How loud would a literal ‘shot heard round the world’ be?
Short answer: To be heard around the world, a 20 Hz sound would have to be nearly 400 dB, which is impossibly loud.
Long answer: Sounds get weaker with distance. When you make a noise the soundwave travels outward in a shell, which is an expanding sphere with the source at the center. By the inverse square law and conservation of energy, the surface area of that sphere grows and the loudness of the sound decreases, which is one reason why sound volume fades at a distance. The other reason is atmospheric attenuation. Sound doesn’t carry forever. As the wave travels it will lose a little bit of the energy to heating the air, eventually dissipating beyond detectability.
So what if we wanted to make a sound that could be heard on the other side of the earth?
My first thought is that the inverse square law isn’t directly applicable. A super loud transglobal soundwave isn’t really expanding like a sphere but more like a circular ripple on the surface of the pond, where the pond is like the atmosphere. With the distances we’re considering the ‘roof’ of the atmosphere doesn’t really matter. In this case, the soundwave spreads out with just an inverse law, not inverse square, so it will carry farther with less loss. Additionally, after this ripple crosses half of the world it will start to refocus itself, eventually recombining at the antipode (that’s one of my favorite words, it means the exact opposite side of the earth).
You can see the same effect scaled down in videos of waves in water droplets in zero gravity:
For example, if there was a bang at the North Pole, that wave would travel down over the northern hemisphere and would spread and weaken along the way, but after passing the equator the wave would start to refocus. Eventually it would converge at the South Pole, and in the absence of attenuation the wave would have the same volume it had at the North Pole!
As a tangent, the same concept can be used in ‘sonic therapy’ to focus energy at a point inside the body. By placing a speaker and a patient at the foci of an ellipse soundwaves can be focused onto a kidney stone, causing it to crumble. Check out this video from a elliptical ripple-tank:
The attenuation is a different matter. Attenuation of sound depends on frequency, and low frequencies carry much farther than high ones. Humans can hear between 20 Hz and 20 kHz, and in moderate humidity a low frequency of 20 Hz has an attenuation of about 0.02 dB/km, while a high frequency of 20 kHz has an attenuation of 500 dB/km . Basically, bass carries.
So how loud would it have to be?
It turns out that if you want to be heard at the antipode you only need to worry about attenuation since the sound is refocusing itself. For a monotonic 20 Hz pitch to travel half of earth’s circumference of 20,000 km with attenuation of 0.02 dB/km, this sound needs to be at least 400 dB at the source.
How loud is this? Remember that decibels are a logarithmic scale – adding +10 dB means the sound is 10x louder! The threshold for pain is about 120 dB, and immediate damage and deafness can occur at 140 dB . This 400 dB sound is 1026 times louder than sounds that can cause deafness! The power in this sound is substantially greater than the power output from the sun’s surface! To be a literal ‘shot heard round the world’ it would literally be the last thing heard by most of the world because of the ensuing mass deafness.
So I suppose the answer isn’t “400 dB” but rather “it’s impossible.” It’s safe to say we’re well outside the regime described by elementary acoustics.
Of course, it’s only impossible if you want the sound to be heard. As a general rule of thumb attenuation scales with frequency, so we can make a sound that can carry farther by lowering the pitch to infrasound (another favorite word – it means sound that’s too low in pitch to be heard, just like how infrared light is just past visible red on the EM spectrum). As I said before, bass carries.
In fact, this is exactly what happened when the island volcano of Krakatoa erupted in the Pacific in 1883. The sound it made was recorded as 172 dB from 100 miles away; sailors within this range were immediately deafened by the blast.
It wasn’t audible to humans on the opposite side of the planet, but the pressure of the passing infrasound wave was detectable in mercury barometers. In fact, the passage of this wave was recorded up to seven times in some places, meaning it went around the world three and a half times! 
special thanks to /u/therationapi
image credit: Wikimedia Commons
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