f the reflected wave depends on The surface conductivity The surface dielectric constant The general shape of the ground profile (e.g., flat, convex, or concave) The roughness of the ground The presence of buildings and trees If the surface of the ground is rough, the reflected energy is scattered. Smoothness can be judged by an application of the so-called Rayleigh criterion: For small angles of radians between the reflected ray and the ground plane, this criterion states that the undulations shall not be greater than for the surface to be treated as smooth. Of course, the irregular shape of the terrain can result in many reflected waves arriving at a reception point. Despite all these difficulties, it is instructive to consider the formulation for a simple case, with the reflected path longer than the direct by 8 and a reflection coefficient of mag- nitude p and a phase change of Q. From a summation of vectors (applying the cosine rule), the received field is reduced from the free-space value by equation 7.84. A,(dB) = -10log[1+p2-2pcos(BS-)] (7.84) For horizontal polarization and angles of incidence below 10 degrees at frequencies above 400 MHz, it is reasonable to assume P = 1 and Q=T. Tropospheric Refractive-Index Effects A further influence on the received field strength is the refractive index of the troposphere. Irregularities in this region bend, scatter, and reflect the wave. This is the main cause of time variability, particularly on a long-distance path. The refractive index, n, of the atmosphere differs by only a small amount from unity, and it is more convenient to use the so-called N factor. This factor is dependent on the pres- sure, P (millibars), temperature, T(K), and relative humidity, W, of the atmosphere, as given by equation 7.85. Chapter 7 Electromagnetic Waves (7.85) Under "normal" conditions, the refractive index of the atmosphere decreases with height by an amount that slightly bends the wave back toward the Earth. However, it is still more convenient to think of the wave traveling in s