Transfer matrix of a truncated cone with viscothermal losses: application of the WKB method

Jean Kergomard,Augustin Ernoult

doi:10.1051/aacus/2020005

Abstract

The propagation in tubes with varying cross section and wall visco-thermal effects is a classical problem in musical acoustics. To treat this aspect, the first method is the division in a large number of short cylinders. The division in short conical frustums with uniform averaged wall effects is better, but remains time consuming for narrow tubes and low frequencies. The use of the WKB method for the transfer matrix of a truncated cone without any division is investigated. In the frequency domain, the equations due to Zwikker and Kosten are used to define a reference result for a simplified bassoon by considering a division in small conical frustums. Then expressions of the transfer matrix at the WKB zeroth and the second orders are derived. The WKB second order is good at higher frequencies. At low frequencies, the errors are not negligible, and the WKB zeroth order seems to be better. This is due to a slow convergence of the WKB expansion for the particular case: the zeroth order can be kept if the length of the missing cone is large compared to the wavelength. Finally, a simplified version seems to be a satisfactory compromise.

Highlights

The calculation of the transfer matrix of wind instrument resonators is an old problem
The errors are not negligible, and the WKB zeroth order seems to be better. This is due to a slow convergence of the WKB expansion for the particular case: the zeroth order can be kept if the length of the missing cone is large compared to the wavelength
The WKB method leads to the possibility to compute the transfer matrix of a truncated cone without division of its length

Summary

Introduction

The calculation of the transfer matrix of wind instrument resonators is an old problem. A classical difficulty is the effect of boundary layers in tubes with variable cross section area, which depends on the radius For this reason, no analytic expression for the transfer matrix of a truncated cone with visco-thermal losses was established yet. The aim of the present paper is to use the well known Wentzel–Kramers–Brillouin (WKB) method, in order to reduce the computing time by limiting the division of a truncated cone to one segment. For this purpose, it is necessary to state the wave equation with visco-thermal effects without term of first-order (space) derivative. Reducing the computing time is not useful for a single input impedance computation, but this becomes useful for applications such as optimization

Objectives

Discussion

Conclusion