Advertisement

Upper and lower bounds for sloshing frequencies by intermediate problems

  • David W. Fox
  • James R. Kuttler
Original Papers

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.

Keywords

Lower Bound Mathematical Method Eigenvalue Problem Numerical Estimate Ideal Fluid 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    N. W. Bazley and D. W. Fox,Truncations in the method of intermediate problems for lower bounds to eigenvalues, J. Res. Nat. Bur. Stds.65 B 105–111 (1961).Google Scholar
  2. [2]
    N. W. Bazley and D. W. Fox,Lower bounds to eigenvalues using operator decompositions of the form B * B, Arch. Rat. Mech. Anal.10, 352–360 (1962).Google Scholar
  3. [3]
    D. W. Fox and W. C. Rheinboldt,Computational methods for determining lower bounds for eigenvalues of operators in Hilbert space, SIAM Rev.8, 427–462 (1966).Google Scholar
  4. [4]
    S. Gould,Variational methods for eigenvalue problems, 2nd ed., Toronto 1966.Google Scholar
  5. [5]
    J. R. Kuttler and V. G. Sigillito,Lower bounds for sloshing frequencies, Quart. Appl. Math.27, No. 3, 405–408 (1969).Google Scholar
  6. [6]
    H. Lamb,Hydrodynamics, 6th ed., Dover, New York 1945 (pp. 444–450).Google Scholar
  7. [7]
    P. Henrici, B. A. Troesch, and L. Wuytack,Sloshing frequencies for a half-space with circular or strip-like aperture, Z. angew. Math. Phys.21, 285–318 (1970).Google Scholar
  8. [8]
    J. T. Stadter,Bounds for eigenvalues of rhombical membranes, SIAM J. Appl. Math.14, 324–341 (1966).Google Scholar
  9. [9]
    B. A. Troesch,Free oscillations of a fluid in a container, Boundary problems in differential equations (R. E. Langer, ed.), University of Wisconsin Press, Madison 1960, pp. 279–299.Google Scholar
  10. [10]
    A. Weinstein and W. Stenger,Methods of intermediate problems for eigenvalues, Academic Press, New York 1972.Google Scholar

Copyright information

© Birkhäuser Verlag 1981

Authors and Affiliations

  • David W. Fox
    • 1
  • James R. Kuttler
    • 1
  1. 1.Milton S. Eisenhower Research Center, Applied Physics LaboratoryThe Johns Hopkins UniversityLaurelUSA

Personalised recommendations