The definitions for the reflected and transmitted electromagnetic waves from a wall given above neglect the effects of internal reflection of the electromagnetic wave within the wall. These reflections give rise to additional components of the propagating wave as illustrated in Figure 3-4.
Now let us define the remainder of the reflected waves, and
transmitted waves in sequence, such that is the
.05emth transmitted wave across a boundary and
is the corresponding internal reflection where
. Thus for
being odd, the transmitted wave will be in region I, and for
being
even in region III of Figure 3-4. For horizontal
polarization we can derive
and for vertical polarization
We can therefore derive expressions for as
for a horizontally polarized incident wave, and
for a vertically polarized one.
Combining these electric fields along with the phase difference between two adjacent rays being emitted from the surface of the dielectric, Burnside and Burgener [87] derive a reflection coefficient and a transmission coefficient for the total reflected and transmitted fields as
and
where is the reflection coefficient defined in
(3.14).
is the phase delay incurred by
propagation through the dielectric slab,
the delay in the
reflection propagation direction between successive reflections, and
the phase difference for the primary transmitted field from the
incident field. As the width of the dielectric slab shrinks to zero,
the phase differences terms disappear from the equations, and thus
and
as expected for an infinitely thin dielectric slab.
However, as indicated in Appendix A, the reflection
and transmission coefficients are for horizontally polarized fields
only with
being chosen appropriately for the electric or
magnetic field. Since we are dealing with the electric field, and may
have vertically polarized fields we can modify these expressions to
give
and
where
However, the derivation of (3.24) and (3.25) assumes that the incident wave is a plane wave, and hence there is no attenuation due to expansion as the distance from the transmitter increases. As can be seen from (3.1) this is not the case for a spherical wave front. When the equations (3.22) and (3.23) are modified to reflect this factor, the infinite series used to formulate (3.24) to (3.27) is no longer a geometric series as the attenuation factor is not constant over each internally reflected segment.