Can They Hear Me In China?

“BOO!” my son yells, leaping out from a shrub.  And then he dissolves into a fit of laughter.

This is a game he likes to play, whenever he gets the chance.  As soon as we’d parked and he got out of the car, he ran up the sidewalk towards the front door of our condo.  And then he ducked back behind the hedge, lurking.  The game, now, is for me to walk towards the door.  Then he’ll jump out and shout “boo” and try to make me jump.

“Did you know I was there, daddy?” he asks.

Of course I did, I think.  You hide in the same place every time.  “Kind of,” I tell him.  “I guessed where you were.”

He blows that off.  “I was loud, wasn’t I?”

“Yes, you were,” I answer, unlocking the door.

“Was I loud enough for them to hear me in China?”

How Do We Hear?

Obviously, we hear with our ears.

howdowehear

Sound waves, which are really just pressure waves in the atmosphere, strike the outer ear and are channeled into the ear canal.  These pressure waves vibrate the eardrum, which in turn vibrates the bones of the inner ear (the malleus, the incus, and the stapes), amplifying the vibrations and transmitting them into the inner ear (or cochlea).  Hairs in the cochlea are stimulated by these vibrations, creating an electrical signal that transmits along the auditory nerve to the brain.

Yes, this is terribly simplified.

How Loud Are You?

Strictly speaking, “loud” is a matter of perception – the same pressure wave can result in different experiences of “loudness”.  However, this perception is tied to the intensity of the pressure wave, just as the perceived pitch of a sound is tied to the frequency of the wave.

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A wave

Using the above image of a wave, the intensity is how high the peaks and how low the valley is – the higher the peak, the more intense the wave.  Another way to think of intensity is how much energy the wave carries – the taller the wave, the more energy (just like how bigger ocean waves hit harder than small ones).  Frequency, on the other hand, is how fast the wave moves – the closer together the peaks, the faster the wave moves and the higher the frequency.  Generally speaking, we perceive intensity as loudness (because the pressure wave hits the ear harder) and we perceive frequency as pitch (because the pressure wave stimulates the bones in the ear faster).

“Loudness” is measured in decibels (dB), because one decibel is the “just noticeable difference” in sound intensity for the human ear – assuming the pressure wave generated is in the 1,000 Hertz (Hz) to 5,000 Hz range we are best at hearing.  Every 10 dB represents multiplying the intensity of the pressure wave by 10 – that is, a 10 dB sound is 10 times more intense than a 0 dB sound, a 40 dB sound is 10,000 times more intense than a 0 dB sound, and a 100 dB sound is 10,000,000,000 times more intense than a 0 dB sound.

We generally can’t hear anything below 0 dB, and normally speak in the 60 to 65 dB range.  A jackhammer 50 feet away is about 95 dB, a power mower 3 feet away is around 107 dB, and loudness causes pain starting around 125 dB.  Sounds at 140 dB and greater can cause permanent damage with even short exposure.

How Far Away Can We Hear?

This gets tricky, because the answer is “no further than when the perceived volume falls to 0 dB”.  Tricky, because sound obeys the inverse square law which states that for any source power P generated at the center of a sphere, the intensity of at the surface of that sphere is P/4πr2 (although a good approximation is P/r2, since the math gets easier).  According to Hyperphysics, r is pretty much always measured in meters for these purposes (because sound intensity is actually measured in watts per meter squared, so it keeps the units the same).

Since sound intensity can be transformed into decibels, it’s really not a stretch to directly apply the inverse square law to decibel measurements.  So, a 60 decibel conversation would be perceived as 60 decibels at 1 meter away, 60/(2*2) = 15 decibels at 2 meters, 60/(3*3) = 6.6 decibels at 3 meters, 60/(4-4) = 3.75 decibels at 4 meters, less than 1 decibel at 8 meters, and so on.  Realistically, at this point, it’s probably safe to call it “inaudible” (even though you could technically detect it).

How Loud Would You Have To Be For Someone To Hear You In China?

All right, here’s where the math gets… entertaining.  I live in Cincinnati, Ohio, which is (according to Google) 10,969 kilometers from Beijing.  Measuring along the curved surface of the Earth, that is.  But, to keep things simple, we’ll ignore that.  So, 10,909 kilometers is 10,909,000 meters.  To be heard in Beijing, we’d have to generate enough decibels to result in a greater than 0 dB sound 10,909,000 meters away.

For laughs, let’s aim for a 60 dB sound.  That way, our sound can be clearly understood.  The radius is 10,969,000.  So, the equation looks like this:  x/10,969,0002 = 60.  Solving for x gives us x = 60(10,969,0002), or x = 7,219,137,660,000,000 dB.  This is a nonsensical level of perceived volume, and would render you deaf in ludicrously tiny fractions of a second.

What could generate that?  Well, we’d have to reverse engineer the decibels into watts of power, which converts to 721913765999988 watts per meter, or about 721.9 terawatts of power.  Now, you can roughly convert watts to Joules per second, so that’s roughly the explosion of a 200 kiloton nuclear weapon.

Assuming I did my math correctly, which I’m not guaranteeing.  What I can guarantee is that there is no way you’d want to be standing anywhere near something loud enough in Cincinnati that you can hear it in China.

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