# Reflection Coefficient — propagation over a plane earth

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.

# References

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

• 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. – Puk Aug 10 '19 at 9:23
• @Puk, thank you for your comment. – Wei-Cheng Liu Aug 10 '19 at 9:40