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: 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 \[ {A_r} = {C_1}\left [ {mr{J_0}\left ( {pr} \right )\cos \left ( {mz} \right ) + pz{J_1}\left ( {pr} \right )\sin \left ( {mz} \right )} \right ]{e^{ - iwt}},\] 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.
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E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, (MacMillan Co., New York, 1948), American edition
H. Bateman, Partial Differential Equations of Mathematical Physics, (Dover Publications, Inc., New York, 1944), first American edition
E. Kamke, Differentialgleichungen Lösungsmethoden und Lösungen, (Chelsea Publishing Co., New York, 1942)
A. G. Webster, Partial Differential Equations of Mathematical Physics, (Dover Publications, Inc., 1955), reprint of second corrected edition
A. Sommerfeld, Partial Differential Equations in Physics, (Academic Press, New York, 1949)
M. G. Salvadori and R. J. Schwarz, Differential Equations in Engineering Problems, (Prentice-Hall, Englewood Cliffs, New Jersey, 1954)
P. Moon and D. E. Spencer, Foundations of Electrodynamics, (D. Van Nostrand Co., Inc., Princeton, New Jersey, 1960)
P. Moon and D. E. Spencer, Field Theory for Engineers, (D. Van Nostrand Co., Inc., Princeton, New Jersey, 1961)
P. Moon and D. E. Spencer, J. Franklin Inst., 256 (1953) 551
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Article copyright:
© Copyright 1965
American Mathematical Society