An acoustical interpretation of the zeroes of ultraspherical polynomials

Georges Le Vey

doi:10.1016/j.crme.2016.01.005

An acoustical interpretation of the zeroes of ultraspherical polynomials

Georges Le Vey ¹

¹ IRCCyN, UMR CNRS 6597, École des mines de Nantes, 4, rue Alfred-Kastler, 44307 Nantes cedex 1, France

Comptes Rendus. Mécanique, Volume 344 (2016) no. 6, pp. 434-438.

Abstract
Résumé

In 1887, T.J. Stieltjes gave an electrostatical interpretation of the zeroes of Jacobi polynomials. This was extended later to Laguerre and Hermite polynomials by G. Szegö. An analogous interpretation is given here for ultraspherical polynomials in terms of piecewise cylindrical acoustical resonators.

En 1887, T.J. Stieltjes a donné une interprétation des zéros des polynômes de Jacobi, en termes d'équilibre d'un système de charges électriques. L'extension aux polynômes de Laguerre et de Hermite en fut donnée ensuite par G. Szegö. Nous donnons ici une interprétation analogue des zéros des polynômes ultrasphériques en termes des fréquences de résonance de résonateurs acoustiques cylindriques par morceaux.

Received: 2015-11-12
Accepted: 2016-01-11
Published online: 2016-03-31

DOI: 10.1016/j.crme.2016.01.005

Keywords: Acoustics, Acoustical resonators, Ultraspherical polynomials, Orthogonal polynomials roots, Resonance frequencies, Plane wave propagation
Mots-clés : Acoustique, Résonateurs acoustiques, Polynômes ultrasphériques, Racines des polynômes orthogonaux, Fréquences de résonance, Propagation par ondes planes

Author's affiliations:

Georges Le Vey ¹

¹ IRCCyN, UMR CNRS 6597, École des mines de Nantes, 4, rue Alfred-Kastler, 44307 Nantes cedex 1, France

@article{CRMECA_2016__344_6_434_0,
     author = {Georges Le Vey},
     title = {An acoustical interpretation of the zeroes of ultraspherical polynomials},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {434--438},
     publisher = {Elsevier},
     volume = {344},
     number = {6},
     year = {2016},
     doi = {10.1016/j.crme.2016.01.005},
     language = {en},
}

TY  - JOUR
AU  - Georges Le Vey
TI  - An acoustical interpretation of the zeroes of ultraspherical polynomials
JO  - Comptes Rendus. Mécanique
PY  - 2016
SP  - 434
EP  - 438
VL  - 344
IS  - 6
PB  - Elsevier
DO  - 10.1016/j.crme.2016.01.005
LA  - en
ID  - CRMECA_2016__344_6_434_0
ER  -

%0 Journal Article
%A Georges Le Vey
%T An acoustical interpretation of the zeroes of ultraspherical polynomials
%J Comptes Rendus. Mécanique
%D 2016
%P 434-438
%V 344
%N 6
%I Elsevier
%R 10.1016/j.crme.2016.01.005
%G en
%F CRMECA_2016__344_6_434_0

Georges Le Vey. An acoustical interpretation of the zeroes of ultraspherical polynomials. Comptes Rendus. Mécanique, Volume 344 (2016) no. 6, pp. 434-438. doi : 10.1016/j.crme.2016.01.005. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2016.01.005/

References
Cited by

[1] J.-P. Dalmont; J. Kergomard Lattices of sound tubes with harmonically related eigenfrequencies, Acta Acust., Volume 2 (1994) no. 5, pp. 421-430

[2] G. Le Vey Graph modelling of musical wind instruments: a method for natural frequencies computation, Acta Acust. Acust., Volume 101 (2015) no. 6, pp. 1222-1233

[3] J.-P. Dalmont; G. Le Vey New lattices of sound tubes with harmonically related eigenfrequencies, ISMA'2014 (2014)

[4] T.J. Stieltjes Sur les racines de l'équation $X_{n} = 0$ , Acta Math., Volume 9 (1887) no. 1, pp. 385-400

[5] G. Szegö Orthogonal Polynomials, Colloquium Publications, vol. XXIII, American Mathematical Society, Providence, RI, USA, 1939 (1975)

[6] R. Koekoek; P.A. Lesky; R.F. Swarttouw Hypergeometric Orthogonal Polynomials and Their q-Analogues, Springer Monographs in Mathematics, Springer Verlag, 2010

Cited by Sources:

Comments - Policy