In a hypothetical situation I'm still sitting in a coffee shop but a gravitational wave similar to the three reported by LIGO passes through me from behind. The source is much closer though, so this one is perceptible but (let's hope) not yet destructive. (If a perceptible event would destroy the Earth, please put my coffee shop in a large spacecraft if necessary)
Let's say the orbital plane of the black holes is coplanar to the floor of my coffee shop so that the wave is aligned vertically/horizontally. If I stand up and extend my arms, will I sense alternating compression and pulling as if I were in an oscillating quadrupole field (pulling on my arms while compressing my height, and vice versa)?
The term "strain" is used do describe the measurement, but would I feel the effect of this strain as a distributed force gradient, so that my fingers would feel more pulling than my elbows?
If I had a ball on the end of a stretchy rubber band, would it respond to this strain (especially if $k/m$ were tuned to the frequency of the wave)? Would I see it start oscillating?
There is an interesting and somewhat related question and answer; How close would you have to be to the merger of two black holes, for the effects of gravitational waves to be detected without instruments? but I'm really trying to get at understanding what the experience would be like hypothetically.
This answer seems to touch on this question but the conclusion "…if the ripples in space-time were of very long wavelength and small amplitude, they might pass through us without distorting our individual shapes very much at all." isn't enough. If it were strong enough to notice, how would a passing gravitational wave look or feel?
Best Answer
Let me try to answer in a few separate steps. (I'll try to make it simple and people should correct me where I oversimplify things.)
What is the effect of a gravitational wave on a physical object?
Let's start with just two atoms, bound to each other by interatomic forces at a certain effective equilibrium distance. A passing gravitational wave will start to change the proper distance between the two atoms. If for example the proper distance gets longer the atoms will start to experience an attractive force, pulling them back to equilibrium. Now, if the change of GW strain happens slow enough (for GW frequencies far below the system's resonance) everything will essentially stay in equilibrium and nothing really happens. Stiff objects will keep their length.
However, for higher GW frequencies, and especially at the mechanical resonance, the system will experience an effective force and will be excited to perform real physical oscillations. It could even keep ringing after the gravitational wave has passed. If they are strong enough, these oscillations are observable as any other mechanical oscillations.
All this stays true for larger systems like your example of a ball on a rubber band or for a human body. It is also how bar detectors work.
How would a human experience this?
So, a gravitational wave exerts forces on your body by periodically stretching and compressing all the intermolecular distances inside it. That means you will basically be shaken from the inside. With reference to the stiffer parts of your body the really soft parts will move by the relative amount that is given by the GW strain $h$. The effect can be enhanced where a mechanical resonance is hit.
I guess you would experience this in many ways just like sound waves, either like a deep rumbling bass that shakes your guts, or picked up directly by your ears. I assume that within the right frequency range the ear is indeed the most sensitive sense for these vibrations.
Is it physically plausible that you could be exposed to high enough GW amplitudes?
Lets take the GW150914 event where two black holes, of several solar masses each, coalesced. Here on Earth, an estimated 1.3 billion lightyears away from the event, the maximum GW strain was in the order of $h\approx 10^{-21}$ at a frequency of about $250\,\mathrm{Hz}$. The amplitude of a gravitational wave decreases with $1/r$, so we can calculate what the strain was closer by:
Lets go as close as 1 million kilometres, which is about 1000 wavelengths out and so clearly in the far field (often everything from 2 wavelengths is called far field). Tidal forces from the black holes would be only about 5 times higher than on Earth, so perfectly bearable.
At this distance the strain is roughly $h\approx 10^{-5}$. That means that the structures of the inner ear that are maybe a few millimetres large would move by something in the order of a few tens of nanometres. Not much, but given that apparently our ears can pick up displacements of the ear drum of mere picometres that's probably perfectly audible!