Volume 4, Number 2, 2020
|Number of page(s)||11|
|Published online||12 May 2020|
Technical & Applied Article
Transfer matrix of a truncated cone with viscothermal losses: application of the WKB method
Magique 3D Team, Inria Bordeaux Sud Ouest, 200 Avenue de la Vieille Tour, 33405 Talence Cedex, France
2 Aix Marseille University, CNRS, Centrale Marseille, LMA UMR 7031, 13013 Marseille, France
* Corresponding author: firstname.lastname@example.org
Accepted: 23 April 2020
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.
Key words: Conical tubes / Visco-thermal effects / WKB method
© A. Ernoult and J. Kergomard, Published by EDP Sciences, 2020
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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