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z n z=x (x,y) z=0 x
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FIGURE 4.14. Geometrical presentation of weaker rough terrain described by the perturbation method.
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function (see Fig. 4.14) z & x; y 4:52
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We choose z 0 so that (4.52) represents the deviation from the average height h& x; y i 0: Moreover, the perturbation method is valid when the phase difference due to the height variation is small, that is, when [4,9] jk & x; y cos yi j ( 1      @&   @&    ( 1;  (1 @x @y
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The boundary condition for the electric eld at this surface requires that the tangential components of E vanish at the surface z & x; y , that is, E n 0 4:54
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where n is the vector normal to the surface z & at point x; y . If the surface pro le (4.52) and the position of sources are known, then the problem is to determine the eld in semi-space z > 0, given that the boundary conditions are known [9]. Let us consider the in uence of roughness as a small perturbation, that is, the total eld is E E 0 E 1 4:55
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where E 0 is the eld that could be derived for the condition & 0, which a priory is well known using knowledge of specular re ection from smooth terrain obtained from two-ray model. The second term E 1 that describes the eld perturbations can be obtained from the wave equation using boundary conditions in (4.54). To present the solution of the perturbation term, let us consider two special cases, which are practical with regard to over-the-terrain propagation channels. Let a vertical dipole be located at point O as shown in Figure 4.15. Its re ection from a at surface at z 0 and at the point O1 according to the re ection theorem must also be directed vertically. By introducing the spherical coordinate systems fR; #; jg and fR1 ; #1 ; j1  jg for each dipole, we can present the components of
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FIGURE 4.15. The geometry of a vertical dipole eld scattered from a rough terrain.
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the non perturbed eld E 0 as
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0 Ex 0 Ey 0 Ez
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' ei ot k0 R ei ot k0 R1 2 k0 p sin #1 cos #1 cos j R R1 & ' ei ot k0 R ei ot k0 R1 2 2 sin j k0 p sin #1 cos #1 k0 p sin # cos # R R1 & ' i ot k0 R ei ot k0 R1 2 2 2 e 2 k0 p sin #1 k0 p sin # R R1
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2 k0 p sin # cos #
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Then, in the plane z 0 R R1 ; # p #1
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0 0 Ex Ey 0; 0 2 Ez 2k0 p sin2 #
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Here p is the modulus of the momentum of the vertical dipole that is well known from the literature (see, for example, [4,9,11]). Because in practical terrain propagation case, the source and the observation point are far from the surface z 0, we can present simple formulas for the perturbed part of the total eld due to the terrain roughness in the case where the incident wave lies in the xy-plane (i.e., when ik R 0 j 0, p e R0 qe ik0 x sin # , where q is constant):
1 Ex 1 Ey 1 Ez 2 k0 2q 2p
' & 0 2 k0 @& @ e ik0 r x sin # 0 0 2 dx dy 2q sin # 0 2p r @y @z  2  & 2 k0 @& @ & @2& 2 2 sin # 2q ik0 0 cos # sin # 2p @x @x02 @x02 ' ik0 r x0 sin # e 2 dx0 dy0 k0 cos2 # sin # r
ik0 cos2 # sin #
' 0 @& @ e ik0 r x sin # 0 0 dx dy r @x0 @z
q Here r x x0 2 y y0 2 z z0 2 , in which x, y, z are the coordinates of the observed point. For small grazing angles (# ! p ), that is, in the case of slipped 2 incident waves, which is very actual in mobile and personal communication, these formulas can be signi cantly simpli ed, for example,