Ten-rays model
Experimentally, it has been demonstrated that the ten ray model simulates or can represent the propagation of signals through a dielectric canyon, in it which the rays that travel from a transmitter point to a receiver point bounce many times.
As example for this model it is assume: a rectilinear free space with two walls, one upper and the other lower, from which two vertical bases are positioned at their ends, these are the transmitting and receiving antennas that it's locate in such a way that their heights don't surpass the limits of the top wall; Achieved this the structure acts as free space for its functioning similar to that of a dielectric canyon of signals propagation, since the rays transmitted from the transmitting antenna will collide each side of the upper and lower walls infinity of times (for this example up to 3 reflections) until reaching the receiving antenna.
[1] For the mathematical modeling of the propagation of ten rays, One has in account a side view and this starts off modeling the two first rays (line by sight and his respective reflection), Considering that antennas have different heights, Then
, and they have a direct distance d that separates the two antennas; The first ray is formed applying Pitágoras theorem: The second ray or the reflected ray is made in a similar way to the first, but in this case the heights of the antennas to form the right angled triangle for the reflection of the height of the transmitter are added up.
In the deduction of the third ray it is necessary find the angle between the direct distance
To model the rays that collide with the wall twice, is used the Pythagoras theorem because of the direct distance
, this divides on the angle formed between the direct distance and the reflected ray.
For the eighth ray is calculate a series of variables that allow to deduce the complete equation, which is composed by distances and heights that were found by likeness of triangles.
Finding the distance between the height of the wall of the second shock with respect to the first shock, is obtain: Deducing also the distance between the height of the wall of the second shock with respect to the receiver: Calculating the height of the wall where occurs the first hit: Calculating the height of the wall where occurs the second shock: With these parameters is calculate the equation for the eighth ray: For the ninth ray, equation is the same as the seventh ray due to its characteristics: For the tenth ray, the equation is the same as the eighth ray due to its reflected ray shape: Is considered a signal transmitted through free space to a receiver located at a distance d from the transmitter.
Assuming there are no obstacles between the transmitter and the receiver, the signal propagates along a straight line between the two.
[2] The trajectory losses of the ten-ray model in free space is defined as:
