## REFLECTION FROM A PLANAR INTERFACE

Plane-Wave Incidence

Radio waves are often reflected from smooth, flat surfaces, such as building walls or ground. When the reflecting surface is not a perfect conductor, part of the radio energy penetrates the surface, and part of the energy is reflected. Reflection of a plane wave from a uniform half space with a planar inter­face can be analyzed exactly by matching boundary conditions at the interface, and the derived reflection coefficients can then be used in other practical applications.

Consider the idealized geometry in Fig. 2. A free-space plane wave propagating at an angle в0 to the surface normal

 and

 (k/k0)2 cos90 — !(k/k0)2 — sin2 90 (k/k0)2 cos90 + V'(k/k0)2 — sin2 00

 (15)

 Tv =

 For the further simplification to a dielectric (a = 0) reflecting medium, Eqs. (14) and Eq. (15) reduce to

 — V’ V — SI

 sin2 90

 cos 9,

 (16)

 rh =

 cos 90 + л/er — sin2 90

 and

 — V’ V — SI

 sin2 9n

 er cos 9,

 (17)

 er COS 9n + л/Єr — si

 • 2 „ sin 90 Figure 2. Geometry for plane-wave reflection from a homogeneous half space.

 where er = e/e0. As incidence angle в0 varies from 0 (normal incidence) to пУ2 (grazing incidence), rh varies smoothly from (1 — Ver)/ (1 + Ver) to —1. However, for vertical polarization, the reflec­tion coefficient rv equals 0 at the Brewster angle, 0B = tan—1(Ver). At this angle, all of the incident energy is refracted into the dielectric. An examination of Eq. (15) reveals that the presence of nonzero conductivity a (which yields a complex k) prevents rv from going to zero. However, if the imaginary Figure 3. Far-field radiation from a vertical electric dipole over a homogeneous half space. The reflection coefficient Tv is a function of the incidence angle в.

part of k is small, there is nevertheless a pseudo-Brewster angle (14) where |rv| goes through a minimum.

Dipole Sources

Consider now a vertical electric dipole source located at a height h over a reflecting half space, as in Fig. 3. In the far field the electric field has only a в component Ee, which can be written as the sum of a direct and a reflected ray:

_ jeOflQIl sin в є_#оГ ^ A cos g + T^e-jk0hcoSe^

4n r

where Tv is given by Eq. (13). For the special case of a per­fectly conducting ground plane (a = oo), the reflection coeffi­cient equals 1, and Eq. (18) reduces to

I = jtoV’fyll sm6>e^or cos (k0h cos в) (19)

2n r

Equation (19) has a maximum at the interface, в = ет/2.

The dual case of a vertical magnetic dipole source is shown in Fig. 4. The source is a small loop of area A and current 1, and the loop axis is in the vertical direction. The electric field is horizontally polarized, and in the far field the ф component Eф is

_ f]0k0IA sin в ^_jk^r cos g _^^ jkQhcosg (20) ф 4n r h

where Гь is given by Eq. (12). For the special case of a per­fectly conducting ground plane (a = oo), the reflection coeffi- Figure 4. Far-field radiation from a vertical magnetic dipole over a homogeneous half space. The reflection coefficient rh applies to hori­zontal polarization.

cient equals — 1, and Eq. (20) reduces to „ jnklA sin в

Ефа=оо = ————— ~———— Є 3 0 Sin(&0/l COS 6>) (21)

2n r

Equation (21) has a null at the interface, в = пУ2.

For realistic (finite) ground parameters, the reflection coef­ficients for both vertical polarization Tv and horizontal polar­ization rh equal —1 at grazing incidence (в = пУ2). Hence the direct and reflected rays cancel in Eqs. (18) and (20), and the electric field is 0:

Eeв =n/2 = 0 and Ефв =п/2 = 0 (22)

In reality, only the space wave (the inverse-distance field that occurs for в > 0) is 0 at the interface. The ground wave is the dominant field component near the interface, and it will be discussed in detail later. It has a more rapid decay with dis­tance, but it does not equal 0 at the interface.