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A two-ray propagation channel is the next step up in complexity from a
free-space channel and is the simplest case of a multipath propagation environment. The
free-space channel models a straight-line *line-of-sight * path from
point 1 to point 2. In a two-ray channel, the medium is specified as a homogeneous,
isotropic medium with a reflecting planar boundary. The boundary is always set at *z
= 0*. There are at most two rays propagating from point 1 to point 2. The first
ray path propagates along the same line-of-sight path as in the free-space channel (see the `phased.FreeSpace`

System object™). The line-of-sight path is often called the *direct
path*. The second ray reflects off the boundary before propagating to point 2.
According to the Law of Reflection , the angle of reflection equals the angle of incidence.
In short-range simulations such as cellular communications systems and automotive radars,
you can assume that the reflecting surface, the ground or ocean surface, is flat.

The figure illustrates two propagation paths. From the source
position, *s _{s}*, and the receiver
position,

`rangeangle`

function and
setting the reference axes to the global coordinate system. The total
path length for the line-of-sight path is shown in the figure by You can easily derive exact formulas for path lengths and angles in terms of the ground range and object heights in the global coordinate system.

$$\begin{array}{l}\overrightarrow{R}={\overrightarrow{x}}_{s}-{\overrightarrow{x}}_{r}\\ {R}_{los}=\left|\overrightarrow{R}\right|=\sqrt{{\left({z}_{r}-{z}_{s}\right)}^{2}+{L}^{2}}\\ {R}_{1}=\frac{{z}_{r}}{{z}_{r}+{z}_{z}}\sqrt{{\left({z}_{r}+{z}_{s}\right)}^{2}+{L}^{2}}\\ {R}_{2}=\frac{{z}_{s}}{{z}_{s}+{z}_{r}}\sqrt{{\left({z}_{r}+{z}_{s}\right)}^{2}+{L}^{2}}\\ {R}_{rp}={R}_{1}+{R}_{2}=\sqrt{{\left({z}_{r}+{z}_{s}\right)}^{2}+{L}^{2}}\\ \mathrm{tan}{\theta}_{los}=\frac{\left({z}_{s}-{z}_{r}\right)}{L}\\ \mathrm{tan}{\theta}_{rp}=-\frac{\left({z}_{s}+{z}_{r}\right)}{L}\\ {{\theta}^{\prime}}_{los}=-{\theta}_{los}\\ {{\theta}^{\prime}}_{rp}={\theta}_{rp}\end{array}$$