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After a differentiation of (7.32) over the y, we nally get  3 @2j @j 1 @j k2 @e 1 2 @j @e 2 cot y 2 0 2 sin 2y k 2 @y 2 @y 2k0 sin y @j 2 0 @y @y @y 7:34
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If inhomogeneities do not exist, that is, @e=@j @e=@y 0, then from (7.34) the integral becomes tan y sin j j0 C0 7:35
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where j0 and C0 are the integration constants obtained. Equation (7.35) for the selected C0 and j0 describes a circle with its center at the origin of the coordinate system. Hence in References [11,12] a rule followed that is: In the homogeneouslylayered ionosphere (independent of y and j) the ray trajectory lies in the plane of a great circle. However, in the presence of large-scale irregularities the derivative dj=dy 0, but de=dy ( 1. This enables terms proportional to (dj=dy 3 and (dj=dy @e=@y) to be neglected. Thus the ray trajectory in a weak inhomogeneous plasma is de ned by the equation: @2j @j 1 2 @e sin y 2 cot y 2 @y 2 @j @y 7:36
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As was shown in References [11,12], analyzing Equation (7.36), it results that: the existence of large-scale inhomogeneities causes the ray trajectory to deviate from the plane of the great circle. The same effect, de ned as refraction in the troposphere, is discussed in Section 6.2. The Curved Smooth Screen Model. The problem of multi-mode re ection from ionospheric layers can be resolved by using (7.31). However, there exists a simpler method based on the model of a curved smooth screen moving with respect to the observer. This model is correct if ope ) o0 , where ope is a critical frequency of plasma for each ionospheric layer, and L ) dF . The geometric optic approximation is applied in this method. The simplest model of a curved mirror is the smooth screen with a sinusoidal shape, which moves with a constant speed without changing shape. Let us consider a rectangular coordinate system fx; y; zg with z-axis in the vertical direction and the base on the ground surface. The receiver and transmitter are at the origin of the system, the altitude z is a function of time t and coordinate x; z z x; t , as is shown in Figure 7.6, according to References [11,12]. The coordinates of the re ected points are de ned from the following condition: Z X; t @Z X; t X @X 7:37
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FIGURE 7.6. Modeling of the ionospheric layer by a sinusoidal layer.
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If ionospheric inhomogeneities have the shape of a running wave with amplitude d, wavelength , and period T, then Z X; t Z0 1 d sin 2p X= t=T and from (7.37) it follows that
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2 2pZ0 d= cos 2p X= t=T 1 d sin 2p X= t=T X
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Here, Z0 is the average height of the re ected level. For relatively small oscillations of the re ected level (d ( 1) ordinates X of the re ected points are found from the solution of the transcendental equation [11,12]
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2 2pZ0 d= cos 2p X= t=T X
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From (7.40), it follows that the number of solutions depend on the position of the screen and on the amplitude d. As was shown in References [11,12], the solution is unique if
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Inequality in (7.41) represents the condition of a single ray re ection from the smooth surface. If the condition for (7.41) is not valid, there are several rays forming the interference picture at the receiving point. From formulas (7.37) and (7.38) we can calculate the incident angle y0 of the radio wave at the receiving point cot y0 X=Z 2pZ0 d= cos 2p X= t=T 7:42
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The maximum ray re ection from the vertical axis, which is a half width of the angle spectrum g, equals g 2pZ0 d= 7:43
Analysis of (7.40) presented in References [11,12] has shown that even very small oscillations of the height of equal plasma density level lead to the multimode re ection of radio signals, to the growth of the angular spectrum width of waves, and to interference. Amplitude and Phase of Re ected Waves. If A0 is the amplitude of the signal re ected from the level of equal plasma density, then the amplitude A of a radio wave re ected from the smooth surface at the point X is [11,12] A X; t A0 1 Z0 =r 7:44a
where r 1 dZ=dX 2 3=2 = d2 Z=dX 2 is the radius of curvature of the screen at the point (X; Z). Relationship (7.44a) describes the focusing and defocusing effects of re ected radio waves caused by large-scale inhomogeneities. The phase at the receiving point is F X; t 2oR=c p=4 7:44b
where R is the optical path length from the re ector to the receiving point, factor p=4 corresponds to the change of wave phase when re ection from the ionosphere takes place. When oblique propagation of radio waves takes place for large distances the ray is re ected from an ionospheric surface of about ten thousand square kilometers. Such a surface contains many inhomogeneities, with about 20 100 km scales, on which radio wave scattering takes place. In the geometricaloptics approximation, the wave scattering on the ionospheric density uctuations is the same as the scattering on the rough surface. If now the radio wave falls on the bottom boundary of the ionosphere under the angle y0 , then the angle y y X; Y; Z will be changed according to Snell s law e X; Y; Z 1=2 sin y X; Y; Z sin y0 7:45
during the eld penetration to the upper altitudes. At the turning point Z Zm X; Y , the angle y p=2 and re ection takes place. The height of a turning point is de ned as a minimum root Z Zm of Equation [11,12] 1 e X; Y; Z cos2 y0 7:46a