Guest Author: Sreekar

As part of a course on Engineering Electromagnetics in the Fall 2019 semester, we had to make a poster on a topic of our interest which involved electromagnetics, which we had to present at our department, on a fine Saturday afternoon (19.10.2019).

# How did we come up with this topic?

With my exams of the course having gone really bad (I still have nightmares about it :( ), I was determined to make my group’s poster one of the best that will be put up on poster day. The three of us, Sreekar, Nithin Varma and I searched the internet for a topic that was “unconventional” and interesting. After a week or so of following links on Wikipedia, I came across this article - Whistler (radio), which caught my eye. It was a topic that most would have not heard about and would be very much interested in learning about it. Bazinga!

Meanwhile, Sreekar had come up with Birefringence, which also seemed interesting. None of us had heard about it before. With these two topics in mind, we asked our instructor (Dr. Deepa Venkitesh) for her opinion and she asked us to go with the former.

Before I start explaining the technical details of the poster, do take a look at the poster we made (click on it for a PDF).

Our Poster

So here it goes.

# Technical Details of the Poster

## History and Discovery

In World War I, soldiers used long communication lines to maintain contact with their base. The people on the receiving side of the line, often could hear sounds that were very similar to that of “falling grenades”, when there were none falling. One keen observation was that this particular sound was heard during the times of lightning strikes.

Here’s a short sound clip of what they heard:

Courtesy: NASA Van Allen Probes

The first one, in which you would have noticed a tone of descending frequency, is called a Whistler, while the second, which consists of rising and falling tones of frequency is a Chorus. This post will talk more on whistlers as they are more extensively studied among the two.

Whistlers are Very Low Frequency (VLF) emissions in the atmosphere that are characterised by a tone of rapidly descending frequencies.

Being long-wavelength (~ 100 km!!) EM Waves, it’s understandable that they couple into communication lines that are as long as 100 km. This explains the strange noises heard by soldiers in WWI.

## Sources and generation of whistlers

A natural question to ask is how these waves are generated in the first place. The literature on this is limited, but there is considerable evidence to say that the source of whistlers must be not very different from lightning.

The only known characteristic of whistlers that distinguishes it from ordinary lightning is their relatively high energy content. After accounting for attenuation losses, the estimated electromagnetic energy in a whistler producing stroke is about ten times as great as that in the average lightning flash. A sufficiently strong lightning flash should be able to produce whistler-mode signals.

But here’s the thing - we also detect these waves at locations where there may be no lightning at all. In fact, these waves can also be detected at places whose magnetically conjugate point has experienced a lightning strike. That means that there must be some mode of propagation that is making this possible. The theory behind it’s propagation can be intuitively understood using the basic concepts of a waveguide.

## Waveguides

To the uninitiated, here’s the wikipedia definition of a waveguide:

A waveguide is a structure that guides waves, such as electromagnetic waves or sound, with minimal loss of energy by restricting expansion to one dimension or two. Without the physical constraint of a waveguide, wave amplitudes decrease according to the inverse square law as they expand into three dimensional space.

tl;dr It literally is in the name - It “guides” waves. One very popular waveguide is the ubiquitous optical fibre.

The simplest idealised case of a waveguide is a parallel plate waveguide, in which the wave is constrained between two conductors. Solving Maxwell’s equations and applying boundary conditions will give us a solution which has a wave that propagates in a direction tangential to the plates.

Now, consider the earth and the ionosphere. The earth is a good conductor, in the sense that at the frequencies we are talking about (KHz), the loss tangent, is about 600.

The ionosphere, as its name suggests, has a lot of ions, making it a good conductor as well. What we now have is a waveguide through which EM waves, more specifically VLF waves, can be guided through.

## Propagation of Whistlers

Energy from a lightning source near the earth’s surface travels in the earth-ionosphere waveguide and enters the ionosphere continuously along the lower surface of the ionosphere. Wave components that enter the ionosphere at the location of a duct are then trapped and “conveyed” to the opposite hemisphere along the same magnetic line of force, which are aligned exactly along the duct, where they emerge from the ionosphere and re-enter the earth-ionosphere waveguide.

So…, a bunch of frequencies is generated by a lightning strike at a location, at the same time. All of them take the same path and reach another location. So, ideally we should be receiving all the frequencies at the same time. But we hear a tone of descending frequency. Why?

The reason lies in the ionosphere. The velocity of propagation of a wave through the ionosphere is dependent on it’s frequency. This leads to a dispersion of waves, with each frequency traveling at a different speed.

Graph of $\frac{1}{\sqrt{f}}$ vs $t$ (time of arrival)

Source: Robert A. Helliwell, Whistlers and Related Ionospheric Phenomena, Stanford University Press, 1965

Due to the dispersion of waves in the ionosphere depending on their frequencies, different frequencies are received at different times, with the higher frequencies being received first, as seen from the above graph. This is why a whistler is a tone of descending frequency.

## Detection of Whistlers

The theory’s great. But how do we practically detect and hear these sounds?

Since these are electromagnetic waves, we have to first “capture” these EM waves, then transduce them, i.e. convert them to audio to actually hear the audio clips you heard before.

The wavelength of these waves are extremely huge (100 km for a 3KHz wave). This means that it is not possible to have a full-wavelength antenna, for capturing the wave accurately. A typical VLF receiver is about 60ft long - there is very little signal current for reception. Using a cascaded combination of high gain, high input impedance (for better sensitivity) amplifiers and low pass filters, we can make up for the difference.

Source: http://www.home.pon.net/785/equipment/build_your_own.htm

## Whistler Spectrograms - a vast ocean of information

What’s a spectrogram?

A spectrogram is a visual representation of the spectrum of frequencies of a signal as it varies with time.

Source: https://en.wikipedia.org/wiki/Spectrogram

Basically it is a graph, with x-axis as time (usually), y-axis as frequency and the intensity/amplitude at a point $(f, t)$ is given by it’s color.

Whistler spectrogram

This is a real-life spectrogram of a whistler. At around 0355:29, you can a see a band of frequencies being generated. This is called a tweek.

Tweek in a spectrogram

At near 0355:31, you see that the “tone of descending frequencies” has started. Notice how there’s a time delay between the tweek and the start of the whistler. This tells us that it is detected elsewhere (remember, the magnetically conjugate point?), meaning it has travelled (or the whistler has been detected at the same place as the tweek.)

According to Robert A. Helliwell, in his book Whistlers and Related Ionospheric Phenomena, the idealised spectrogram for a whistler is:

Ideal whistler spectrograms

Source: Robert A. Helliwell, Whistlers and Related Ionospheric Phenomena, Stanford University Press, 1965

There’s a lot more in the above diagram than just ideal spectrograms. The figure shows us a wealth of information.

If you look closely, we can see the ideal spectrogram containing all the “reflections” of the whistler.

Hold On! Where did reflections come from?
Well, as I had mentioned that it propagates, I had mentioned that the tweek generated would travel into an ionospheric duct, then come back into the earth-ionosphere waveguide. Well, EM waves, in fact any wave, simply don’t do nothing when they travel from one medium to another.

Because of the change in refractive indices of the media, some of the energy carried by the waves will be reflected into the first medium, and the rest will be transmitted into the second (there’s also absorption of energy, but I’m ignoring that as it is very less compared to the other two).

In this part of the above figure, we can see the tweek, and also a number of whistlers, each dragged out more than the previous. Such whistlers are detected at the same place of origin of the tweek. They are called as Odd-Hop Whistlers, since they have “hopped” or “jumped” (i.e. reflected) an odd number of times.

Why are they getting dragged out more? Because, each undergoes a greater amount of dispersion than the previous as they travel a lot more distance after each reflection before detecting them.

Similarly, in this part of the figure, we can see all the Even-Hop Whistlers, and these are detected at the magnetically conjugate point to the point of origin of the tweek.

I had mentioned that the waves get dispersed in the ionosphere, right? Well, later studies showed that according to dispersion theory, every whistler should have a characteristic frequency (not necessarily the maximum frequency detected), that has the minimum time delay. This frequency is called the nose frequency of the whistler. This nose frequency is very apparent in the whistlers whose frequency range about 10KHz.

Spectrogram of a nose whistler

Source: Robert A. Helliwell, Whistlers and Related Ionospheric Phenomena, Stanford University Press, 1965

In fact the spectrograms shown before are actually the lower half of the spectrogram of the nose whistler.

## Practical Applications

The nose frequency of a whistler is very closely related to the electron distribution of the ionosphere. So, one of the main applications of whistler waves is to map the electron distribution of the ionosphere.

Studies of the interaction of whistler-mode waves with the charged particles of the atmosphere have indicated that it maybe possible to accelerate charges by means of man-made whistler-mode signals. This has been proposed to help protect satellites from the charged particles of the Van Allen radiation belt, during their launch.