So, how exactly do oblique radio rays bend as they transit the ionosphere and return to earth? Let’s find out.

By now we know that some radio waves will bend in the ionospheric plasma. Your amount of bending depends on the index of refraction. With μ=1 there is no bending and the signal continues up in a straight line. On the other hand, if your μ=0, the signal reflects straight back. We measure the critical frequency f_c by sending a signal perpendicular up until it stops reflecting, as shown in yellow above.

Our textbooks tell us that the ionosphere can only bend signals below the critical frequency, f < fc. But, we also know that signals maybe 2 or 3 times f_c can be bent back to Earth. What gives?

The answer lies with oblique radio rays, or signals slanted at less the 90°. Back in the 1600’s,  Dutch astronomer Willebrord Snellius discovered how light bends as it passes through different media. Snell’s Law applies to all electromagnetic waves including radio.

Consider the ionosphere as a series of layers with increasing free electron density up to N_MAX as shown above. As height increases, the index of refraction μ decreases from μ₀=1 (free space) down to say μ₃=0.5 at maximum density. As your radio wave travels across each layer boundary, the decreasing μ causes the angle of incidence θ to increase.

Eventually, at the same point that your found f_c for a vertical ray, your oblique ray f_ob (in orange, above) bends over and returns down.

Oblique Radio Rays do the work on HF

This is the essential story of how radio waves bend in the ionosphere, and eventually how we calculate the Maximum Useful Frequency. These formulas become more complex as we consider curved earth and magnetic fields, but the basic story remains.

In the top left, you see the formula for the refraction index, μ. You can see that it decreases with electron density and increases with frequency. Over on the right, you can estimate your critical frequency as a function of electron density.

If you are interested, you can play with these formulas in a spreadsheet and get a better handle on how it all works. Electron density is measured in free electrons per cubic meter and is around 10¹² in the F layer. Frequency is measured in Hz.