How fast does force propagate through matter? 
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Is it possible for information to be transmitted faster than light? 

Consider the following thought experiment. You have a long perfectly rigid beam (for the sake of simplicity, suppose it is one light-second long) which is placed on a fulcrum in the middle, so it is like an extremely long see-saw. There are two participants on either side of the beam.
Suppose participant A flips a coin and, based on the flip, chooses to place either a light or heavy boulder on his end of the beam. Participant B, one light-second away, sits down on his end of the beam.
What happens to participant B for the first second of his sit? If participant A randomly decided to place a light boulder on his end, then participant B would lower the beam with his weight; conversely, if there's a heavy boulder on the other end, he'll stay up in the air. Either way, he will know in that first second what the outcome of participant A's coin flip was, and therefore gain one bit of information faster than the speed of light! This is, of course, patently impossible.
My guess, as you can probably tell from the title of the question, is that this entire hypothetical situation cannot happen because there is not really such a thing as a "perfectly rigid body". The reason that one end of a lever moves at all in relation to the other is because the electromagnetic forces between the atoms in the beam push each other up. But what does this "look like" for absurd lengths like one light-second? Does the upward motion travel like a wave through the beam? Does the speed of this wave depend on the material, the magnitude of the forces involved, or something else entirely? Is there a name for it? And what, if anything, does participant B feel in that first second, and what would the beam "look like" to an external observer who can see the whole beam at once? I can't intuitively visualize any of this at all.
Standard disclaimer: I only know high-school level physics, if that helps when aiming answers.
 A: You're correct that there is no such thing as a rigid body in reality. Any time a force is applied to an object at one point (such as a boulder being placed on one end of the see-saw), it only immediately applies to the molecules that it is touching. The displacement of those molecules propagates to the rest of the object as a "deformation wave," which is basically the same thing as a sound wave (though there are differences to the way sound is carried in fluids like air).
In your example, what you would see as a distant observer is the boulder being placed on one end of the see-saw and the person sitting down on the other end simultaneously. Both ends would dip downward due to the force on them, so you would see curves begin to appear in the bar. Those curves would travel as waves along the bar toward the center, pass each other in the center, and then would eventually reach the opposite end from where they started, where they would have the effect you'd normally expect them to have (i.e. person B might be pushed back up). Note that since the speed of sound is much slower than that of light, it would in general take longer than a second for the effect to propagate from one end of the bar to the other.
A: For longitudinal waves, the effect moves with the speed of sound inside the matter which is equal to $$c=\sqrt{\frac{E}{\rho}}$$ where $E$ is the modulus of elasticity and $\rho$ is the mass density. For steel this is like $5000\;{\rm m/s}$.
For beam transverse waves, it depends on which harmonic is excited and how many wavelengths fit in the length of the beam. For a beam of length $L$, with 2nd area moment $I$ and cross sectional area $A$ the 1st harmonic effect speed is $$c=\frac{\pi}{L}\sqrt{\frac{E\,I}{A\,\rho}}$$
