Have you ever considered what actually causes the sound of thunder? The past summer brought with it a few thunder storms, which got me thinking about a topic I find very fascinating and cool: simulating thunder!
Note that the simulation can be quite heavy for more complex lightning strikes (some are even unrealistically complex) and for longer distances from the strike. Calculating the result can take quite a bit of time, depending on your computer. A sample frequency of 22 kHz is used and the simulation is monaural. The lightning consists of a single discharge between the cloud and the ground.
If the simulation appears to jam up, please reload the page. Once again, I recommend Chrome for the simulation. Take a look at the source code here - sorry, I was somewhat lazy with the commenting of the code!
What is lightning?
Let's use the definition of lightning given by google:
The occurrence of a natural electrical discharge of very short duration and high voltage between a cloud and the ground or within a cloud, accompanied by a bright flash and typically also thunder.
I believe the geometry of the lightning channel becomes clearer when one considers the part of the following video starting at 1 min 15 s:
Note that the person in the video talks about the lightning consisting of "roughly 50 yard segments". These segments, referred to as the tortuosity of the lightning channel, are usually between 5 and 70 meters long [Rakov et al. 2003].
In our case, let's simplify the lightning as consisting of pretty much straight lines, with a random length of 5 to 70 meters. The lines zig-zag constantly, with a random variation of about 16 degrees between each line. Also, we'll need to add a small statistical deviation in the vertical direction.
Here is an example of what this looks like (calculated in 3D, and projected to 2D):
Lightning channels generated using a simple algorithm loosely based on the statistics given by Rakov et al.
What causes thunder?
Ok, so what causes the sound of the lightning? Let's consider what happens when the discharge happens. We very quickly warm up a channel of air (the orange area in the cut plane image below). As the temperature in the channel rapidly rises to ~24000K [Orville, 1968], the pressure of the air in the channel rises enormously (to about 10^6 Pa). This pushes the the air outwards at speeds exceeding the speed of sound, causing a shock wave expanding at roughly 3000 m/s [Few, 1986].
The shockwave caused by lightning
After this, the air in the channel quickly cools down. The pressure behind the rapidly expanding shock wave will momentarily drop below atmospheric pressure due to the inertia of the outwards traveling air mass. The shock wave will travel some distance (the "relaxation radius"), after which it will dissipate, leaving behind what is called the weak shock wave. This weak shock wave can now be plotted as a function of pressure:
Weak shock wave [Brode, 1955]
Never mind the scales for now (y-axis represents atmospheric pressure), but note this: the pressure wave will propagate towards you from the lightning so that the "sharp" part of it will reach you first.
What is thunder?
Ok, so now we now that the really hot lightning channel causes a traveling pressure wave. We also know that changes in pressure equals sound. So what we're hearing is the pressure waves caused by the rapidly heating air in the lightning channel.
But what causes the rumbling sound? Why does the thunder keep on rumbling for many seconds? What makes a close lightning strike sound (sort of) like a clap, while a distant strike can only be heard as rumble?
Here are some of the reasons:
1. The size of the lightning is huge
Sounds from different parts of the lightning will arrive at different times
Consider a lightning strike some distance from you, as in the image above. If you measure the distance to each part of the lightning, you will see that the distance can vary by miles/kilometres! Considering that sound only travels at about 340 m/s (1,125 ft/s), there will be multiple seconds between when the sound from the nearest part of the lightning strike arrives at your position, as compared to the sounds form the more distant parts.
The situation can also be thought of according to Huygens' principle, which states that any source can be thought of as a series of spherical sources (kind of like in the image above). This is how the simulation, presented at the beginning of this post, works. The lightning is divided into multiple small segments, each modeled as a separate spherical sound source.
2. Sound attenuates by distance
It is, perhaps, obvious that more distant sound sources are quieter. But, additionally, it should be noted that higher frequencies attenuate much faster! Thus, when sound travels a distance, it gets "muffled" by air. Wolfram alpha is a great resource for this, it calculates this attenuation according to ISO 9613-1:1993. This causes distant sounds to "rumble" while the closest sounds are sharp and discernible.
3. Other stuff
There are loads of things at play in real life (for example atmospheric diffraction, which is the reason for the thunder sometimes being completely inaudible even when the lightning strike is clearly visible). If you're interested in learning more about the topic, I found the following books useful:
- Vladimir Rakov & Martin Uman - Lightning, physics and effects
- Geophysics study committee - Earth's electrical environment
- Hans Volland - Handbook of Atmospheric Electrodynamics, Volume 2
4. Future ideas
It would be really cool to make the simulation in stereo, so that the sounds from the lightning channel are panned to their respective position (or even using head related transfer functions!). If someone else is up for the task and has some knowledge of acoustics and DSP, feel free to contact me! 🙂