Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-28T14:05:32.447Z Has data issue: false hasContentIssue false
  • English
  • Français

Singularities of two-dimensional exterior solutions of the Helmholtz equation

Published online by Cambridge University Press:  24 October 2008

R. F. Millar
R. F. Millar
Affiliation:
Radio and Electrical Engineering Division, National Research Council, Ottawa, Canada
Get access Rights & Permissions [Opens in a new window]

Abstract

A technique for locating possible singularities of two-dimensional ex-terior harmonic functions was discussed in a previous paper. In the present work, the method is generalized to exterior solutions of the Helmholtz equation. Although the procedure deviates in some of its details from the earlier exposition, the conclusions are similar. In particular, it is verified that solutions of the Laplace and Helmholtz equations that satisfy the same Dirichlet boundary condition on the same boundary, possess the same convex hull of singularities. The possibility of extending the method to more general equations is raised.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 69 , Issue 1 , January 1971 , pp. 175 - 188
Copyright
Copyright © Cambridge Philosophical Society 1971

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Courant, R. and Hilbert, D.Methods of mathematical physics, volume II (Interscience Publishers, a division of John Wiley and Sons; New York, 1962).Google Scholar
(2)Garabedian, P. R.Partial differential equations (John Wiley and Sons, Inc.; New York, 1964).Google Scholar
(3)Lewy, H.On the reflection laws of second order differential equations in two independent variables. Bull. Amer. Math. Soc. 65 (1959), 37–58.Google Scholar
(4)Millar, R. F.On the Rayleigh assumption in scattering by a periodic surface. Proc. Cambridge Philos. Soc. 65 (1969), 773–791.Google Scholar
(5)Millar, R. F.Rayleigh hypothesis in scattering problems. Electron. Lett. 5 (1969), 416–418.Google Scholar
(6)Millar, R. F. and Bates, R. H. T.On the legitimacy of an assumption underlying the point-matching method. IEEE Trans. on Microwave Theory and Techniques (Correspondence), to appear. MTT - 18 (1970), 325–327.Google Scholar
(7)Millar, R. F.The location of singularities of two-dimensional harmonic functions. I. Theory. SIAM J. Math. Anal. 1 (1970), 333–344.Google Scholar
(8)Millar, R. F.The location of singularities of two-dimensional harmonic functions. II. Applications. SIAM J. Math. Anal. 1 (1970), 345–353.Google Scholar
(9)Miller, R. F.On the Rayleigh assumption in scattering by a periodic surface. II. Proc. Cambridge Philos. Soc. 69 (1970), 217–225.Google Scholar
(10)Müller, C.Zur Methode der Strahlungskapazität von H. Weyl. Math. Z. 56 (1952), 80–83.Google Scholar