Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X

   
 
 

 

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, Inc., New York, 1944), first American edition MR 0010909
  • [3] E. Kamke, Differentialgleichungen Lösungsmethoden und Lösungen, (Chelsea Publishing Co., New York, 1942)
  • [4] A. G. Webster, Partial Differential Equations of Mathematical Physics, (Dover Publications, Inc., 1955), reprint of second corrected edition MR 0073814
  • [5] A. Sommerfeld, Partial Differential Equations in Physics, (Academic Press, New York, 1949) MR 0029463
  • [6] M. G. Salvadori and R. J. Schwarz, Differential Equations in Engineering Problems, (Prentice-Hall, Englewood Cliffs, New Jersey, 1954)
  • [7] P. Moon and D. E. Spencer, Foundations of Electrodynamics, (D. Van Nostrand Co., Inc., Princeton, New Jersey, 1960) MR 0118275
  • [8] P. Moon and D. E. Spencer, Field Theory for Engineers, (D. Van Nostrand Co., Inc., Princeton, New Jersey, 1961) MR 0121018
  • [9] P. Moon and D. E. Spencer, J. Franklin Inst., 256 (1953) 551 MR 0058038

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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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