# Sounding Out Krakatoa

I recently watched an interesting documentary on Krakatoa, which is what inspired this post.

The 1883 eruption of Krakatoa, a volcanic Indonesian island, was one of the largest in recorded history, killing between 30,000 – 100,000 people. Wikipedia gives a good overview of its remarkable destructive power. The sound that emanated from the eruption was perhaps the loudest in recorded history — reports suggest that sailors ruptured their eardrums, and subsequently went deaf, up to 80 miles away from the island. The eruption was heard across huge distances, from Sri Lanka to Australia (see pg 80-87). People on Rodrigues Island, close to Madagascar, were reported to have heard the eruption from across the Indian Ocean. Rodrigues Island is about 5,700 km or 3,800 miles from Krakatoa. Here is a Google map showing their separation (click to enlarge):

In this post, I intend to make a couple calculations to discuss the following:

1. The approximate sound level at Rodrigues Island
2. The maximum distance at which the volcano was probably heard
3. The possibility of acoustic circumnavigation of the world

There are three facts that are needed to discuss the above points:

1. Reports suggest that the sound level of the eruption was approximately a whopping 175 dB at 100 miles from the volcano.
2. Sound intensity falls as $1/r^2$.
3. Acoustic damping in air is generally lower for lower frequencies. (A great little applet where one can calculate the sound absorption coefficient of air can be found here.)

In addition, we can make some decent estimates by using the sound level formula:

$SL (dB) = 10*Log_{10}(I/I_0) - \alpha*r$,

where $I_0$ is $10^{-12} W/m^2$ is the threshold of human hearing, $\alpha$ is the coefficient of sound absorption in $dB/m$ and $r$ is the distance in $m$.

Here is a plot for the sound level in dB as a function of distance from the eruption site with Rodrigues Island marked by the dots (click to enlarge):

There are immediately a couple things to note:

1. I have plotted three curves: the blue curve is a calculation that does not consider any damping from air at all, the yellow curve considers a low but audible frequency sound that includes damping, while the green curve considers an infrasonic sound wave that also includes damping. (The damping coefficient was obtained from this link assuming a temperature of 20C, pressure of 1atm and humidity of 75%).
2. Close to the volcano, this calculation gives us an unrealistically large value for the sound level. It turns out that sound cannot exceed ~194dB because this is the sound level at which rarefaction of air corresponds to a vacuum. Values greater than this correspond to a shock wave.

Keeping these things in mind, the yellow curve probably is the best estimate of the sound level on Rodrigues Island, since it includes damping for an audible signal (humans can’t hear below about 20 Hz). Therefore, we can estimate that the eruption was heard with about 70 dB on Rodrigues Island! This is approximately the sound level of a noisy restaurant.

If we follow the yellow line to about the 40 dB mark, which is an approximate value where someone may still notice the sound, this would be at a distance of about 4,800 miles! This is approximately the distance from Cape Town, South Africa to Baghdad, Iraq.

The last point to address is the seven-time infrasonic global circumnavigation. It turns out that if one follows the green line on the plot out to where it reaches the 0 dB level, this would be approximately at a distance of about 11 million meters. The earth’s circumference is approximately 40 millions meters, however, and if we were to circumnavigate the world seven times, the required distance of travel would be 280 million meters. What went wrong in the calculation?

There is one major factor to consider. Very low frequency sound basically propagates with very little damping through air. For sub-Hz infrasonic sound, values for the absorption coefficient don’t seem to be very easy to find! (If you know of a database for these, please share and I’ll update this post). Let us then consider the case of no damping (the blue curve). The blue curve actually crosses the 0 dB mark at a distance of approximately 85 trillion meters. This way over-steps the mark (corresponding to circumnavigation 2.125 million times!). Even though this is a ridiculous estimate, at least it beat the 280 million meter mark, which suggests that with the right absorption coefficient, we may be in the right ballpark. A quick calculation shows that for realistic values of the absorption coefficient (about half the value of the 5Hz sound absorption coefficient), we would be very close to the 280 million meter mark (in fact, I get about 225 million meters for this absorption coefficient). This tells us that it is indeed possible for low frequency sound to circumnavigate the planet this way!

Interestingly, we can learn quite a bit concerning the sound propagation of the Krakatoa eruption using relatively simple physics.

Note: Throughout, we have neglected one very important effect — that of reflection. Anyone who has been inside an anechoic chamber will be acutely aware of the effects of sound reflection. (In an anechoic chamber, one can actually stand at different spots and hear the interference pattern when playing a sine wave from a pair of speakers). Even with this oversight, it seems like we have been able to capture the essential points, though reflection probably had a non-trivial effect on the acoustic propagation as well.