So you want to have an array of individual very long tubes, with a single sensor at one of ends of each tube? Did I understand correctly?

The following is more of an educated guess than a factual claim. If you think I'm in the right direction I could read up and try to provide an answer. In reality you always have $\omega = \sqrt{\frac{k(M + m)}{Mm}}$ But if $M$ is very large, compared to $m$ you have $M + m \approx M$ so the $(M + m)$ from the numerator kind of cancels out the $M$ in the denominator, so you're left with only $\sqrt{\frac{k}{m}}$. While the box is stationary $M$ is the mass of the box + the mass of the earth, so it is very large, compared to $m$ and when it is in free fall it is just the mass of the box.

Is wikipedia's derivation enough? en.wikipedia.org/wiki/Reduced_mass#Newtonian_mechanics