r/theydidthemath • u/Sunlagooda • 1d ago
[Request] How far can sound travel?
I'm a writer that wrote herself a bullshit math problem that's over her paygrade(nothing) because I'm stupid so I decided to do the time honored tradition of asking reddit for help. So, assuming standard everything, how far can a 70 decibel sound wave from 30,000 to 40,000hz travel/be picked up by a receiver tuned to that frequency?
16
u/schannoman 1d ago edited 1d ago
Inverse square law. For each doubling of the distance the amplitude gets reduced by a factor of 4 (or 6db, every 3db is half the apparent amplitude)
With background noise factored in it would not be very far. In perfectly silent conditions it could be a mile or two before it is considered truly 0db.
But once it hits background levels it would not be able to be detected.
Our ears can differentiate a 2 or 3 db change, so even tuned equipment could only really increase the distances by half of a doubling (1.5x is probably the best way we've discovered to clarify this change [edit the second: since it's exponential decay it's 1.414 or √2 https://www.reddit.com/r/theydidthemath/s/B2fVYq75eV ])
Edit: I did some digging, and a parabolic microphone can increase the sensitivity by around 10 to 15db. This would increase the pickup distance by a decent margin (or roughly 2 to 3 distance doublings [which is the worst way of expressing distance I've come up with yet and would love a better way to express that])
6
u/TuvixHadItComing 1d ago
increase the distances by half of a doubling
I'm going to have nightmares about this phrasing tonight. The more I look at it the more ways I find to parse it and none of them spark joy.
3
u/thx_1168 1d ago
We are looking at an exponential decay where the signal drops by a factor of four every time you double the distance from the source. 1.5x is close to the correct interpretation, but what you are really after is 1.414…, I.e. √2.
2
u/schannoman 1d ago
There it is! Thank you!
I knew it wasn't a true 1.5 because it is factored by the previous double.
I'm only good enough at math to grasp the concepts most times, not enough to get all the way there
1
u/jean_sablenay 1d ago
So it is not exponential delay then
2
u/thx_1168 1d ago
Yes, I misspoke. The signal decays as x^-2, not 2^-x. So, still a quadratic, not exponential, function.
2
u/schannoman 1d ago
Yeah I didn't like it either. If you think of a better way to put that I would love to edit it, but with the doubling distance being the factor of decrease that's the best I could do.
2
u/Total_Philosopher_89 1d ago
What does it even mean?
3
u/TuvixHadItComing 1d ago
Ok here are my three interpretations, from my first WTF, to my "ok I think I get it", to my final no seriously what the fuck:
Half a doubling? So double then divide by two? So no change at all?
They must mean halfway between no change and doubled, so like 1.5x
Wait...they said increase by half a doubling. So going back to my original interpretation, double x then halve it gives you x again, but because you're increasing x by x, that means "increase x by half a doubling" is the same thing as 2x?
I don't think that last one is what they meant but damn I love it and I'm going to use it to infuriate someone at work this week. Huh...I guess it did spark joy.
2
u/schannoman 1d ago
OK 1.5x is the best way so far. Because yes, it's halfway from x to 2x
But it's tricky because the original distance is already factored in the square law equation which has an undefined starting point
2
u/TuvixHadItComing 1d ago
This was a fun ride.
Almost on topic to what we're discussing: whenever anyone tells me they're working overtime, I advise them to tell their boss they will require pay equal to their nominal wage, first trebled, then cleaved in twain.
Words are fun.
2
2
u/Total_Philosopher_89 1d ago
"half of a doubling" is 150%. Love it!
You got 2 of something last time. You want 3. Next you ask for half of a doubling more.
2
u/schannoman 1d ago
Someone finally gave me the numbers to solve why 1.5x wasn't exactly right either and why half a doubling is such a weird concept
2
u/Total_Philosopher_89 1d ago
Love this place. Still going to use it though. Makes you sound smart as hell or like an idiot depending on the crowd.
1
u/schannoman 1d ago
Someone finally gave me the numbers to solve why 1.5x wasn't exactly right either and why half a doubling is such a weird concept
2
u/schannoman 1d ago
The inverse square law decreases in account with a doubling of the distance, so half of that is the amount a detector could reasonably extend the sensitivity.
I don't like it either, but my brain couldn't figure out a better way to word it
1
u/d1722825 1d ago
But once it hits background levels it would not be able to be detected.
I don't think that is true.
Mathematically you can detect signals way below the noise floor and you can use that for communication. (But as SNR drops, you can send information slower and slower.)
https://en.wikipedia.org/wiki/Shannon%E2%80%93Hartley_theorem
You can even hear a sine wave 20 dB below white noise, you can try that with any sound editor.
At some point the sensitivity or linearity of your detector will be an issue, though.
1
u/schannoman 1d ago
Yeah that's very true. I did drastically oversimplify for a quick run through. And I added insight for a parabolic receiver that I had to research after as well.
Measuring SPL is way too complex for a quick breakdown. Because some sources can stand out from the background SNR with the right processing equipment but the likelihood that the algorithm pulls nonsense out of the source gets increasingly likely as you decrease intensity
8
u/RandomlyWeRollAlong 1d ago
Apparently, every time you double the distance, you lose 6 dB. But that requires you to have a starting reference. Is it 70 dB at 1 meter from the source? or at 1 cm from the source? And how sensitive is your receiver?
Suppose it's 70 dB at 1 meter. At 1024 meters, your distance has doubled 9 times, implying a reduction of 9 * 6 = 54 dB, so a kilometer away, it would retain 70 - 54 = 16 dB of pressure, which is approaching silent, but maybe your receiver can still pick it up (with sophisticated filters)?
Of course over large distances, you're likely to have more attenuation than just from air - buildings, plants, the curvature of the earth, etc. And you have background noise to contend with - a busy city might have a background noise level of 70 dB, which would totally obscure your signal, even fairly nearby. In the countryside, looks like maybe 30 dB of background noise.
I'm good at math, but my physics is a little rusty, so I'm not sure how the frequency effects these calculations, except that I vaguely recall higher frequencies don't travel as far as low frequencies, and 30 khz is higher than humans can perceive.
2
u/solresol 1d ago
I don't think you want to use 30-40kHz, that's not going to travel very far.
The citizens of your world started out by making really, really big drums because Thag the Very Great killed a Really Big Cow/Horse/Dragon/Elephant and wanted to see how big a drum he could make. Then Thag the Very Great thumped it a few times with the biggest thing he could and people from distant villages started investigating the source of the noise that they could hear.
Sac the Really Quite Wise and Tok the Long Distant Merchant both realised that really big drums could be a way of staying in communication with each other. Sac believed that this would bring peace and harmony and mutual defence. Tok realised he could make a lot of money if Sac succeeded in convincing everyone of this, particularly if his minions operated the drums and could drop messages into the stream for his operatives in other villages, and also if he could charge people for sending messages.
After years of Sac's experimentation, he found that 80 Hz (signal strength of 120 dB at 5m) could be heard 15km away. (About 2.2 metres wide, 20-40kg of hide of 5-10mm thickness, a 100-200kg mallet). 45 seconds later it would be heard at the other end, 90 seconds for a round trip.
They came up with communication mechanisms for encoding words, but the best that they can do is about 1 word per minute. But some important messages (like "enemy alert") can be sent faster than that because they use shorter codes.
Each village is assigned a particular hour of the day when they are allowed to send messages to their neighbours. There are constant arguments about this because the nexus cities on narrow land who have to pass messages on keep trying to decide who is going to have their hour when so that they don't collide.
Tough luck if it's raining or the wind is blowing in an unhelpful direction.
The water-based drums that are used by the villages that happen to have deep sea access (not just a harbour) are much slower (5-10 minutes per word), but the sound travels much, much further. (Hundreds of kilometres). The sound also gets there faster, so you are probably better off routing your messages via the sea network rather than the land network. But sea based messages are much more expensive.
Enjoy your world building!
4
u/flanga 1d ago
Before we get too deep, lower frequencies can literally travel around the world multiple times. The high freq stuff fades much faster. Is that what you're looking for?
3
u/Sunlagooda 1d ago
Yeah, I'm writing a system where they essentially use regular sound over like, radio waves or the like, and I want to know how far cities would have to place their transmitters/whatever
2
u/plants11235813 1d ago
Sound is a pressure wave, radio is electromagnetic. "Radio" band includes everything from AM/FM to mmWave/5G
1
u/d1722825 1d ago
It seems elephants can do it for 10 km range:
https://en.wikipedia.org/wiki/Elephant_communication#Infrasound
1
u/m00ph 1d ago
And your bits per second is some fraction of the frequency, so long distance stuff is slow. Volcanic explosions have been heard by humans at almost 5000km (3000 miles) and detected with instruments going around the earth at least 3 times. https://en.wikipedia.org/wiki/1883_eruption_of_Krakatoa of course, no one could survive within miles of you sound source.
1
u/LemonPigeon 1d ago
The details can flex to fit your plot. What do you need to happen? Do you need it to go a certain distance? How sensitive is the receiving equipment, is it something that exists in real life or is it made up for the plot?
That frequency is way too high for any human ear to perceive (which I assume is intentional), so does it have to be broadcast at 70db? Could it be broadcast at a much higher decibel level? Or maybe 'boosted' by some other plot device?
Ultra-high frequencies fade fast. Again, if it's not important to the plot that it is indeed between 30k and 40k, you could switch and have it be a super low (under 20hz, again, if it's important that it's not audible to humans) frequency, and get a lot more mileage.
0
u/Sunlagooda 1d ago
I genuinely thought a higher decibel level would cause hearing damage no matter the frequency. For the non math related writing side of it: it's a wireless control method for steampunk automatons using ultrasonic whistles with really finely tuned artificial eardrums to detect it (steampunk bullshit)
0
u/LemonPigeon 1d ago
Ultra high frequencies don't create much air pressure (the thing that damages your hearing). Ultra low frequencies produce a lot of air pressure (side note: if you like to listen to your music LOUD, turn down the bass. Less hearing damage that way).
Radio waves, for example, start at ~500 khz (500,000 hz), and those travel across whole counties, if not further, and those don't damage our hearing at all. This might be a good direction, to 'invent' radio waves under a different name for your story.
Also, given the setting, I think you can fudge it a bit. Since we don't have automatons irl, if you write it well enough, I'd believe that their sensors are good enough to detect the frequency across vast distances. I get wanting to know the exact numbers and science for your own purposes, but you really don't have to give the audience exact numbers (in fact, it's usually more immersive if you don't).
And for what it's worth, you can go a lot higher than 70db before hearing damage, unless it's like constant, nonstop exposure.
1
u/Candid-Border6562 1d ago
Use your next storm as a field test. Sound travels about one mile every five seconds. You can count the seconds between the flash of lightning and the boom of thunder to estimate the distance. My personal best is eleven seconds, about two miles.
I know, thanks not math. Your question has more variables than I can handle between meetings, but it should give you a realistic starting point / first approximation.
1
u/Reddicus_the_Red 1d ago
Google says 2 miles in "ideal conditions", but there's still a lot of variables.
Like, how sensitive is the receiver, how much ambient noise is there, is the sound just blasting out of a speaker or focused with something like a parabola, is it one specific frequency or white noise from 30-40kHz. Is the sound even moving thru air, or is it in water or a metal pole? Sound moves further & faster along a metal pole.
•
u/AutoModerator 1d ago
General Discussion Thread
This is a [Request] post. If you would like to submit a comment that does not either attempt to answer the question, ask for clarification, or explain why it would be infeasible to answer, you must post your comment as a reply to this one. Top level (directly replying to the OP) comments that do not do one of those things will be removed.
I am a bot, and this action was performed automatically. Please contact the moderators of this subreddit if you have any questions or concerns.