Constructive and destructive interference

[Biker]

Homework Statement

Suppose you have two pulses, One is a trough, It's amplitude is 3 cm and the other is a crest and its amplitude is 9 cm. Assuming they have the same wave length when they overlap the energy of the created wave is:
1- 1/16 E
2- 1/4 E
3- E
4- 4E

Homework Equations

$E$ is proportional to $A^2$

The Attempt at a Solution

(Assuming the waves have the same characteristics)[/B]
The teacher gave us this question so I had to follow this reasoning to get to the forth answer
Suppose that the first wave has E energy and the combined wave by superposition we get its amplitude to be 6 cm (down) so you get 4E

But this doesn't make sense or at least doesn't conserve energy. Another solution using conservation of energy, the first wave has E and the second has 9E then the total energy would be 10 E

I could apply this to a simple question, Assume both crest and trough have the same amplitude, when they overlap you will get zero amplitude.. Where did the energy go? It has changed its form to become kinetic energy just as in a spring. So the total kinetic energy is 2E

Another situation is when two crests or two troughs overlap, it loses all its kinetic energy and it will only have potential energy. If you use energy conservation you get 2E. But if you use the E proportional to $A^2$ you get 4E.

So my question here, Does that proportional doesn't apply in superposition situation or my thinking is wrong? I couldn't look for the math behind it because it has integration and partial derivatives in the derivation for the energy equation of a wave. I have only studied derivatives so I wish you could somehow support your argument with some math

 

[CWatters]

The teacher gave us this question so I had to follow this reasoning to get to the forth answer
Suppose that the first wave has E energy and the combined wave by superposition we get its amplitude to be 6 cm (down) so you get 4E

I believe that should be a crest of 6cm (eg 6cm up not down).

I'm not sure I agree with the answer 4E but I assume this is how she got it...

Energy is proportional to amplitude squared so..

3² = 9
6² = 36
36/9 = 4

I think a better answer might be 0.4E...

Before
3² = 9
9² = 81
Total = 90
After..
6² = 36
36/90 = 0.4

I could apply this to a simple question, Assume both crest and trough have the same amplitude, when they overlap you will get zero amplitude.. Where did the energy go?

That's quite a common question. Google something like: destructive interference where does the energy go

If these were two waves on a rope moving in opposite directions then the waves reappear after passing each other. You only get zero amplitude when the two original pulses are exactly coincident, which only occurs for an instant in time.

 

[Biker]

I believe that should be a crest of 6cm (eg 6cm up not down).

Yes sorry, I have conflicted between the names. English is not my primary :C

I believe that should be a crest of 6cm (eg 6cm up not down).
I think a better answer might be 0.4E...

Before
32 = 9
92 = 81
Total = 90
After..
62 = 36
36/90 = 0.4

I think the question asks about the ratio between the created wave and the first wave
not between total energies.
And why do you consider 36 to be the total energy after the overlap instead of 90 which conserves energy

That's quite a common question. Google something like: destructive interference where does the energy go

It was just a question to initiate the conversation, I have already looked for it a week ago and came up with an answer. An answer which is right after the question.

"It has changed its form to become kinetic energy just as in a spring. So the total kinetic energy is 2E"

 

[CWatters]

And why do you consider 36 to be the total energy after the overlap instead of 90 which conserves energy

I should have said "during" rather than "after".

 

[Biker]

I should have said "during" rather than "after".

And during? Shouldn't it be the same? It is like momentarily change in energy..

 

[CWatters]

I guess I'm not explaining very well.

Oscillation/waves involve energy being transformed back and forth between two states, for example a pendulum transforms KE to PE and back again. At all times the total energy is constant. It's not clear what these two states are in your original problem. If we assume these are waves on a rope then the two states are KE (the rope is moving) and PE (tension in the rope).

"E" isn't defined but we have assumed it is the energy in a 3cm high pulse. By the same definition there is 9E in the 9cm high pulse (because energy is proportional to the amplitude squared).

So before they coincide the total energy is 10E in the pulses plus unknown energy elsewhere in the rope.

When the two pulses are coincident the result is a 6cm high pulse due to interference. A 6cm pulse has energy 4E (6cm is twice 3cm but energy is proportional to the amplitude squared). The apparently "missing" 6E is stored as tension in the rope.

After they pass each other the pulses reappear. I think you understand but here is a drawing (not to scale) anyway..

 

[Biker]

I guess I'm not explaining very well.

Oscillation/waves involve energy being transformed back and forth between two states, for example a pendulum transforms KE to PE and back again. At all times the total energy is constant. It's not clear what these two states are in your original problem. If we assume these are waves on a rope then the two states are KE (the rope is moving) and PE (tension in the rope).

"E" isn't defined but we have assumed it is the energy in a 3cm high pulse. By the same definition there is 9E in the 9cm high pulse (because energy is proportional to the amplitude squared).

So before they coincide the total energy is 10E in the pulses plus unknown energy elsewhere in the rope.

When the two pulses are coincident the result is a 6cm high pulse due to interference. A 6cm pulse has energy 4E (6cm is twice 3cm but energy is proportional to the amplitude squared). The apparently "missing" 6E is stored as tension in the rope.

After they pass each other the pulses reappear. I think you understand but here is a drawing (not to scale) anyway..

View attachment 109050

Yes I understand all of that. But by that argument, doesn't it seem to imply that there is an unknown energy "tension in the rope" which accounts for the missing energy?

Isn't tension in the rope is directly related to the potential energy it carries? Why not just consider the simple argument of summing the energy and saying that the created wave must have the same value of energy but distributed differently on KE and PE?

 

[Cutter Ketch]

This is a horrible horrible question.

A) pulses are not waves, and they do not have have a wavelength. Ok forgive that we know what was meant.

B) traveling waves or pulses are things that are defined by their motion, their evolution in time. Taking a snapshot of the configuration at the moment in time when they happen to be on top of each other and saying there is a new created wave with a new energy is ludicrous. If you start the movie again, no 4E traveling "wave" (by which of course I mean pulse) with an amplitude of 6 emerges. What emerges is exactly what you started with.

C) The energy is not A^2. There is no defined amplitude for a snapshot. Take a standing wave on a rope. That is actually the interference of two counterpropagating waves exactly as in this problem. (Only now I actually mean waves, not pulses). I can take a snapshot when the antinodes are at maximum displacement and equal to the amplitude. I can take a picture when the antinodes are zero and the rope is straight. The straight rope does NOT show the amplitude. This picture of the moment of greatest interference is EXACTLY the same as taking a picture of the straight rope and calling it the amplitude. It is NOT the amplitude and has no bearing or connection to the energy.

Horrible horrible horrible

 

[haruspex]

The apparently "missing" 6E is stored as tension in the rope.

At the instant the two pulses are completely aligned, there is no visible amplitude, so no tension. The missing energy will be as KE in the rope. This can be seen by considering the moments before and after alignment. For the pulses you drew, the left hand side will be moving rapidly down and the right hand side rapidly up.

 

[Biker]

This is a horrible horrible question.

A) pulses are not waves, and they do not have have a wavelength. Ok forgive that we know what was meant.

B) traveling waves or pulses are things that are defined by their motion, their evolution in time. Taking a snapshot of the configuration at the moment in time when they happen to be on top of each other and saying there is a new created wave with a new energy is ludicrous. If you start the movie again, no 4E traveling "wave" (by which of course I mean pulse) with an amplitude of 6 emerges. What emerges is exactly what you started with.

A) Yes, It is my mistake. Shouldn't have said wavelength. I translated the question quickly without revising the mistakes I said. Sorry again.

B) Assuming that I understand what you meant, It doesn't emerge. Just looking at it momentarily at the moment of interference.
Could you explain what you mean with no visible amplitude?

 

[haruspex]

Could you explain what you mean with no visible amplitude?

I believe that was my comment.
At the instant of total alignment, the wavefroms completely cancel, producing a flat line. Therefore all the energy is as KE.

 

[Biker]

I believe that was my comment.
At the instant of total alignment, the wavefroms completely cancel, producing a flat line. Therefore all the energy is as KE.

Isn't that only in the case of pulses that have equal amplitude?

 

[Cutter Ketch]

You may be familiar with the look of standing waves, for example the modes of a guitar string. In the lowest mode the ends don't move and the middle of the string moves up and down so the string goes from being arched up and then arched down. In between at a single moment in time the string is straight. If you took a snapshot you couldn't tell if the string was moving or not. You certainly couldn't say anything about the amplitude.

All of the energy is hidden in the picture because you can't see that the string is moving down at a high velocity. You could have taken a picture in between and seen any amount of arch you like. The arch you see in the snap shot is not the amplitude. To find the amplitude you have to watch it a while and see where the maximum is, see where the velocity goes to zero. You can't really define the amplitude without thinking in terms of motion over time, either the maximum displacement over time, or the place where the velocity, the motion over time, goes to zero.

Now as it happens the modes on a guitar string are actually the interference of counter propagating waves exactly like your counter propagating pulses. And the moment when the string is passing through middle and is straight is exactly the moment you have been given to contemplate with the pulses. Your pulses had different amplitudes, but what would this moment look like if they had been equal? Your string would be perfectly flat. The two pulses at that moment would perfectly destructively interfere. Would you conclude that the amplitude was zero? Would you say the energy was zero? No! This is like catching the guitar string in the middle of the swing. It doesn't indicate the amplitude of a pulse any more than the picture of a straight string indicates the amplitude of the vibration. It doesn't indicate the energy because it doesn't show the motion. As haruspex pointed out the string may be straight but different portions of it are in rapid motion. That's where the energy is. The energy certainly isn't related to the square of this instantaneous displacement (or lack thereof).

 

[Biker]

You may be familiar with the look of standing waves, for example the modes of a guitar string...

By all means, really awesome answer. If you mind could you look at what I came up with from your reply?

If we take a snapshot of the string (standing wave). At any moment other than the maximum of amplitude of the wave, It doesn't give us the amplitude because the highest point in the string can either go upward more or downward more to cause a higher displacement

But let's say we have pulses as following,

The red pulse: If you look at the highest point, its kinetic energy is zero implying that this is the highest amplitude this wave can reach.
And the same applies to the blue pulse.
The Superposition of these pulses gives us a pulse with amplitude of $A_{Blue} - A_{Red}$ and the kinetic energy of that point should be zero right?
This implies that this is the maximum height that this point can actually reach. However if you continue on you will see that, points beside that maximum height point will reach a greater displacement(downward) (on the left side of the maximum height) and the opposite happens on the right side.

This is not a normal behavior of a pulse. In a pulse, every point will have enough energy to reach its amplitude and that doesn't apply hear so the E propotional to A^2 doesn't apply. Discrepancy emerges from it not having the same kinetic energy as a normal pulse would do.

Indeed it is a horrible question that why I ask about those in this forum

 

[haruspex]

Isn't that only in the case of pulses that have equal amplitude?

Yes, sorry, I should have made that clear.

 

[CWatters]

At the instant the two pulses are completely aligned, there is no visible amplitude, so no tension. The missing energy will be as KE in the rope. This can be seen by considering the moments before and after alignment. For the pulses you drew, the left hand side will be moving rapidly down and the right hand side rapidly up.

I was really thinking of the case where the pulses aren't equal but you are probably still right. Although if the vertical velocity changes (aka acceleration) I think the tension must also change?

 

[CWatters]

Perhaps see "Energy in a String Wave" down here..
http://hyperphysics.phy-astr.gsu.edu/hbase/Waves/powstr.html
They start by looking at the energy in a small segment of rope (both KE and PE) and integrate over a wave length. Still end up with an A² term in the equation for the energy.

 

[haruspex]

I was really thinking of the case where the pulses aren't equal but you are probably still right. Although if the vertical velocity changes (aka acceleration) I think the tension must also change?

For the equal and opposite pulse case, when fully aligned it is like SHM at the equilibrium point. Instantaneously there would be no linear acceleration. But it would be rotating rapidly about the centre, as at a node in a standing wave.