The input impulse response, or related functions such as the plane-wave or spherical-wave reflection functions, of a conical bore with discontinuities can be calculated from the reflection functions associated with discontinuities by means of a multiconvolution process. This approach, as alternative to the Fourier transform of the acoustical impedance Z(,f), is attractive because of the detailed description it gives of the pressure time evolution inside the bore. However, the calculation may become rather involved if each reflected and transmitted wave is followed individually, and numerical instability may arise whenever the implied reflection functions contain growing exponentials. The first difficulty has been overcome by summing up all outward traveling waves and all inward traveling waves to give a unique outward wave and a unique inward wave. Numerical instability has been avoided by using an iterative convolution algorithm. Internal damping due to viscous and thermal losses has been located at discontinuities and represented by means of damping functions through convolution products. The reflection functions used are presented in the companion article "Conical bores. Part I: Reflection functions associated with discontinuities" [Martinez and Agul16, J. Acoust. Soc. Am. 84, 1613-1619 (1988) ].
Reflection functions associated with discontinuities in conical bores are worked out for the usual cases found in woodwind instruments: taper and diameter discontinuities, open and closed ends, and open and closed holes. Radiation at open ends is considered through the impedance formulation for the open end of a cylindrical tube fitted with an infinite flange. The input impulse response of woodwinds, or related functions such as the plane-wave or sphericalwave reflection functions, can be calculated in the time domain by means of these reflection functions through multiple convolutions as shown in the companion article "Conical bores.
An experimental method to measure the reflection functions for discontinuities in cylindroconical waveguides is presented. It involves a single discontinuity and uses a pulselike incident signal. FIR techniques are used to design the incident signal and signal processing is made in the time domain. Two methods to stabilize the numerical deconvolution are considered, those based on singular value decomposition, and the conjugated gradient method. Results presented are related to taper discontinuities, but the method can be applied to other discontinuities as well.
In unidimensional acoustical systems, the impulse response h(t) at the input section, which describes the pressure evolution originated at this section by the introduction of a flow unit impulse through it, relates pressure p(t) and flow u(t) at the input section by means of the convolution product p=h*u. If damping and radiation are small, it is interesting to find other functions of faster decay than h(t) in order to improve the convolution convergence. As alternatives to h(t), this article studies the impulse responses h′(t) and h″(t), which correspond to the modified systems that result when coupling the original acoustical system input section to a cylindrical anechoic termination and a conical anechoic termination, respectively. These functions h′(t) and h″(t) are related to the plane-wave reflection function Rp−(t) and spherical-wave reflection function Rs−(t), respectively. The comparison of these three impulse responses shows that the use of h′(t) and h″(t), though of faster decay than h(t) in principle, is not always advisable. For conical bores with small truncation, the convolution with h′(t) may lead to numerical instability more easily than that with h(t). On the other hand, the impulse response h″(t) turns out to be divergent for certain geometrical duct configurations, and thus it is useless as a kernel function in a convolution in these cases.
Traditionally the description of the acoustical behavior of a bore has been done in the frequency domain through the acoustical impedance or the reflection coefficient. Whenever the time-domain response of it (impulse response, reflection function) has been needed, it has usually been obtained through the Fourier transform (FT) of the corresponding frequencydomain function. However, there are a number of cases in which this approach is unsuitable because the FT leads to noncausal functions which cannot be understood physically as impulse responses or reflection functions. In such cases the time-domain calculation is unavoidable. This calculation leads to exponentially growing functions which obviously do not accept an FT. This article presents the time-domain calculation of the main basic functions (reflection and transmission functions associated with a single discontinuity, impulse responses of anechoic bores) which are needed to obtain the input reflection function and the impulse responses of a bore. The analytical calculation of an application case is also presented.
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