Superposition for mechanical waves

[aaaa202]

Suppose we send a single mechanical pulse down a wire from one side and an identical one with the top down from the other side - like on the picture. Now suppose these two pulses interfere destructively. Then there is a specific time at which the string is completely at rest. My question is: Where is the energy from the two pulses motion stored at this instant of time? Surely there are no moving masses on the string anymore?

 

[collinsmark]

Suppose we send a single mechanical pulse down a wire from one side and an identical one with the top down from the other side - like on the picture. Now suppose these two pulses interfere destructively. Then there is a specific time at which the string is completely at rest.

Boldface mine.

I would agree that there is an instant in time in which the string contains no deformation. But that is not the same thing as saying the string is completely at rest.

My question is: Where is the energy from the two pulses motion stored at this instant of time? Surely there are no moving masses on the string anymore?

Why would you think that there is no moving mass on the string? Are there not any points on the string moving in a direction perpendicular to the length of the string?

Hint: Draw the string immediately before the instant of zero deformation, and draw it again immediately after the instant. Do you see any points on the string that in both cases (the two cases being immediately before and immediately after) continue moving in the same direction?

 

[aaaa202]

I am trying to understand what happens on the microscopic level mechanically. Is the energy on the flat string stored in kinetic energy? I have tried to draw a representation of what happens, in terms of the picture of the string being made up of a lot of point masses. I can see that when the waves interfere the midst point between them will not move since each wave front is pulling at it with equal force but opposite direction. How you get from this to a complete destruction of each wave is however harder for me to see on the microscopic level. On the other hand it is the only thing that makes sense because of the symmetry.
Can you try to draw a series like mine where you see what happens in terms of the individual point masses, or can you link me to one?

 

[voko]

You could use a simple analogy. A pendulum. When it passes through its neutral point, it looks - instantaneously - as if it were stationary without any energy, while we know that it this same instance its KE is maximal.

 

[aaaa202]

yes but look at the point particles around the point where the waves meet. They have a velocity in respectively the y and -y direction - that is they are moving away from the equilibrium line. How are they brought down to this line in the instant where the two waves have interfered at their respective position?

 

[voko]

 

[collinsmark]

Hello aaaa202,

Others have replied with some very good information. But I would like to respond to your post directly.

I am trying to understand what happens on the microscopic level mechanically. Is the energy on the flat string stored in kinetic energy?

Motion, a.k.a nonzero velocity, implies Kinetic energy. So yes, if there is motion (regarding an object having mass), there is kinetic energy.

I have tried to draw a representation of what happens, in terms of the picture of the string being made up of a lot of point masses. I can see that when the waves interfere the midst point between them will not move since each wave front is pulling at it with equal force but opposite direction.

Yes, that is correct, the midpoint on the sting doesn't move.

But consider, at the instant of zero deformation, what is happening to a point just to the left of midpoint? And a point just to the right?

Again, use superposition for each particular point. Let's start with a point just a little bit to the left of the midpoint, at the instant of zero deformation. Is the right-moving wave (the wave that's just leaving it) causing that point to move up or down? In other words, what is the right-moving wave's contribution to the point's velocity, up or down? Is the left-moving wave (the wave that's just entering it) causing that point to move up or down?

Is the contribution of each wave to that point's velocity in the same direction or opposite directions? We've already established that the waves' positions cancel each other out at the moment of zero deformation, but what about the waves' velocity contributions?

How you get from this to a complete destruction of each wave is however harder for me to see on the microscopic level. On the other hand it is the only thing that makes sense because of the symmetry.
Can you try to draw a series like mine where you see what happens in terms of the individual point masses, or can you link me to one?

I've seen that you have drawn one diagram at the point where the waves almost overlap. But I would like to draw three of them. Immediately before total overlap, At the instant of complete overlap (the instant of zero deformation), and immediately after total overlap.

Now look at a point on the string near the midpoint (but not the midpoint itself), at those three different instances in time. Does the point have a non-zero velocity? If so, it has kinetic energy.