Energy conservation with interfering waves

[SaraF]

Homework Statement

This is a theory question that has puzzled me for a long time: Two waves of exactly the same amplitude, and exactly opposite phase and opposite velocity are traveling toward each other through some medium such as a rope, a slinky, or whatever. They meet and interfere destructively. While they are passing through each other, the displacement of the medium is zero. After they've passed through, they resume traveling in their respective directions as before. My question is, what happens to their energy while they are destructively interfering with each other?

Homework Equations

none

The Attempt at a Solution

The energy can't be potential because there is no displacement in the region where the two waves exist simultaneously. If it's kinetic, it would have to be on a molecular level (again, because no bulk displacement) and that means the medium would heat up. If so, there would be a loss of amplitude when the waves are finished passing through each other and resume traveling in opposite directions. I am really puzzled by this, and would appreciate insight from minds that understand this better than I do.

 

[Mr.A.Gibson]

Good question, the energy is stored is due to the tension on a molecular level.

As the energy move down the rope it moves from particle to particle. If we imagine the rope as a line of single particles. Then as one particle moves the bond to the next particle is stretched like a spring, this provides a force which moves the next particle and so on. In any practical situation this will result in heat as you suggested, the amplitude of the wave on the slinky/rope decreases with time.

In the node of a standing wave the particle is not moving as the resultant force on it is zero. The tension at this point is increased since two forces are acting, if you think of the two particles either side of the node the force on them is increased, the distance they move apart from each other is increased. Hence the bond stretched further and as such more energy is stored in it.

Have a look at the animation:
http://phet.colorado.edu/en/simulation/wave-on-a-string"

Set the damping to zero and the generator to pulse, send down a pulse and then a second pulse when the first pulse reaches the right, this should set up the interference we're after.

You can see at the point around the node the particles are separated by a larger distance than when the wave is traveling without interference.

Quite a challenging thing to describe, hope this description, and the animation helps.

 

[SaraF]

Thanks, that is exactly what I needed. The animation was very helpful, too, in part because of your suggested settings.