Application of a root finding method for tropospheric ducting produced by trilinear refractivity profiles

Knowledge of whether or not all modes of consequence have been found is of paramount importance in numerical waveguide studies. A general root finding method (RFM), developed for VLF/LF propagation in the earth-ionosphere waveguide, locates all complex zeros in physically important rectangular regions of the complex plane. The method, based on phase contour tracing, is applied to the problem of tropospheric ducting in a trilinear refractivity environment. Frequencies from VHF and into the microwave range are considered. Principles upon which the RFM is founded are reviewed and modifications of the modal equation demanded by the methods are discussed. Particularly in the microwave range, overflow can be a problem and methods of overcoming this are discussed. However, the RFM is most easily implemented if the latter modifications can be avoided. Therefore, tables are included to indicate for trilinear profiles the maximum frequency below which no overflow problems would occur. These tables depend upon the maximum number which the computer can handle, the location of the search rectangle, and the refractivity profile parameters. In the present case study of ducting produced by a strong elevated layer (40 M unit deficit at about 200 m) the RFM has been applied successfully into S band with computer run time requirements comparable to those of the more conventional Newton-Raphson method.