Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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Non-separable solutions of the Helmholtz wave equation


Author: Donald S. Moseley
Journal: Quart. Appl. Math. 22 (1965), 354-357
MSC: Primary 35.75
DOI: https://doi.org/10.1090/qam/183970
MathSciNet review: 183970
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Abstract | References | Similar Articles | Additional Information

Abstract: A set of solutions not obtainable by the method of separation of variables is presented for the vector Helmholtz wave equation in circular cylindrical coordinates limited to non-angular dependence. These are constructed of Bessel and trigonometric functions. For example, if A is the vector, the $ r$-component of the simplest member of the set is

$\displaystyle {A_r} = {C_1}\left[ {mr{J_0}\left( {pr} \right)\cos \left( {mz} \right) + pz{J_1}\left( {pr} \right)\sin \left( {mz} \right)} \right]{e^{ - iwt}},$

where $ {C_1}$ is an arbitrary constant, $ m$ and $ p$ are propagation constants, and $ \omega $ is angular frequency. Brief reference is made to three-dimensional solutions in rectangular coordinates.

References [Enhancements On Off] (What's this?)

  • [1] E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, (MacMillan Co., New York, 1948), American edition
  • [2] H. Bateman, Partial Differential Equations of Mathematical Physics, Dover Publications, New York, N.Y., 1944. MR 0010909
  • [3] E. Kamke, Differentialgleichungen Lösungsmethoden und Lösungen, (Chelsea Publishing Co., New York, 1942)
  • [4] Arthur Gordon Webster, Partial differential equations of mathematical physics, Dover Publications, Inc., New York, 1955. Edited by Samuel J. Plimpton; 2d ed. MR 0073814
  • [5] Arnold Sommerfeld, Partial Differential Equations in Physics, Academic Press, Inc., New York, N. Y., 1949. Translated by Ernst G. Straus. MR 0029463
  • [6] M. G. Salvadori and R. J. Schwarz, Differential Equations in Engineering Problems, (Prentice-Hall, Englewood Cliffs, New Jersey, 1954)
  • [7] Parry Moon and Domina Eberle Spencer, Foundations of electrodynamics, The Van Nostrand Series in Electronics and Communications, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR 0118275
  • [8] Parry Moon and Domina Eberle Spencer, Field theory for engineers, The Van Nostrand Series in Electronics and Communications, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1961. MR 0121018
  • [9] Parry Moon and Domina Eberle Spencer, The meaning of the vector Laplacian, J. Franklin Inst. 256 (1953), 551–558. MR 0058038, https://doi.org/10.1016/0016-0032(53)91160-0

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Additional Information

DOI: https://doi.org/10.1090/qam/183970
Article copyright: © Copyright 1965 American Mathematical Society


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