How does a string store information during wave superposition?

Question

Given we have two Guassian wave pulses in the same medium (string) but in opposite directions. The principle of superposition states that they should pass through each other without being disturbed, and that the net displacement is the vector sum of the individual displacements.

Like this:

My question is, how does the string have 'memory' of the information on the shape and velocity of both pulses separately? For instance, consider if the interference were destructive. Then when the string is perfectly flat, it cannot have memory of what brought it to this state; similar to how a particle does not continue to accelerate after a force has stopped acting: it has no memory. I understand that although the displacement is zero, the velocity is non zero and at this point the string has kinetic energy. How can the string know from the velocities of various points which waves will emerge?

How do both the waves emerge unchanged?

I think my confusion stems from a lack of conceptual clarity on the principle of superposition, I may not completely understand it. Any help is appreciated.

Note to moderators: I request that this question not be treated as a duplicate. While there are similar questions here and here, none of the answers are able to justify how this happens in terms that make sense to me.

Answer 1

Let's look at it this way:

As an observer, move along one wave on an infinite string, so that it appears motionless to you. It is just a bump on the line.

Consider the other, moving wave. You can see it move towards the bump, move above it, and move past it. That moving wave has always kept its shape, even when it was travelling over the bump. So, there is nothing special happening when the two waves encounters - each one sees the other one as a bump on its way; that's superposition. No need here for any more 'memory' than when there is no bump (no other wave).

In other words, the need for a 'memory' of sort only arises because when you see the two waves superposing, you do not recognize their previous shapes, you do not 'see' in the overall shape the separated waves that you previously identified. But the way we make sense of what we perceive can be tricky - and in that case, if it may seem to you that some 'memory' is needed, then even a single travelling wave that you always can 'see' would still need this 'memory' to keep its shape. Again, nothing special here with regard to superposition.

Answer 2

Wave superposition happens because each point in a string of superimposed state, gets a pair of velocities, from a first and second impulse :

Actually, it's this individual point speed addition rule that makes wave interference work.

Answer 3

As for your conceptual clarity on superposition- imho, you already have it. Its the question which is counter intuitive.

To begin with lets make things clear-the string has no memory of its past motion. In classical waves on a string, wave evolution is differential-so the future of a wave gets determined only from the differential instant before-and nothing before that. No history is needed. All instants after the initial condition get determined from the differential evolution. This stems from the determinism of the classical motion.

Its an intriguing question as to how waves can emerge from nothing. For e.g. the pass through of two exactly opposite waves. From a matehematical POV, its clear-the superposition principle. Since the evolution of any individual wave isn't affected by the presence of another, they in fact should not stay put post superposition.

But why does the superpositioned wave bother to split back into constituent waves? Afterall, a wave of the same shape as the superpositioned wave that doesn't split is also a solution. In other words, in the extreme case of two exactly opposite waves, why doesn't the string stay flat post superposition,forever, just like an initially flat string would ?

You may have guessed the answer by now. The difference between a flat string and a flat state of superpositioned pulses is in their energies. A superpositioned flat string simply can't stay so forever since its buzzing with the energy of its constituents. Same goes for any general superpositioned state.

To extend the analogy you provided, just because an object is travelling with constant velocity, it doesn't mean its travelling with $0$ velocity.

The energy that we talk about is not just one number-rather its how much of it is distributed in the various constituents of just any one pulse. These shape constituents are just the harmonic modes while the energy constituents are normal modes.

Its so much easier when one sees the motion of just one harmonic mode of a string. There this energy is evident in the form of impending motion-even when the string is flat one doesn't expect it to stay so.

One could then ask-for a general superposition, maybe its possible that for a brief instant the string stays flat and still-how would it then separate back into its constituents? The answer is that such a state is impossible if the string wasn't initially also flat and still. So its energy would drive the string to separate back into its constituents.