January 4, 2025
In late spring 2024, I revisited the topic of lumped element modelling of microacoustical effects via three different approaches. One was the Padé approximant approach, where a paper was published previously (https://doi.org/10.1121/10.0005824) for the slit geometry. As there are several mistakes in this paper, I had previously looked at correcting these, but deciphering exactly where these mistakes were made had proven difficult in the past as some information was missing from the article. So, I started from scratch, and redid all of the work in the article, found the errors that related to both implementation of circuit and fitting of circuit component values, and finally I could correct all of these and come up with a new circuit and correct values. This was first done for the slit geometry, but then there was a straight-forward path towards doing it also for the circular cross-section tube and for the equilateral triangular cross-section tube. My new circuits are so-called 'positive-real', whereas the previous lumped circuit for slit violates passivity, and so the new circuits can be used for both steady-state solutions and for time-domain solutions.
While working on this approach, I also revisited an alternative approach, namely the continued fraction method. This had been demonstrated previously for the circular cross-section tube (https://doi.org/10.1121/1.4861237), but having never really worked with continued fractions, I did not get into the details at the time of publication. However, in working with the Padé approach, I found it useful to compare the two method, and in recreating the continued fraction work, I noticed that just as the Bessel functions related to the circular cross-section can be described via an infinite continued fractions, so can the hyperbolic trigonometric functions related to the slit and the equilateral triangular cross-sections. Hence, I could now finally establish complete lumped circuits for these cross-sections created via continued fractions, and these circuits, when truncated to a desired order related to the maximum frequency of interest, are guaranteed to be positive-real.
Having now both the Padé approximant circuits to a particular chosen order and the continued fraction circuits to any order as it is trivially expanded, for all three prototypical cross-section, I decided to publish the continued fraction approach first, as it is very elegant, and wait with the Padé approximant paper to 2025, as the method is both less relevant in cases such as the three present ones where a continued fraction can be established and also due to having to clear up the mistakes in a previous paper in a respectful but rigorous manner.
It should be mentioned that at least two alternative approaches are also relevant. One is the Partial Fraction Fitting approach, which has the upside of being applicable to any cross-section, but with the downside of being numerical, not analytical, and so new component values must be calculated for each new sizing change, change in medium, or similar. Another method to be investigated is the Mittag-Leffler theorem approach, but it is too early to say if this will lead to unique circuits.
In conclusion, the continued fraction approach is probably the only one needed for anyone wanting to model these microacoustical effects with lumped circuits having constant and positive component values across a wide frequency range, in contrast to previous methods that either obtained such circuits as a low-frequency approximation, or as a high-frequency approximation with frequency-dependent component values.
My paper "Circuit models for thermoviscous acoustics in waveguides of various cross sections via continued fractions" was published in JASA in December 2024, which coincidently was the last printed version of the publication:
With the following statement from ASA after the publication notice, I am allowed to attach the paper below: "Under ASA policy, you have the right to post this article on your personal web page or your employer's website, provided you include the appropriate copyright and a web link to ASA's official online abstract, as indicated by the URL above."
Copyright notice:
Copyright 2024 Acoustical Society of America. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the Acoustical Society of America.
The following article appeared in J. Acoust. Soc. Am. 156, 3930–3942 (2024), and may be found at https://doi.org/10.1121/10.0034550
For those of you who read Danish, I have made an overview of the work done in the following presentation:
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