How loud would it be if every cat in the world meowed at once?

I saw this tumblr post shared on facebook the other day. I tell you what, questions like this already get me distracted, but even worse is where there is a WRONG response on it. Now I have to do the math… Hope you enjoy 😀

The question as posed is actually pretty boring. If every cat meowed at the same time, you wouldn’t hear it at all unless some cats were near you. Volume drops geometrically with distance. You’ll notice you also do not hear any of the thousands of jet engines running right now around the world or any of the construction sites (unless you happen to be near either). Let me propose a different question which I think gets at the heart of what is being asked: What if every cat in the world surrounded you and meowed at the same time? How loud would it be? First we have to correct this commentor’s understanding of decibels.

A decibel (dB) is a unit of sound, but it is logarithmic not linear. If 1 cat 1 meter away meows at 65dB, 10 cats does not make 650dB. 10 cats would be 75dB. 100 cats would be 85dB, and so on. So let’s use the original commentor’s estimate of 2 billion cats in the world. A billion is 10^9, so add 90dB, plus a little more for 2 billion, call it 92dB. So in total, 65dB+92dB is 157dB. Something like the sound of a jet engine from just 20m away. Still deafening, but not close to the proposed calculation. BUT there is a big problem here. Sound drops off with distance. This calculation assumes every one of those 2 billion fur balls is about a meter away from your face. Impossibru. Let’s be [a little more] realistic and say they are all standing in a circle around you. First let’s figure out the meow’s power instead of just how loud it is at a particular distance.

Next we need to figure out how closely we can jam those cats together. I’m assuming normalish house cats. I did a theoretical calculation assuming the cross-sectional area and found that it agree almost exactly with the cats and estimated area in this image I googled. CPD confirmed!

Now let’s turn that sea of discrete cats into a nice uniform soup of meowy catness by giving it an average power density.

Next let’s figure out how big of a radius we need for all those cats! A billion cats needs a circle of about 10km radius. Fun facts we learn on the way. I left this as a function of Number of Cats for plotting later.

Now the tricky part of the calculation is the calculus, but fortunately sound is just like electric charge! Anything radiative follows the Inverse Square Law (gravity, light, electric charge, etc), so I borrowed some math already done. Thanks to my brother-in-law Alex for helping me find this!

Finally we put that math in a form I can use. That’s a cool looking equation, huh? 😀

And here are the results! In short, it would be about 97dB. That’s something like the sound of a loud motorcycle passing right by you. Not deafening, but uncomfortably loud. Interestingly you don’t even need all the cats in the world. After about 10,000 cats, they start getting too far away to matter. In the lower numbers of cats it matters a little bit how tall you are. Anyway, hope that was fun! If you find errors in my calculations please let me know!