Electrical and Computer Engineering ETDs

Publication Date

Spring 4-4-1969


Several different forms of integral solutions exist to the problem of dipole radiation in presence of a half space with plane boundary, all of which can be easily shown to agree with one another by transformation. For the evaluation of these integrals, a number of assumptions are usually made on some parameters in the expressions such as the medium property, the wavelength and so on. Unfortunately, however, even with these simplifying (but meaningful) assump­tions, practically useful results for the dipole radiation problem are still lacking. Also it appears that there is no theory of dipole radiation when the plane boundary is roughened.

This paper sets out to find steady state solutions in us­able form for an arbitrarily oriented dipole source. The smooth boundary case is considered first, which is followed by a statistically rough boundary case. Use is made of a method which may be called a "plane wave approach." This method differs from other classical plane wave methods in that here the incident Hertzian wave is uniquely decomposed so that the reflected and the transmitted waves are found for each component Hertzian wave using the Fresnel coefficients. The integrals are evaluated in the geometrical optics approx­imation using the stationary phase method. The results are either new (the rough boundary case) or of new form (the smooth boundary case).

In the case of the smooth boundary, the applicability of such results is carefully reviewed. In both the vertical and the horizontal dipole cases for reflection as well as transmission, it is found that the source and the observa­tion points cannot simultaneously approach the boundary. An exception to the above is the case of an horizontal di­pole when the observation point lies on or near the dipole axis for the reflected field. The results also indicate that in the case of the horizontal dipole the total Hertz potential everywhere need not of necessity have a vertical component in addition to the component in the direction of the dipole, as has been hitherto generally believed. In fact, one finds that a Hertz potential, in order to be a solution to the horizontal dipole problem, must have at least two compo­nents in the rectangular coordinate system, so that there are altogether four permissible resolutions of the Hertz potential.

For the formulation of Hertz potentials for the roughened plane boundary, the vector Helmholtz integral is utilized, of which we give a somewhat more general derivation. To accommodate the vector nature of scattering including the effect of polarizations, dyadic reflection and transmission coefficients are used at the boundary. The rough boundary is slightly rough and considered to be a stationary random process with a gaussian height distribution. The stationary phase method is applied with respect to the mean plane of the rough boundary for the evaluation of the integrals. This was motivated by the physical fact that the density of the stationary points of such a rough boundary is the greatest in the neighborhood of the stationary point of the mean plane. Results are ob­tained for the expected values of Hertz potentials, electro­magnetic fields, and power. Each of these results involves a factor representing the effect of roughness, which in the limit of a smooth plane boundary correctly reduces to unity. A method for experimentally determining the r.m.s. slope of a class of natural surfaces by using overflight data is out­lined.

Document Type


Degree Name

Electrical Engineering

Level of Degree


Department Name

Electrical and Computer Engineering

First Committee Member (Chair)

Ahmed Erteza

Second Committee Member

Arnold Herman Koschmann

Third Committee Member