# Wave propagation around the globe

I've been able to detect the pressure wave of the tonga volcanic eruption with the atmospheric pressure sensor here in switzerland (about 17Mm away). I've detected a swing of about +1.2hPa and -1hPa.

I think I've also detected a second wave a few hours later (I'm not sure anymore if time is in UTC or UTC +1)

My first thought was: Oh yeah, that came "the other way around" but than it dawned on me that I'm basically every possible distance between ~17Mm and ~23Mm away from tonga. So the first wave is the shortest distance. But after that I should get some interference of waves that travelled different directions around the world.

So my question is: How does a pressure wave traveling within the atmosphere around the globe behave?

• How long/far does the inverse square law hold true?
• as the circle of the wavefront get smaller again after going around 1/4 of the circumference, does the amplitude of the signal rise agin?
• would there be a significant "spike" detectable exactly opposite of the eruption?

PS: I know, that's more than one question, but I don't see a way to phrase it differently. Feel free, to edit to improve!

• Are you sure that any of the data shown is related to the Tonga eruption? If so, how so? Feb 28, 2022 at 7:10
• The pressure wave spreads out in a circle, so its direction of travel is along a radius. On the sphere this means it is traveling along a great circle path from Tonga to you. This gives you the 17Mm and 23Mm distances. There are other paths from Tonga to you, but they are not great circle paths. Feb 28, 2022 at 7:11
• @AndreaAlciato that the first one is related is for sure. The timing is correct (also, official stations in switzerland detected it as well) and the "waveform" is unique. Changes due to weather look different. Feb 28, 2022 at 7:58
• Feb 28, 2022 at 8:30
• This is an amazing subject. This Master Thesis gives a taste and references how complicated the treatment of those waves can be. Feb 28, 2022 at 12:18

Long-range acoustic propagation in the atmosphere is a complicated subject with lots of research in it. Here are the answers to your specific questions, though:

• How long/far does the inverse square law hold true?

The inverse square law describes the geometric spreading of energy over an expanding sphere, and thus applies as long as the wave can spread out in three dimensions. The atmosphere is pretty thick, but not compared to the size of the earth. Thus, the energy will eventually start to spread in an effectively two-dimensional waveguide, and the "inverse square law" would become an "inverse law". The energy of a pressure wave is proportional to the square of the pressure, so this "inverse law" would lead to the pressure amplitude decreasing with square-root of the radius.

• As the circle of the wavefront get smaller again after going around 1/4 of the circumference, does the amplitude of the signal rise again?
• Would there be a significant "spike" detectable exactly opposite of the eruption?

All of the discussion above is for a flat world. On a sphere any sort of "inverse X law" would become rather complicated to describe, but your intuition is correct. At the antipodal point there is likely to be a pressure spike. However, this spike is not likely to be anywhere near the amplitude of the original spike for a number of reasons. First, the Earth is not perfectly smooth or spherical, and so different parts of the wavefront will be traveling different distances to reach the antipodal point. Second, the atmosphere does not have a uniform temperature (and hence wave speed), and so different parts of the wave front will propagate at different speeds, and all at different times. Third, the air is not static, and so the wavefront will be translated in space throughout its propagation and become relatively incoherent due to turbulent scattering. Fourth, the air is absorptive, and so some (or a lot) of the initial energy will have dissipated as heat.

All of these effects become less important for long-wavelength signals such as infrasound. (The importance of these effects are why you don't hear any conversations from the opposite side of the globe.) For a very rough estimate, you would expect the antipodal spike to be potentially observable if the wavelength was much larger than the largest inhomogeneity length scale of interest, which I imagine is the length scale of mountain ranges. Let's go with 100 km for that length scale. The speed of sound in air is on the order of 0.3 km/s, and so a 100 km wavelength would be associated with a signal with a period of about 300 s (about 0.003 Hz). The eruption appears to have a similar frequency, and so this (admittedly very rough) approximation suggests that the antipodal point likely exhibited the spike you are suggesting.

• Awesome answer! Thank you. About the "Frequency": Sadly, my sampling rate was only 5 minutes. But the first "full wave" was around 30 minutes long. The second one around 40minutes. So 1800s to 2400s, in that range. 500uHz. 600km Wavelength... Feb 28, 2022 at 13:22