Analog model for thermoviscous propagation in a cylindrical tube

Abstract

Modeling acoustic propagation in tubes including the effects of thermoviscous losses at the tube walls is important in applications such as thermoacoustics, hearing aids, and wind musical instruments. Frequency dependent impedances for a tube transmission line model in terms of the so-called thermal and viscous functions are well established, and form the basis for frequency domain analysis of systems that include tubes. However, frequency domain models cannot be used for systems in which significant nonlinearities are important, as is the case with the pressure-flow relationship through the reed in a woodwind instrument. This paper describes a cylindrical tube model based on a continued fraction expansion of the thermal and viscous functions. The model can be represented as an analog circuit model which allows its use in time domain system modeling. This model avoids problems with fractional derivatives in the time domain.

Passive time-domain numerical models of viscothermal wave propagation in acoustic tubes of variable cross section

“…Such a balance follows from any passive circuit representation, and not merely the Foster forms shown in Fig. 4-the Cauer form given by Thompson et al 15 would give a similar balance. Even more generally, one could make use of any rational positive real approximations to Z v and Y h , though the circuit realisation is an invaluable tool and, as will be seen in Sec.…”

“…Such methods fall under the umbrella of geometric or structure-preserving integration techniques, dealt with in detail in the case of lossless ordinary differential equations by Hairer and Wanner, 12 and by various authors in the present case of partial differential equations for systems in acoustics. 13,14 For the Zwikker-Kosten model, the use of such methods is complicated somewhat by the need for a passive finite-order approximation to the viscothermal immittances; one such approximation has been presented recently by Thompson et al, 15 and another, suitable for optimization, by the present authors. 16 In this article, such approximations are used to design an energy-balanced timestepping method, and to derive numerical stability conditions.…”

“…0. Such an approach leads naturally to circuit representations, 15,24 which may be truncated to finite order in preparation for a numerical design.…”

“…In particular, a continued fraction expansion leads to a Cauer structure for both immittances. 15 Approximation error decreases with truncation order, but convergence can be slow, requiring a large number of elements. This is an important consideration in a time domain implementation, as the order of the approximation to viscothermal losses is a bottleneck in terms of computational costs.…”

A transmission-line model of back-cavity dynamics for in-plane pressure-differential microphones

“…For such designs, TL segments can be crafted to take advantage of prior studies where careful attention has been given to physics at the boundary layer. 11,13,22 …”