Eigenfrequencies of an elliptic membrane
HTML articles powered by AMS MathViewer
- by B. A. Troesch and H. R. Troesch PDF
- Math. Comp. 27 (1973), 755-765 Request permission
Abstract:
The first few eigenfrequencies of a homogeneous elliptic membrane, which is fixed along its boundary, are given in a graph. It is explained in detail, how more accurate results can readily be obtained for special purposes. The known expansion of the eigenfrequencies for small and large eccentricities are summarized. As an application some nodal patterns for a membrane with a double eigenvalue are presented.References
-
Tables Relating to Mathieu Functions, Nat. Bur. Standards, Appl. Math. Ser., vol. 59, Washington, D. C., 1967.
- S. D. Daymond, The principal frequencies of vibrating systems with elliptic boundaries, Quart. J. Mech. Appl. Math. 8 (1955), 361–372. MR 75686, DOI 10.1093/qjmam/8.3.361
- A. Fletcher, J. C. P. Miller, L. Rosenhead, and L. J. Comrie, An index of mathematical tables. Vol. I: Introduction. Part I: Index according to functions, 2nd ed., Published for Scientific Computing Service Ltd., London, by Addison-Wesley Publishing Co., Inc., Reading, Mass., 1962. MR 0142796 J. G. Herriot, The Principal Frequency of an Elliptic Membrane, Department of Mathematics, Stanford University Report, 31 August 1949. Contract N6-ORI-106. M. J. King & J. C. Wiltse, Derivatives, Zeros and Other Data Pertaining to Mathieu Functions, Johns Hopkins University Radiation Laboratory, Techn. Report No. AF-57, Baltimore, Md., 1958.
- E. T. Kirkpatrick, Tables of values of the modified Mathieu functions, Math. Comput. 14 (1960), 118–129. MR 0113288, DOI 10.1090/S0025-5718-1960-0113288-4 E. Mathieu, "Mémoire sur le mouvement vibratoire d’une membrane de forme elliptique," J. Math. Pures Appl., v. 13, 1868, pp. 137-203.
- N. W. McLachlan, Theory and application of Mathieu functions, Dover Publications, Inc., New York, 1964. MR 0174808
- Josef Meixner and Friedrich Wilhelm Schäfke, Mathieusche Funktionen und Sphäroidfunktionen mit Anwendungen auf physikalische und technische Probleme, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, Band LXXI, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1954 (German). MR 0066500 G. Pólya & G. Szegö, Isoperimetric Inequalities in Mathematical Physics, Ann. of Math. Studies, no. 27, Princeton Univ. Press, Princeton, N.J., 1951. MR 13, 270.
- Maria Josepha De Schwarz, Determinazione delle frequenze e delle linee nodali di una membrana ellitica oscillante con contorno fisso, Atti Accad. Sci. Fis. Mat. Napoli (3) 3 (1960), no. 2, 17 (Italian). MR 0126994
- Irene A. Stegun and Milton Abramowitz, Generation of Bessel functions on high speed computers, Math. Tables Aids Comput. 11 (1957), 255–257. MR 93939, DOI 10.1090/S0025-5718-1957-0093939-3
- G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, England; The Macmillan Company, New York, 1944. MR 0010746 J. C. Wiltse & M. J. King, Values of the Mathieu Functions, Johns Hopkins University Radiation Laboratory, Techn. Report No. AF-53, Baltimore, Md., 1958. J. W. Wrench, Jr., Private communication.
Additional Information
- © Copyright 1973 American Mathematical Society
- Journal: Math. Comp. 27 (1973), 755-765
- MSC: Primary 73.65
- DOI: https://doi.org/10.1090/S0025-5718-1973-0421276-2
- MathSciNet review: 0421276