Asymptotic Dipole Radiation Formulas

Abstract

The analysis of the radiation from dipoles as given hy Sommerfeld and by von Hoerschelmann is deficient in one respect: it does not give the true <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup> asymptotic expressions for the radiation leaving at a considerable angle from the horizontal. The correct asymptotic formulas have already heen easily supplied by an appeal to the Reciprocal Theorem; lately M. J. O. Strutt <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> has got them directly from the boundary conditions and H. Weyl <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> has derived the correct asymptotic formula for a vertical dipole at the surface of the earth by a method quite different from Sommerfeld's. In the present paper it is shown how they can be got by merely improving the rigor of Sommerfeld's analysis.