k 1 1 p & 2 sin2 j Z2 m ; 2 r 0 r0 2 or its strict mathematical presentation

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m 0; 1; 2; . . .

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&2 Z2 1 p m p m 1 1 1 1 k sin2 j k sin2 j r 0 r0 r 0 r0

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RADIO PROPAGATION ABOVE FLAT TERRAIN

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is an equation of ellipses with semi-axes: 1 am sin j r p m r 0 r0 k r0 r0

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along the x-axis:

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and along the y-axis:

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r p m r0 r 0 bm k r0 r0

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These ellipses are the real boundaries of the zones of specular re ection from a at ground surface. For small grazing angles j ! 0 am ) bm , all ellipses are elongated along the x-axis, that is, along the direction of wave propagation. Earlier these ellipses were de ned as the Fresnel zones, but now they are described when the specular re ection from the ground surface is taken into account. Approximate size of the re ecting areas in [4] were also estimated from along the x-axis: and along the y-axis: 4 p lR sin j p 2bR 4 lR 2aR 4:37a 4:37b

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where R is the minimal value between the two distances, r0 and r0 . So far we have considered the situation when the antennas, transmitter, and receiver are above the earth surface. What happens if one of the points, for example, point A, lies close to the ground plane, that is, at z 0 In this second case zA % 0 as shown in Figure 4.9, the term r r in the exponent of expfi f expfi k r r g in integral (4.20) has a minimum at point A, that is, when x xA , y yA . Also if we assume that yA % 0 and introduce the polar coordinate system r; a with a center at point A (see Fig. 4.9), then x xA r cos a; The r and r can be related as: r % r0 r cos a cos j 4:39 y r sin a 4:38

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if % ikr0 i k r 1 cos a cos j

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Using these expressions, one can again obtain for the fast oscillating term in the integral (4.19), the signi cant area where re ection occurs. The boundaries of this area are described by the following equation: p k r 1 cos a cos j m ; 2 m 0; 1; 2; . . . 4:40

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ELECTROMAGNETIC ASPECTS OF WAVE PROPAGATION OVER TERRAIN

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or r m p=2 k 1 cos a cos j 4:41

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Expression (4.41) describes a family of ellipses with their foci at the point r 0 (point A). Their large semi-axis is elongated along the x-axis and is described by am m p 2 k sin2 j 4:42a

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and their small semi-axis is elongated along the y-axis and equals bm m p 2 k sin j 4:42b

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These ellipses are strongly elongated in the direction of the source, as shown in Figure 4.9. In this case the distance from point A to each successive ellipse is d a p m p 2 k 1 cos j 4:43

and in the opposite direction to the source this value is d a 0 m p 2 k 1 cos j 4:44

In the case of a wave incident with a small grazing angle j j % 0 , several initial Fresnel zones will embrace most of the radio path between points O and A (the

z O(0,0,z0)

r r0 f

A(xA,0,z)

XA (x,0,0)

FIGURE 4.9. The Fresnel-zone re ected area presentation for both antennas located near the ground surface.

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XA x

FIGURE 4.10. Specular re ection of the incident ray from a smooth at terrain.

source and observer, respectively). Estimations carried out in Reference [4] showed that the area in front of an observer placed at point A, located near the Earth s surface, is very important for propagation. At the same time, the area behind the observer is not that signi cant. The conditions of propagation and hence of communication between points O and A become more effective with an increase in grazing angle j or, of course, with a decrease of the range between the source and the observation point. In the third case, the source and the observation point occurs when both are located near the earth s surface (let us say, in the plane z 0, as shown in Fig. 4.10). In this case the position and the con guration of the Fresnel zones are determined by the earlier introduced condition of equality of phase of eld oscillations, that is, k r r const. But from this condition we can once more obtain the equations for the ellipses with their foci at points O (source) and A (observer). Because the minimum value of such a constant can be achieved for r r xA, the boundaries of the Fresnel zones are determined by the following conditions: p k r r k xA m ; 2 The large semi-axis of each ellipse is am r r xA m p 2 2 4 k 4:46 m 0; 1; 2; . . . 4:45

Behind points O and A these ellipses are close to each other and are very elongated along the x-axis. The small semiaxis is bm r r sin a 2 r r m p m p m xA % l xA k 4 k 8 4:47