconveniences associated with numerical FTs. To overcome these difficulties, Martínez et al. ͑1988a͒ proposed a direct method in the time domain through a numerical multiconvolution process which has proven to be useful. However, that algorithm has a limitation: the time step ⌬t used in the calculation has to be small enough so that each of the spacings between discontinuities along the bore ͑changes in diameter, in taper, open and closed tone holes etc.͒ is an exact multiple of c⌬t ͑where c is the phase velocity of sound in air neglecting dissipation͒. Smith ͑1992͒ worked with a similar idea to simulate wave propagation in strings. There is a brief discussion of a band-limited interpolation method to treat the case where ⌬t is not an exact multiple of c⌬t, but there is no discussion of how this method is generalized to model wave propagation in conical waveguides with discontinuities r...

Discrete-time modelling of musical instruments

“…The tube in wind instruments can be cylindrical or conical bore, and it can more or less expand in diameter towards the radiation end of the tube (called the bell for clearly expanding cases). Modelling of the instrument bores has been studied in [19,61,70,77,231,246,248,283,287,303,304,306,307]. Conical and exponentially expanding bores are more demanding to simulate than the cylindrical ones [19,70,77,231,248,267,283,287,303,304,306,307].…”

“…Modelling of the instrument bores has been studied in [19,61,70,77,231,246,248,283,287,303,304,306,307]. Conical and exponentially expanding bores are more demanding to simulate than the cylindrical ones [19,70,77,231,248,267,283,287,303,304,306,307].…”

From a Model of Lossy Flared Pipes to a General Framework for Simulation of Waveguides

“…Second, for (M2) with ǫ = 0, the corresponding continuoustime impulse response h 2 involves decreasing exponentials (cf. [24], [33]) and digital versions involve first order filters so that the computational cost for one piece of pipe is twice greater for cones than for cylinders. To preserve a similar global computational cost and since the discontinuity of h 2 is smoothed by exponential responses, the number of pieces of pipe is chosen as K = 64 so that δz 2 = 2δz 1 .…”

Wave-based Simulation of Wind Instrument Resonators

“…In their original "multiconvolution algorithm", the spacing between discontinuities is constrained to be a multiple of the spatial sampling interval. This limitation was later removed by Barjau et al [27]. There are strong similarities between the wave digital modeling approach and the multiconvolution approach.…”