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Eigenfrequencies of an elliptic membrane. (English) Zbl 0271.35017

MSC:
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35P05 General topics in linear spectral theory for PDEs
74K15 Membranes
35J25 Boundary value problems for second-order elliptic equations
70J10 Modal analysis in linear vibration theory
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References:
[1] Tables Relating to Mathieu Functions, Nat. Bur. Standards, Appl. Math. Ser., vol. 59, Washington, D. C., 1967.
[2] S. D. Daymond, The principal frequencies of vibrating systems with elliptic boundaries, Quart. J. Mech. Appl. Math. 8 (1955), 361 – 372. · Zbl 0064.37702 · doi:10.1093/qjmam/8.3.361 · doi.org
[3] 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, Second edition, Published for Scientific Computing Service Ltd., London, by Addison-Wesley Publishing Co., Inc., Reading, Mass., 1962. A. Fletcher, J. C. P. Miller, L. Rosenhead, and L. J. Comrie, An index of mathematical tables. Vol. II. Part II: Bibliography. Part III: Errors. Part IV: Index to Introduction and Part I, Second edition, Published for Scientific Computing Service Ltd., London, by Addison-Wesley Publishing Co., Inc., Reading, Mass., 1962.
[4] J. G. Herriot, The Principal Frequency of an Elliptic Membrane, Department of Mathematics, Stanford University Report, 31 August 1949. Contract N6-ORI-106.
[5] 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.
[6] E. T. Kirkpatrick, Tables of values of the modified Mathieu functions, Math. Comput. 14 (1960), 118 – 129. · Zbl 0098.32010
[7] E. Mathieu, ”Mémoire sur le mouvement vibratoire d’une membrane de forme elliptique,” J. Math. Pures Appl., v. 13, 1868, pp. 137-203.
[8] N. W. McLachlan, Theory and application of Mathieu functions, Dover Publications, Inc., New York, 1964. · Zbl 0128.29603
[9] 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). · Zbl 0058.29503
[10] 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. · Zbl 0044.38301
[11] 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).
[12] Irene A. Stegun and Milton Abramowitz, Generation of Bessel functions on high speed computers, Math. Tables Aids Comput 11 (1957), 255 – 257. · Zbl 0084.12101
[13] G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, England; The Macmillan Company, New York, 1944. · Zbl 0063.08184
[14] J. C. Wiltse & M. J. King, Values of the Mathieu Functions, Johns Hopkins University Radiation Laboratory, Techn. Report No. AF-53, Baltimore, Md., 1958.
[15] J. W. Wrench, Jr., Private communication.
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