There is also a typo in my post where there's 10^-7 instead of 10^-5, but the calculations are done correctly.
That corrected assumption gives:
d_audible / d_actual = 4 x 10^-17 x 410 Mpc = about 500,000 km
----
<rest of post deleted since it was based on an incorrect premise>
> 10^-7, that is, 4 times 10^-17 times larger than the gravitational wave detected
which doesn't sound right. (Like saying "X is 0.1 times larger than Y" when X = 10Y.) The ear displacement is 2.5×10^16 larger than the gravitational wave, no?
(Also I think 10^-7 should be 10^-5 in the quoted phrase, as per your reply above.)
Yeah, you're right, I had a typo in one of the intermediate numbers (the rest of the calcs used the correct number). Turns out I was just wrong overall though, not because of a typo but because of an incorrect assumption. Energy goes as 1/r^2, but amplitude is only 1/r, so the correct value is much smaller (it's about the Earth-Moon distance).
> 10^-7, that is, 4 times 10^-17 times larger than the gravitational wave detected which doesn't sound right. (Like saying "X is 0.1 times larger than Y" when X = 10Y.) The ear displacement is 2.5×10^16 larger
A is 2.5x10^16 larger than B is equivalent to B is 4x10^-17 larger than A, although I get what you mean by the phrasing being weird. "times the size of" would've been better than "larger".
For other readers, this poster's correction is right, here's an astronomy SA thread on it. https://astronomy.stackexchange.com/questions/34761/why-does...
I swear, the one time I try posting outside my areas of expertise :D
Agree. We should list of all kinds of similar endeavors. I volunteer to be dropped at PSR J0108-1431
But as noted in other posts the post you're replying to was based on an incorrect assumption - there's a corrected value in one of the comment threads, and the distance for it to be audible is much closer than originally posted, roughly the Earth-Moon distance.
Gravitational waves obey the inverse-square law much like most radiation does (and subject to some constraints about weird spatial geometry, but most of their propagation is going on in open flat-ish space, so we can ignore that).
GW150914, the first gravitational wave observed, had an amplitude of about 4 times 10^-22 [1], i.e., differences changed by a factor of about that. Typical sound displaces the eardrum by on the order of half a micron or so [2], with the threshold of hearing at about 100 nm. The inner ear's shape is curved, but linear length is on the order of 10 mm [3] (it curves around so the total length is longer, but the gravitational wave would be transverse along its length).
A 100 nm displacement on a 10 mm length is a relative change of (100 x 10^-9) / (10 x 10^-3) = 10^-7, that is, 4 times 10^-17 times larger than the gravitational wave detected. That gravitational wave was emitted at a distance of about 410 Mpc [1], and so we can solve:
(d / 410 Mpc)^2 = 4 x 10^-17
d^2 = 4 x 10^-17 * (410 Mpc)^2
d = 2 parsec.
Granted, this is at the limit of hearing for a very brief sound (the sound was only in the human audible range for about a tenth of a second). You'd need to be perhaps 10 times closer - about 20,000 AU - for it to be a loud sound under these assumptions.
Of course you wouldn't be around to hear it for very long because you're 20,000 AU from one of the most energetic events in the cosmos, but hey, you'd hear a brief "click". Totally worth being vaporized.
[1] https://en.wikipedia.org/wiki/First_observation_of_gravitati... [2] https://biology.stackexchange.com/questions/79963/how-far-do... [3] https://www.verywellhealth.com/inner-ear-anatomy-5094399