What are good sources for information about the propagation of RF that details the equations behind the curtain?

I've been able to find some simple stuff on wikipedia but nothing fancy that I'm sure exists. There's also a lack of clarity on what these formals output.

The Free-space path loss formula is a good example of this in wikipedia.

What does the number you get out of this mean? Is this the effective transmission power at that distance?

Are there any introductory sources to the papers that cover how all of this works (to the best of our current knowledge)? I'm particularly interested in:

  1. Losses
  2. Refraction and reflection
  3. Scattering

I am not interested in approximations of the losses a link will experiance. I am interest in the way the waves will 1) move through space and get reflected off of things (the paths they take) and 2) what kind of losses these mediums and deflections introduce (the losses experianced). Equations like the Friis equation only estimate how the wave might look in total when it gets to the other station. It does not take into account many factors like solar indexes, time of day, path the radio will travel, etc.

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    $\begingroup$ FSPL is, as the name says, a loss. I kind of think that name is self-explanatory - how much of your input power do you lose in free space. $\endgroup$ – Marcus Müller Apr 8 '17 at 17:57
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    $\begingroup$ At this point, it'd be very interesting to know what your current level of RF knowledge is – on this site, that might include "hey, I already learned for ham exam XY and ABC". I don't really know how to help you – all these things are part of an EE degree (at least, if you go into EE directions where that's relevant), but I don't think throwing EE material at you is a great idea – since I don't know whether you'd like the math that EEs have to study before that, for example.There's certainly lots of amateur radio material out there,too,but we don't know what of that you've already read,either. $\endgroup$ – Marcus Müller Apr 8 '17 at 17:59
  • $\begingroup$ My issue is what unit is FSPL calculating? mW/m, dB of attenuation, (kJ/fortnight)/furlong? Saying it's "a loss" doesn't tell you what kind of loss. $\endgroup$ – Joshua Katz Apr 8 '17 at 18:09
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    $\begingroup$ you'd never say "I've lost 10", because, yes, that would make no sense. But it also wouldn't make much sense to say "I lost 10 kg", because it makes a goddamn difference if you weigthed 60kg before or 180 kg! So, technically, losses are always an amount of the thing that was there before, so you'd say "there's now only 80% of me left", and that's a loss. Seriously, though, it feels a lot like I'm talking to someone who's not quite in a "technical mood" right now – you complain about wikipedia not saying it's unitless, though the formula has the cancels the units right under your nose,if you … $\endgroup$ – Marcus Müller Apr 8 '17 at 20:01
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    $\begingroup$ What does "on a simulation level" mean? And can you edit your question so it clearly asks just the question you want answered, without "Edit:" at the end? $\endgroup$ – Phil Frost - W8II Apr 11 '17 at 2:00

Propagation is a very complex subject. To predict using purely mathematical means how a transmission from point A to B is more complex than trying to predict the throw of dice. There are stochastic processes, huge complexity, and many unknowns involved.

Simplifications must be made, and the valid simplifications depend on your objective, the frequency, and the application, among other things. If you want to understand all the physics relevant to propagation, the best I can do is refer you to many concepts to further research.

The most general solution is Maxwell's equations. All of the other concepts discussed here are solutions to these equations with certain assumptions added. If you want an exact model of how propagation occurs in all conditions, this is it. These equations are the foundation of classical electromagnetism, and consequently radio, optics, and electronics.

Of course to understand exactly the path from A to B would require a precise understanding of the geometry of everything in the path and near it, as well as the exact electromagnetic properties of every material within. That's not not realistically possible for complicated paths.

When the paths are simple we have things like the Friis transmission equation. All waves will be subject to this equation, though there may be other things contributing to the path effects. For things like space communications, or microwave links with nothing in the path's Fresnel zone, the Friis equation is an accurate and precise calculation of path losses. There's very little to be understood about this model besides the inverse square law.

Adding a bit of complexity, there's the two ray ground reflection model. This is relevant for many VHF and microwave links where the ground will reflect the signal and create multipath interference due to interference between the waves, but with otherwise clear paths.

Of course multipath interference can occur due to reflections off buildings and other things. You might try to use ray tracing to calculate the interference, though unless the reflecting surfaces are very much larger than the wavelength, diffraction will make this calculation inaccurate. A better calculation uses a field solver to approximate a solution to Maxwell's equations.

Environments often contain lossy materials, resulting in absorption. The ground is one such common lossy material. Steel structures will incur losses due to hysteresis. Resistive losses also exist in metal structures, among other things. Dielectric loss can occur, especially at higher frequencies. Water is a common factor in dielectric loss. At high enough frequencies, gasses in the atmosphere can cause absorption.

In practice, modeling the contour of the Earth, buildings, and trees in the path is infeasible, so some sort of propagation model will be used which approximates typical environments. Often these are a combination of educated guesses about the environment's properties and empirical data.

Skywave propagation such as on HF is very complicated due to conditions which are constantly changing. Here again, Maxwell's equations describe the behavior, but the chaotic and complex nature of the ionosphere makes a simple solution impossible. Many topics in the field of optics are relevant here. In practice empirical models are used, for example VOACAP.

I hope this gives you some things to research.

  • $\begingroup$ Thank you so much. This is the first answer that was not "Go to college". If I see you at Dayton I'll thank you personally. I've been reading a few papers on my own and it seems like you are right on the optics front. That's the "way to do it" or at least the way the real guys do it. $\endgroup$ – Joshua Katz Apr 13 '17 at 16:11

You're talking about Maxwell's equations and the Electromagnetic wave equation

Path loss and scattering are functions of the medium through which the waves are passing, which is reflected in the variables in these equations.

Wikipedia is not your best source for information about this, however. University courses are much better.

Here is the search you want to do:


One of the best results is this paper:


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    $\begingroup$ I'm familiar with Maxwell's equations but that is not what I'm asking. I understand those and the implications of those. What I'd like to find is the papers that detail equations that model different losses that I've descibed above. Maxwell's equations don't talk about how a radio wave passing through a charged plasma of a specific frequency is obsorbed, reflected, or refracted. It also doesn't talk about how a wave reacts to a collision with the ground a specific angle or what kind of scattering will happen after that. That is what I'm interested in. $\endgroup$ – Joshua Katz Apr 11 '17 at 1:51
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    $\begingroup$ "I understand those and the implications of those": So you understand where diffraction, path loss and everything related to wave physics come from? Cool, so what's the question? "Maxwell's equations don't talk about how a radio wave passing through a charged plasma of a specific frequency is obsorbed, reflected, or refracted.": Yes, they (and their implications, which you claim you understand) do. But you'd have to learn how to interpret them to answer this question. That would, exactly as SDsolar says, best be covered by literature and uni courses. $\endgroup$ – Marcus Müller Apr 11 '17 at 10:45

In addition to what has already been recommended, I would recommend:

If you are interested in empirical methods for estimating propagation loss in various environments, you might take a look at the Okumura-Hata model.

You can also take the MIT undergraduate course in Electromagnetics and Applications online for free at MIT OpenCourseware. You can browse the graduate courses there as well. I think you will find the material you are looking for covered in 6.641 and elsewhere.


I am converting my comments into an answer with two publications that I believe are very worthy in answering the OPs questions.

"Introduction to RF Propagation" by John S. Seybold, Wiley Press, 2005. A good introductory book but the focus is in commercial applications (Radar, microwave, etc.) other than HF. Contents of book:

  • Electromagnetics and RF Propagation
  • Antenna Fundamentals
  • Communications Systems and the Link Budget
  • Radar Systems
  • Atmospheric Effects
  • Near Earth Propagation Models
  • Fading and Multipath Characterization
  • Indoor Propagation Modeling
  • Rain Attenuation of Microwave and Millimeter Wave Signals
  • Satellite Communications

And, the second book is: "Ionospheric Radio Propagation" published by the National Bureau of Standards, U.S. Department of Commerce, editor/author: Kenneth Davies. 1965. This is like the encyclopedia of Ionospheric HF models for a variety of conditions. Key topics covered (not all) include:

  • Earth's Atmosphere, Geomagnetism, and the Sun
  • Theory of Wave Propagation
  • Synoptic Studies of the Ionosphere
  • Oblique Propagation (mainly HF propagation)
  • Signal Strength
  • Ionospheric Disturbances
  • Ionospheric Propagation Predictions
  • Scatter Propagation with VHF
  • Propagation of Low and Very Low Frequency Waves

"A Guide to the Wireless Engineering Body of Knowledge, 2nd ed." (IEEE Press, Wiley) is a reference that I use. Chapter 4, "Radio Engineering & Antennas", covers what you're looking for. Also, there are a number of lectures available on online if you search for them.


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