Abstract
This article presents very good, rigorous, numerical estimates for the “sloshing” eigenvalues of two families of two-dimensional regions. For each region the eigenvalues are proportional to the square of the frequencies of small lateral oscillations of an ideal fluid under the influence of gravity in a canal or in a horizontal cylindrical tank that has the region as cross-section. Our estimates are obtained by using conformal transformations that carry rectangles onto the regions, and then by employing intermediate problems with a generalized special choice to find upper and lower bounds to eigenvalues. The transformed problems are analogous to certain nonuniform string eigenvalue problems we estimate in a similar way.
Résumé
On calcule des bornes inférieures et supérieures aux fréquences d'oscillation latérale d'un liquide idéal sous l'influence de la gravité dans un réservoir cylindrique ou dans un canal. Les carrés des fréquences sont proportionnnels aux valeurs propres d'un opérateur aux dérivées partielles dans un ensemble de deux dimensions qui prend la forme de la section du liquide. Pour déterminer les bornes aux valeurs propres, on se sert des fonctions analytiques qui transforment les sections en rectangles ainsi que de problèmes intermédiaires choisis spécialement sous un aspect général. Les problèmes aux valeurs propres résultant des transformations sont analogues à certains problèmes de cordes vibrantes pour lesquels on peut calculer des bornes aux valeurs propres de façon semblable.
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This work was supported by the Department of the Navy, Naval Sea Command under Contract No. N 00024-81-C-5301.
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Fox, D.W., Kuttler, J.R. Upper and lower bounds for sloshing frequencies by intermediate problems. Z. angew. Math. Phys. 32, 667–682 (1981). https://doi.org/10.1007/BF00946978
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DOI: https://doi.org/10.1007/BF00946978