tly modify the surface impedance, in ways described in Causebrook (1978a). This modification can result in attenuation varying as shown in Figure 7.28; that is, instead of a simple monotonic curve, there is an initial rise and a significant dip at equal to about 100. In practice, it is desirable to have a calculation system that allows for The curvature of the Earth A terrain that varies in height A surface impedance modified by clutter Hufford (1952) gives an integral equation that, at least in principle, provides a general solution for attenuation of radio waves. This can be written as in equation 7.98. A(R) =1- (7.98) Chapter 7 Electromagnetic Waves Fig. 7.29 Profile data (quantized form). (n-1) In equation 7.98, R is the distance of path over which attenuation is required; is the angle in radians between the line from the receiver to the surface at range r and the tangent to the surface at this point; S is the distance between terminals subtracted from the sum of distances from terminals to surface at r (see Figure 7.29); n is the relative surface impedance at r. The integral contains quantities that do not have "mathematically closed" forms, SO an analytical solution is unobtainable. To overcome this, it is first possible to divide the distance, R, into N quanta of length A to give equation 7.99, where In is given by equation 7.100. A(r)=1-EQnIn (7.99) (7.100) Qn is a part of the integrand that can be considered constant in the interval, A,to be given by equation 7.101, with quantities defined by Figure 7.29. (7.101) The next barrier to a simple solution is that A(r) is initially an unknown function. Up to a distance of 24 this presents little problem because terrain variations will not greatly modify the result. Thus the Sommerfeld-Norton function can be used to determine attenu- ations A1 and A2 at distances A and 24, respectively, by equation 7.102. (7.102) Also, I can be determined analytically by putting A(r) equal to a power series expansion of the Sommerfeld-Norton formula. Further values of In and An