My professor asked an interesting question at the end of the last class, but I can't figure out the answer. The question is this (recalled from memory):
There are two travelling wave pulses moving in opposite directions along a rope with equal and opposite amplitudes. Then when the two wave pulses meet they destructively interfere and for that instant the rope is flat. Why do the waves continue after that point?
Here's a picture I found that illustrates the scenario
I know it's got to have something to do with the conservation laws, but I haven't been able to reason it out. So from what I understand waves propogate because the front of the wave is pulling the part of the rope in front of it upward and the back of the wave is pulling downward and the net effect is a pulse that propogates forward in the rope (is that right?). But then, to me, that means that if the rope is ever flat then nothing is pulling on anything else so the wave shouldn't start up again.
From a conservation perspective, I guess there's excess energy in the system and that's what keeps the waves moving, but then where's that extra energy when the waves cancel out? Is it just converted to some sort of potential energy?
This question is really vexing! :\
Best Answer
What you cannot see by drawing the picture is the velocity of the individual points of the string. Even if the string is flat at the moment of "cancellation", the string is still moving in that instant. It doesn't stop moving just because it looked flat for one instant. Your "extra" or "hidden" energy here is plain old kinetic energy.
Mathematically, the reason is that the wave-equation is second-order, hence requires both the momentary position of the string as well as the momentary velocity of each point on it to yield a unique solution.