In Section 2.1.2 "Propagation Over a Plane Earth" of [1], the author wrote:

The reflection coefficient, $R$, of the ground depends on the angle of incidence, $\theta$, the polarization of the wave, and the ground characteristics; it is given by $$ R = \frac{\sin \theta - z}{\sin \theta + z}, \tag{2.1-3} $$ where $z = \frac{\sqrt{\epsilon_0 - \cos^2 \theta}}{\epsilon_0}$ for vertical polarization, $z = \sqrt{\epsilon_0 - \cos^2 \theta}$ for horizontal polarization, $\epsilon_0 = \epsilon - j 60 \sigma \lambda$, $\epsilon =$ the dielectric constant of the ground relative to unity in free space, $\sigma =$ the conductivity of the earth in mhos per meter, and $\lambda$ is the wavelength.

I would like to know how to derive equation (2.1-3)? Thanks in advance.


[1] W. C. Jakes, Jr., Microwave Mobile Communications. New York: Wiley, 1974, reprinted by IEEE Press.

  • $\begingroup$ The units don't seem to check out, so some of the quantities likely have implied units (not to mention the strange factor of 60). Regardless, I think these are just a special case of Fresnel's equations for reflection coefficients when reflection is from a conductive medium. $\endgroup$ – Puk Aug 10 '19 at 9:23
  • $\begingroup$ @Puk, thank you for your comment. $\endgroup$ – Wei-Cheng Liu Aug 10 '19 at 9:40

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