What do these waves look like as they pass through us? Acoustic-like compression and expansion of particles as molecules temporarily reorient toward a “down” that is ever-so-slightly off from the Center of mass of the Earth? Also, I assume that these waves are very gentle sinusoids? Could the opposite — a high-amplitude gravitational square wave — be possible? What would it do to the things it passes through?
Yes, that's basically right. The gravitational way has a direction (say, z) in which its propagating. Within the plane perpendicular to that direction (x-y), a circular ring of particles will at at one moment experience squeezing in one direction (x) and stretching the perpendicular direction (y). As the wave passes through and you move from the peak of the wave to the trough, the directions reverse, so the first dir…
If this analogy holds, then can it be taken further? Acoustic waves dissipate their energy insofar as they trigger plastic deformation in a material. Could gravitational waves plastically deform... spacetime itself? Or would they just be deforming the material? Or is gravitational energy not dissipated into other forms of energy at all?