The Diffraction of a Circularly Symmetrical Electromagnetic Wave by a Co-axial Circular Disc of Infinite Conuctivity

A disc of infinite conductivity, whose radius is $a$, and whose center is at the origin, lies in a plane which is perpendicular to the $z$-axis of cylindrical coordinates, $r$, $z$, $\phi$. A circularly symmetrical electromagnetic wave of wave-length $\lambda$ impinges on the disc, and the resultant field is required. The solution depends upon solving an integral equation of the first kind. When $\frac{2\pi a}{\lambda }<1\mathrm{}$, this equation reduces to an integral equation similar to Abel's which may be solved explicitly. As an illustration the solution is obtained for the diffraction of a wave due to an oscillating electric dipole whose axis is the axis of $z$. It is mentioned that these equations have been used in determining the powerflow into the earth below a vertical antenna which is grounded by a circular disc lying on the surface.