h antennas at ground level, another mechanism of propagation tends to dominate. This is a wave which is guided along the boundary between the surface of the 6.6.3 The Sommerfeld-Norton theory Earth and the air above it. The best term to be applied to this mechanism is the surface wave. The Sommerfeld-Norton theory (Jordan and Balmain, 1986, pp. 635-649; Terman, 1943) gives a method for prediction of attenua- 6.6.2 Reference field strength tion over a homogeneous smooth earth. This method will not be fully described here but, for a vertical antenna, it is convenient to Equation 6.49 gave the field strength for a short monopole on a define this calculation via the surface impedance concept. Thus, for perfectly conducting plane. In practice, allowance must be made a ground conductivity o, relative permittivity Er, and for a wave of length A, the relative surface impedance at grazing incidence, may be given by Equation 6.95. 1. The power gain g of a particular antenna in a given direction, relative to the short monopole, assuming a perfect ground (6.95) plane. The imperfect conductivity of the ground plane, which causes a significant fraction of transmitter power to be lost as heat in the ground, and not radiated as electro-magnetic energy. This is taken into account by the transmitter antenna system efficiency For a given distance d a parameter W may be defined by Equation 6.96. 3. The reduction in field strength for the real path relative to that for a perfectly conducting flat plane. This is taken into account (6.96) by the path attenuation factor A. The surface wave 6/23 Length of monopole in wavelengths Figure 6.27 Relative power gain of a monopole antenna From this, the complex surface wave attenuation factor A is given quanta of length A to give Equation 6.99, where In is given by by Equation 6.97 where erfc denotes the complimentary error func- Equation 6.100. tion (Aboramowitz and Stegun, 1964). (6.99) (6.97) A(r)=1-," 6.6.4 The realistic terrain - A solution via an integral equation (6.100) A realistic terr