Continued fractions and passive ladder networks for acoustic impedances: the baffled piston and the horn
Abstract
Lumped-element networks give transparent, passive models of acoustic impedances whose low- and high-frequency limits can be read from the circuit, but a systematic, arbitrarily accurate construction for non-rational impedances is lacking. Continued fractions, one- and two-point Padé approximants and their identity with ladder networks and Cauer synthesis are introduced in tutorial form and applied to two examples: the radiation impedance of a baffled circular piston and the throat impedance of a horn with a piston mouth. The even contraction of the piston's continued fraction has positive coefficients, certified by interval arithmetic for the first 100 pairs, and yields a ladder of positive resistances, masses and compliances; with 14 elements the error is 4e-4 up to ka = 5. Beranek's four-element network is, to about 2%, the order-two two-point Padé approximant. A horn modelled as a cascade of waveguide sections is a continued fraction whose tail is the mouth impedance; the infinite exponential horn is a periodic continued fraction, an exact high-pass ladder. The piston networks are passive at every certified order (up to 202 elements) and are extended without changing existing elements; for a finite horn the error of the mouth network passes one to one to the throat impedance.
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BibTeX
@misc{aarts2026continued,
title = {{Continued fractions and passive ladder networks for acoustic impedances: the baffled piston and the horn}},
author = {Ronald M. Aarts},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.20371},
eprint = {2610.20371},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.20371},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Continued fractions and passive ladder networks for acoustic impedances: the baffled piston and the horn
AU - Ronald M. Aarts
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.20371
UR - https://arxiv.org/abs/2610.20371
AB - Lumped-element networks give transparent, passive models of acoustic impedances whose low- and high-frequency limits can be read from the circuit, but a systematic, arbitrarily accurate construction for non-rational impedances is lacking. Continued fractions, one- and two-point Padé approximants and their identity with ladder networks and Cauer synthesis are introduced in tutorial form and applied to two examples: the radiation impedance of a baffled circular piston and the throat impedance of a horn with a piston mouth. The even contraction of the piston's continued fraction has positive coefficients, certified by interval arithmetic for the first 100 pairs, and yields a ladder of positive resistances, masses and compliances; with 14 elements the error is 4e-4 up to ka = 5. Beranek's four-element network is, to about 2%, the order-two two-point Padé approximant. A horn modelled as a cascade of waveguide sections is a continued fraction whose tail is the mouth impedance; the infinite exponential horn is a periodic continued fraction, an exact high-pass ladder. The piston networks are passive at every certified order (up to 202 elements) and are extended without changing existing elements; for a finite horn the error of the mouth network passes one to one to the throat impedance.
ER -