Series for the Wave Function of a Radiating Dipole at the Earth's Surface

T HE wave function above the earth of a vertical dipole placed at the surface of a plane earth is 1 tj / Z) = fu . W + i u i r°° Jo M ? ~ ^ + k M - itf ' (1> where r and z are the horizontal and vertical distances from the dipole. ki and fo are constants depending upon the electrical properties of the air and ground, respectively.2 We shall be concerned with the value of this function at the surface of the earth. Setting z -- 0 gives us an integral for IIi(r, 0) which is the function of r to be investigated here. Although the electric and magnetic intensities are the properties of an electromagnetic field which have the greatest physical significance, writers on this subject often deal with the wave function because of its simpler form and because in many cases of practical interest it is nearly proportional to the electric intensity. However, the electromagnetic field may be obtained from the wave function by differentiation. If the real parts of He~iul and Ee~iwt represent the electric and magnetic intensities the field above the earth produced by the dipole is IIr = II, = 0, = 1 2