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When they talk about the arms of LIGO or Virgo stretching by a thousandth of the width of a proton, they always emphasize that this is the wave's amplitude, not wavelength...

The wavelengths are apparently quite long, with correspondingly low frequencies... Articles usually explain what this means in terms of what is going on out there in space, with the actual black holes and such, but...

What difference does the wavelength and frequency make for us here on Earth? For the detectors?

The detectors are said to only be capable of detecting gravitational waves within a certain range of wavelengths.... Why?

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  • $\begingroup$ This question received very little answers, here's why I think why that is. You ask several questions. The first ones (about frequency/wavelength) are very searcheable; I recommend the wikipedia article for that. Asking questions that are very searcheable discourages answers. However, your last question is interesting. With the current detectors, why is there an optimum frequency range? I didn't find information about that. My best guess is that in noise suppression there is a trade-off. Interestingly, the future GW-detector-in-space is described as having a very wide frequency sentitivity $\endgroup$ – Cleonis Aug 14 '20 at 4:46
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Detectors like LIGO, VIRGO, LISA that use interferometry have arms of a fixed basic length. At the end of those arms there are test masses that are free to move, but the displacement of those test masses will be very small compared to the length of the arms.

These types of detectors will be sensitive to wavelengths similar to that arm length for reasons similar to how the length of an antenna for a electromagnetic system influences which part of the EM spectrum it can detect. If the arm is much shorter than the wavelength, then the variations caused by the wave passing through are too small to see. Instead of measuring something where you see peak-to-peak amplitude differences, you will just be sampling points of the wave that are much less than one wavelength apart and therefore having relatively small difference in amplitude. If the arm is much longer than the wavelength, you are under-sampling the wave and cannot resolve it.

In any case, the magnitude of the effect from gravitational wave on the test masses at the ends of the arms is very, very, very small by human-sized scales.

So I think you're mixing in your mind the amplitude of the effect (very small) with where on the wave you sample the signal (could vary depending on the wavelength and the size of your detector, but typically "large").

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If a wave is of huge intensity (amplitute), it makes the test bodies move accordind to its "frequency" (which is the case, I think, observed so far). It is like a driving force to a free body.

As the observed frequencies are small enough, we observe, at best, a "coherent" states of gravitons, i.e. when their number is huge. No way to observe a single graviton with the energy $\hbar \omega$.

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