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Asymptotic dipole expansions for small horizontal angles

Published online by Cambridge University Press:  24 October 2008

F. H. Murray
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Extract

An approximate calculation of the electromagnetic field of a vertical dipole at the surface of a conducting earth, for small angles of the radius vector with the horizontal, was given by Sommerfield; the case of large angles with the horizontal has been studied by a number of writers. It is proposed here to develop formulae for the vertical dipole by a method which takes into account the singularities of the integrand of a certain integral more accurately than is done by Sommerfield; the analysis is developed especially for small horizontal angles and small numerical distances.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 28 , Issue 4 , October 1932 , pp. 433 - 441
Copyright
Copyright © Cambridge Philosophical Society 1932

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References

* Ann. der Phys., 28, p. 665 (1909)Google Scholar; 81, p. 1135. See also Weyl, H., Ann. der Phys., 60, p. 481 (1919)CrossRefGoogle Scholar; Wise, H., Bell System Tech. Journal, 8, p. 662 (1929)CrossRefGoogle Scholar; Strutt, M. J. O., Ann. der Phys. (5), 1, p. 721 (1929)CrossRefGoogle Scholar; 4, p. 1 (1930).

† Wise, loc. cit., Bateman, , Electrical and Optical Wave Motion (1915), p. 75.Google Scholar

* Riemann-Weber, , Differentialgleichungen der Physik (1927), ii, p. 550.Google Scholar

* Wise, loc. cit., p. 666.

* Wise, H., Proc. Inst. Radio Engineers, 09. 1931, 19, pp. 1684–9Google Scholar, has developed another expansion for this problem, valid for large |k 1r|; the two expansions have somewhat different regions of rapid convergence.

* Watson, , Theory of Bessel Functions (1922), equation (8), p. 78.Google Scholar