Alternatives to the impulse response h(t) to describe the acoustical behavior of conical ducts

Abstract

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.