People who know me know that I have an interest in physics.  A few weeks ago, I saw this post at Mazui fansubs that calculated the size of some fireworks fired in To Aru Kagaku no Railgun.  On a similar note, if you’ve watched Sora no Woto episode 12, you know that Kanata hears a signal from quite a long distance away.  Some people have expressed skepticism that she could have heard it at all.  Inspired by that post from Mazui, I decided to investigate the conditions myself.  I do not claim to be an expert in physics, but I’m going to give this my best shot with the information I have at hand.  I invite all corrections to the following.  Now, let’s get underway.

Now now, don't jump the gun.

The first thing we must establish is the sound level of Rio’s trumpet.  Obviously, if she doesn’t play loud enough, the sound will never travel the distance to reach Kanata’s ears.  We need to start with the maximum loudness.  I found this post at Trumpet Master that states the maximum sound level for a trumpet is 140 decibels (dB).  The figure is impressive, but there’s two qualms with using this figure.  First, as the maximum, it’s unlikely that you can get much music out of the instrument.  In my head, I just imagine someone blowing as hard as possible to make the loudest sound.  Second, the pain threshold for the human ear is about 120 dB.  Above this, we might hear sound, but we can’t really interpret it well.  It sounds distorted.  Thus, we can’t directly use the maximum, but it does mean that we can be liberal when selecting a point to experiment.  I think 100 dB is a reasonable value, so that is what I will use.

Rio does play a pretty mean trumpet.

The second thing to establish is the speed of the tank Rio rides.  The fastest armored vehicle of World War II that I could find is the M18 Hellcat.  The speed tops out at a blistering 60 miles per hour.  The measurement is quite impressive, but there’s a fault with the choice.  This is not a tank in the traditional sense.  They used it to take out tanks, but it lacked much of the armor and protective features we think of in a typical tank.  Therefore, I also selected the Cromwell to use in this experiment.  This British tank had a maximum speed of 40 miles per hour, though they restricted it to 32 in practice to maintain structural integrity.  The difference in speeds makes a difference, but probably not for the reason you think.  A faster tank actually puts Rio’s trumpet playing skills at a disadvantage.  Why?  Because I’m going to assume that Rio moves at top speed all the way to the front.  This means a faster tank must start further away from the battle.

Really, these bulky things don't look too fast...

With these parameters established, we also need to think about the conditions.  I’m not going to spend weeks of my life becoming an expert in the physics of sound just to write a post.  Therefore, we’re going to assume ideal conditions for this experiment.  This means that the energy of the sound waves is conserved, it continuously travels, and no amplification or interference occurs.  We know from the episode that once the Colonel escapes, he plans for the battle to commence in 10 minutes.  From the time the troops start moving until the time that Rio arrives, 3 more minutes elapse.  The total time is 13 minutes.  As stated earlier, Rio must move at top speed all the way to the front.  This means at 60 miles per hour, she starts 20900 meters away, and at 40 miles per hour, she starts 13900 meters away.  Her trumpet plays at 100 dB.  You can find the formulas here, or you can just plug it into a prebuilt calculator here.  Let’s get started.

Clear as mud, right?

So what does it all mean?  First, the starting distance doesn’t make much of a difference.  However, I think we can agree that even the small change in decibels could potentially make or break something like this.  Second, to interpret this properly, you need some sort of baseline to compare it too.  A lot of sites state that 15 dB is about what you can expect from a whisper.  That gives you an idea of what exactly she hears.  The work means that it is plausible with ideal conditions.  Kanata could indeed hear Rio’s signal.

And there was much rejoicing!

We do need to note two things.  First, the curve for diminishing intensity remains the same no matter where you start.  What this means is that if Rio plays 10 decibels higher, it stays 10 decibels higher across the board until it reached Kanata.  That difference roughly correlates to doubling the loudness, which greatly increases the chances of this plan working.  Second, decibels are objective.  We got to these formulas by taking the averages of many people.  What an actual person experiences is subjective.  Some are more sensitive to sound, and others less.  This explains why Kanata hears the signal while Kureha has no idea what’s happening.  It also means that Kanata’s genetics can pick up a little slack by making her more sensitive to sound.  With this, I think I have pretty well wrapped up the post.  Again, if you have anything to add or I’ve made mistakes, please comment.  I’m actually really interested in this topic.