Most amateur propagation in the troposphere relies on diffraction. There will seldom be a free-space path between the two stations, even from hilltop to hilltop. Typically some terrain objects intrude into the path. The object with the greatest effect is the Earth’s bulge.

A typical path is shown below. The Earth bulge is shown in red.

Were it not for diffraction, the signal loss between transmitter and receiver would be total.

Diffraction works using Huygen’s principle.

## Huygen’s principle

Huygen’s principle has it that as a wave is launched from the transmitting antenna and spreads, every point at every radius launches a secondary wavelet. Each wavelet spreads and each point on each wavelet sets up another wavelet, and so on. The received signal at the receiver is the vector sum of all wavelets captured by the receiving antenna.

This principle means that it is impossible to block all energy from the receiving antenna. Some wavelets will always get through no matter how severe the obstruction. Of course, the energy received may be infinitesimally small and unusable. But it does mean that reception is possible behind hills and in the shadow of the Earth’s bulge.

## Diffraction loss due to spherical Earth

The Earth is a sphere. No surprises there. The horizon is for most a few tens of kilometres distant, so unless we use very high antennas, the Earth quickly becomes the dominant intrusion into the path.

The path loss is given by four terms:

Loss = F(FSL) + F(X) + G(Y1) + G(Y2) dB

FSL is the Free Space Loss as discussed elsewhere on this site. X is a term including primarily the distance but also the type of ground and polarisation. Y1 and Y2 are antenna height gain terms. Both X and Y terms also consider the Earth’s effective radius.

The above formula can be expanded to be calculated numerically, or it can be calculated using nomograms. Nomographs are graphs that enable the independent variables to be set and the dependent variable to be read off. The nomograms in Rec. ITU-R P. 526 are given for k=1 and for k=4/3. k=4/3 represents the normal troposphere when k is exceeded for 50% of the time.

See the page on tropospheric lifts for more on how k changes with time.

An edited nomograph is shown below.

## Correction for low percentage time

The final loss over free space, E_{0}, can be corrected for low percentages of time by using a hypothetical frequency in the nomograph equal to f/k^{2}, where f is the operating frequency. Assuming that k would be very large for low percentages of time, and selecting a value of 3, the hypothetical frequency would be 144/9. This gives about 16MHz for operation at 144MHz.

The application of the hypothetical frequency to enable computation at low percentages of time shifts the path loss over a 150km path from over 150dB for most of the time, to a very tolerable 30dB for low percentages. The loss over a 300km path is ‘only’ 65dB! Losses quoted are losses over Free Space Loss (so the values must be added to the loss calculated using the Free Space Loss equation).

## Application

So, what is this telling us?

Referring to the page on this web site on system value, we see that at 144MHz for SSB, there’s a budget of 166dB available.

The free space loss over a 300km path is:

FSL = 32.4 + 20log f + 20log d, or 125dBi

The total path loss is therefore 125dBi plus 65dB, or 190dBi.

The system value is 166dB meaning that we need to find a mere 24dBi to make the path work! That’s easily found if both stations are using Yagi antennas with gains of at least 12dBi – not a great problem considering that the typical radio amateur Yagi has 11 elements with a gain of 15dBi. There’s even 6dB in hand!

So, this shows why radio amateurs can communicate at 144MHz over distances of more than 300km (for small percentages of time).

## Greater distances

If the mode changed to FT8, with its system value of 186dB, the same 300km could be achieved for greater percentages of time or for low percentages of time using dipole antennas.

And likewise, if one or other of the stations went onto a hilltop with full Fresnel Zone clearance in the foreground and significant height compared to the foreground terrain, the resulting height gain would reduce E_{0}. A height of 300m at one end might give a height gain there of 25dB. This moves success over a 300km path to around 50% of the time.

Finally, if both stations have height gain of around 25dB, this ‘extra’ 50dB extends the range to around 550km for 50% of time. This is the state typically existing during VHF NFD.

- See RECOMMENDATION ITU-R P.526-10,
*Propagation by diffraction*, available at https://www.itu.int/en/ITU-R/Pages/default.aspx.