The field of a dipole above an infinite corrugated plane

We investigate the excitation and propagation of the three dimensional electromagnetic field over an infinite corrugated plane which is approximated by an anisotropic impedance boundary condition. Emphasis is placed upon effects of surface anisotropy which are not evident in two dimensional treatments. In particular we consider the excitation by a magnetic point dipole in detail. It turns out that the fields are determined by a scalar wave function which satisfies a mixed boundary condition involving a linear combination of the wave function, its normal derivative and its second order tangential derivative. The exact formal solution is first derived, and then the radiated far field and the surface wave far field are evaluated separately. Both the phase and the amplitude of the excited surface wave are dependent upon the direction of observation. Numerical results are given. The physical significance of this solution is discussed. A comparison is made between this problem and the theory of ship waves.