Today in class (11/1/19) I described a puzzle that I didn’t have a clear explanation for. The puzzle involved two pulses of the same shape but opposite elevation rushing toward each other from either end of a string. There comes a point in time where they merge and the string appears to be perfectly flat.
The question was what happened to the energy?
Now, sitting here in my home, having foisted my children off on some very nice neighbors, and with a little extra diet coke in my system, I have a clear answer. The answer is that in this situation the energy is maximally kinetic. The string is indeed completely relaxed, but the parts of the string that are wiggling up and down are in fact doing so in such a way that the kinetic energy is equal to the total energy at this moment.
It’s a nice puzzle because it really tempts you to think of the energy as going away, even though that can’t possibly be the case. The reason you might think it went away is (a) visually, the snapshot of the string with no bumps looks like a string that isn’t vibrating, but of course, the snapshot is incomplete. It fails to tell you about the transverse motion of the bits that make up the string. (b) If you are overly reliant on formulas, you might recall that the total energy density at some position along the string is proportional to the square-amplitude, but that only applies for a coherent wave with a pure frequency.