FACTORS INFLUENCING TRANSMISSION 37] 
First of all, the incident radiation field is resolved 
into nearly plane wave components, each forming a 
narrow pencil of rays striking the reflecting surface 
within a small area which we shall call the reflection 
point. There are two types of such rays depending 
on their state of polarization. If the electric vector 
is parallel to the reflecting plane, the rays are said 
to be horizontally polarized and if the electric vector 
is parallel to a vertical plane through the rays, they 
are said to be vertically polarized. When consider- 
ing a very irregular surface, the reflected field may 
show extreme complexity even though the incident 
wave is linearly polarized. Increasing roughness 
may result in diffuse reflection which is ineffective 
in reinforcing the direct wave. The existence of 
diffuse reflection depends primarily on the size of the 
irregularities of the surface in comparison with the 
wavelength of the incident radiation and on the 
grazing angle of the incident field. This problem 
will be discussed in more detail later. 
Plane Reflecting Surface 
Consider first the simplest case, when a plane 
wave strikes a plane surface such as that of an 
absolutely calm sea. The incident ray is then split 
into two parts. One is the reflected ray, which is 
returned to the atmosphere, and the other is the 
refracted ray, which is absorbed by the sea. At the 
point of reflection, the ratio of any scalar quantity 
in the reflected wave to the same quantity in the 
incident wave is defined as the reflection coefficient 
of the sea for plane waves of given frequency. Thus 
defined, the reflection coefficient can and will be 
different for the various components of the field. 
For simplicity, let us assume the reflecting plane 
to be the ry plane of a rectangular coordinate system, 
the xz plane to coincide with the plane of incidence, 
and the reflection point to be the origin of the 
coordinate system. 
“PLANE OF INCIDENCE 
Yella dddddddd, 
Figure 10. Geometry of reflection and refraction. 
For horizontal polarization, the electric vector for 
the incident wave then is 
E; = Eye?! [t—(1/e) (zcos y—z sin )] (15) 
where y is the grazmg angle, f the frequency of the 
radiation and c the velocity of light in free space. 
The electric vector of the corresponding reflected 
field is given by the similar expression 
E, = Ey pew —J2nf (t-(1/c) (x cos p +2 8in )] (16) 
where p and ¢ are real constants. The ratio of the 
reflected to the incident field at the reflection point 
(2 = x = 0) is seen to be 
R = pe, (17) 
By definition this is the reflection coefficient for 
horizontally polarized waves. Thus the reflection 
coefficient is a complex quantity, the amplitude of 
which is the reflection coefficient of the wave 
amplitude and the phase is the lag in pkuse of the 
reflected wave with respect to the incident wave at 
the point of reflection. 
The reflection coefficient for vertically polarized 
radiation is defined in the same way. It is found, 
however, that the expression of p and ¢ in terms of 
the grazing angle yy and the ground constants 
are quite different for the:two types of polarization. 
For an arbitrary position of the plane of polariza- 
tion, the wave must be separated into its vertically 
and horizontally polarized components, and the 
proper reflection coefficient applied to each compo- 
nent separately. 
The quantities p and @ are determined by the 
boundary conditions for the electric vector at the 
reflecting surface, namely, that the tangential 
components of the electric vector on the two sides 
of the boundary surface shall be equal. This brings 
in the ground constants, that is, the conductivity 
and dielectric constant of the reflecting body. How 
these boundary conditions are applied may be 
illustrated by the simple example of horizontally 
polarized rays reflected from a surface of infinite 
conductivity. In the surface itself, the sum of the 
incident and reflected field strength must always 
be such that the currents set up in the body just 
suffice to produce the reflected field. Within a 
reflecting body of infinite conductivity, an infinitely 
weak field is sufficient, and hence the boundary 
condition is such that the reflected field, at the 
reflection point, shall be equal in magnitude and 
opposite in phase to the incident field, so that the 
resultant field is zero. Hence for infinite conduc- 
tivity and horizontal polarization 
Peele py =) adn 18024 9 (18) 
Fresnel’s Formulas 
For finite conductivity, the reflection coefficient 
may assume a variety of values. The general formu- 
las, as derived from electromagnetic theory, are 
given in equations (19) and (20). For horizontal 
polarization 
Pe sin y — Ve, — cos (19) 
sin y + Ve, — cos ’ 
