ity, giving equation 7.18. (7.18) For N1<N2 Snell's law gives a1 < A2, SO for (a1 + a2): Greater than /2: rv is positive (i.e., no phase change). Less than /2: rv is negative (i.e., a T phase change). Equal to /2: rv is 0 and A1 equals the Brewster angle. Case 2 An electric vector perpendicular to the plane of incidence (equivalent to a horizontally polarized transmitter), as shown in Figure 7.2, gives equations 7.19 and 7.20. (7.19) (7.20) Fundamental Laws of Electromagnetism Medium 1 n1, n1 Medium 2 n2, N2 Fig. 7.2 Reflection and refraction at a boundary with the magnetic vector in the plane of incidence. These results allowed for a relative magnetic permeability of nonunity, but conductivity was zero. For propagation over the Earth's surface, it is more appropriate to approximate magnetic permeability to unity but allow for the finite conductivity of the medium. It is also reasonable to assume unity for the refractive index of air in order to obtain formulas for reflections off the Earth's surface. Then, for a wave of angular frequency W, equation 7.21 can be obtained. (7.21) Then the reflection coefficient modulus and phase are given by equations 7.22 and 7.23. (7.22) (7.23) Examples of the use of these formulas are given by Figures 7.3 and 7.4 for the land and by Figures 7.5 and 7.6 for the sea. Huygen's Principle/Fresnel Diffraction An earlier solution to propagation phenomena came via Huygen's principle, which states that every point on a wave front acts as a source of a secondary wavelet. This provides an explanation for some propagation phenomena. Most important, it helps to quantify diffraction. Diffraction effects can be formulated via the wave equation and Kirchhoff's integral (Longhurst, 1967), but the original derivation by Fresnel was via Huygens principle. A semi- infinite opaque screen is assumed to exist between transmitter and receiver, as shown in Figure 7.7. Then, Fresnel's integral states that the attenuation is related to free space, in Chapter 7 Electromagnetic Waves Horizontal Vertical Po