Two-ray ground-reflection model

The two-rays ground-reflection model is a multipath radio propagation model which predicts the path losses between a transmitting antenna and a receiving antenna when they are in line of sight (LOS). Generally, the two antenna each have different height. The received signal having two components, the LOS component and the reflection component formed predominantly by a single ground reflected wave.
Mathematical derivation.
From the figure the received line of sight component may be written as
and the ground reflected component may be written as
where formula_3 is the transmitted signal, formula_4 is the length of the direct line-of-sight (LOS) ray, formula_5 is the length of the ground-reflected ray, formula_6 is the combined antenna gain along the LOS path, formula_7 is the combined antenna gain along the ground-reflected path, formula_8 is the wavelength of the transmission (formula_9, where formula_10 is the speed of light and formula_11 is the transmission frequency), formula_12 is ground reflection coefficient and formula_13 is the delay spread of the model which equals formula_14. The ground reflection coefficient is
where formula_16 or formula_17 depending if the signal is horizontal or vertical polarized, respectively. formula_18 is computed as follows.
The constant formula_20 is the relative permittivity of the ground (or generally speaking, the material where the signal is being reflected), formula_21 is the angle between the ground and the reflected ray as shown in the figure above.
From the geometry of the figure, yields: 
and
Therefore, the path-length difference between them is
and the phase difference between the waves is
The power of the signal received is 
where formula_27 denotes average (over time) value.
Approximation.
If the signal is narrow band relative to the inverse delay spread formula_28, so that formula_29, the power equation may be simplified to
where formula_31 is the transmitted power.
When distance between the antennas formula_32 is very large relative to the height of the antenna we may expand formula_33, 
using the Taylor series of formula_35:
and taking the first two terms only,
The phase difference can then be approximated as
When formula_32 is large, formula_40,
and hence
Expanding formula_43using Taylor series
and retaining only the first two terms
it follows that
so that
which is accurate in the far field region, i.e. when formula_48 (angles are measured here in radians, not degrees) or, equivalently,
and where the combined antenna gain is the product of the transmit and receive antenna gains, formula_50. This formula was first obtained by B.A. Vvedenskij.
Note that the power decreases with as the inverse fourth power of the distance in the far field, which is explained by the destructive combination of the direct and reflected paths, which are roughly of the same in magnitude and are 180 degrees different in phase. formula_51 is called "effective isotropic radiated power" (EIRP), which is the transmit power required to produce the same received power if the transmit antenna were isotropic.
In logarithmic units.
In logarithmic units : formula_52
Path loss : formula_53
Power vs. distance characteristics.
When the distance formula_32 between antennas is less than the transmitting antenna height, two waves are added constructively to yield bigger power. As distance increases, these waves add up constructively and destructively, giving regions of up-fade and down-fade. As the distance increases beyond the critical distance formula_55 or first Fresnel zone, the power drops proportionally to an inverse of fourth power of formula_32. An approximation to critical distance may be obtained by setting Δφ to π as the critical distance to a local maximum.
An extension to large antenna heights.
The above approximations are valid provided that formula_40, which may be not the case in many scenarios, e.g. when antenna heights are not much smaller compared to the distance, or when the ground cannot be modelled as an ideal plane . In this case, one cannot use formula_58 and more refined analysis is required, see e.g.
As a case of log distance path loss model.
The standard expression of Log distance path loss model is
The path loss of 2-ray ground reflected wave is
where
and
for formula_64 the critical distance.
As a case of multi-slope model.
The 2-ray ground reflected model may be thought as a case of multi-slope model with break point at critical distance with slope 20 dB/decade before critical distance and slope of 40 dB/decade after the critical distance. Using the free-space and two-ray model above, the propagation path loss can be expressed as
formula_65
where formula_66 and formula_67 are the free-space and 2-ray path losses.