power and waves

[CPL.Luke]

I got this prolem on a homework a while ago and I couldn't quite make sense of the book answer (and my professor can't really comment on it)

 

say that you have two identical waves traveling along a string, what is the average power of the two waves.

 

I said that it would be twice the power of one wave, because if you imagine two strings that are joined in a Y formation, and you send a wave down the two top ends of the Y, then the two waves join and travel down towards the bottom of the Y.

 

because you put a certain amount of power (p) into each wave at the top you should get twice the power at the bottom (2p).

 

but his was not the solution in the book, the book said that the amount of power of two identical waves traveling down the string is 4 times the power of just the one wave, because the formula for the power in a wave contains a maximum amplitude squared term. this means that because the amplitude is doubled when there are two waves the average power output must be 4 times that of one wave.

 

I know that the latter interpretation should be correct, but I can't figure out how to reconcile that with my original interpretation.

[Spyman]

I think a point on the wave, (like the peak), is only a potential, (current force), and the total power of the wave is dependent of the time the force is in action, (the wavelength).

 

If you have two waves with identical amplitude but different wavelength, which one carries the most power ?

 

(Also, if I remember correct, the area of a wave represents the power.)

[m4rc]

If two identical waves interfere constructively with each other at a particular location, you will get twice amplitude and four times the energy at that spot. However it is impossible to add two distinct waves in such a way that constructive interference occurs everywhere. Destructive interference will happen somewhere and the total amount of energy is conserved.

[CPL.Luke]

but what if your ading to identical waves?

[Spyman]

but what if your ading to identical waves?

Will the new wave be doubled in size ?

[m4rc]

If you add two identical waves then the resulting wave will interfere constructively everywhere and will have twice the energy as two separate waves.

 

If you have two waves with distinct sources, then it is impossible to combine them constructively everywhere.

 

To produce two waves that overlap perfectly, then you will need have the source of the second wave will located on the first. Because the source of the second wave is interacting with the first wave, it will be harder to generate the second wave. Think about it like the stretching of a spring. More energy is required to stretch it from 1 cm to 2cm than was required to stretch it from 0 to 1cm (the energy of a spring is given by E=1/2 k x^2 ).

[CPL.Luke]

but what about in my setup with the Y?

 

at least temporarily two identical waves will be traveling down the bottom of the string, and because I was only putting 2P amount of power in those waves only have 2P power available to them and yet they should be exerting 4P power

[m4rc]

After thinking about it for a while I am not convinced that for a y-shaped string, the resulting wave would be a superposition of the the two original waves. The reason for this is that there would be a change in the tension in the ropes. If two strings meet and continue as one, then the tension in the one string will be twice that of the individual strings. The amplitude of the oscillation in the rope will depend on both the vibrational energy and the tension in the rope. (you can see this on a guitar string while you tune it)

 

I intend to confirm this by monday.

[J.C.MacSwell]

After thinking about it for a while I am not convinced that for a y-shaped string, the resulting wave would be a superposition of the the two original waves. The reason for this is that there would be a change in the tension in the ropes. If two strings meet and continue as one, then the tension in the one string will be twice that of the individual strings. The amplitude of the oscillation in the rope will depend on both the vibrational energy and the tension in the rope. (you can see this on a guitar string while you tune it)

 

I intend to confirm this by monday.

 

If the strings all meet at 120 degrees the static tension would be all equal, though I suspect waves on two of them meeting at the apex simultaneously would not completly constructively interfere down the third string but also have two secondary waves rebounding back down the two original strings.

[Spyman]

CPL.Luke, maybe this analogy with electrical waves can help:

 

Two identical power generators are identical geared to a wheel so for every turn on the wheel you get an electric wave out from each generator.

 

There is a connection box for the generators with an output wire to a load, lets say a resistor heater.

 

The power used for turning the wheel will match the heat output from the heater and each of the generators will supply half of the power.

 

1) Connect the generators in parallel and measure the power for one turn.

 

2) Connect the generators in serial and measure the power for one turn.

 

Will the power used differ between the two setups and how much ?

(Hint: P=U*I and U=R*I -> P=U²/R)

[CPL.Luke]

hmm well as for bounce back, as long as the oscillation of the center point has twice the amplitude of the original two waves, hen the wave traveling down the other string should have twice the amplitude.

 

we could also do away with the string and look at power in an electromagnetic wave

 

the formula for this power is E^2/c and we look at this and see that if you superimpose two waves on each other than the power becomes 4 times the power of one wave.

 

you could accomplish this through some kind of lensing system I'm sure, but then you have the problem that you put 2P energy in and now it appears that your getting 4P energy out.

[Spyman]

as long as the oscillation of the center point has twice the amplitude of the original two waves, hen the wave traveling down the other string should have twice the amplitude.

but then you have the problem that you put 2P energy in and now it appears that your getting 4P energy out.

So obviously your setup with the Y-formation will not double the amplitude at the center point.

 

we could also do away with the string and look at power in an electromagnetic wave

 

the formula for this power is E^2/c and we look at this and see that if you superimpose two waves on each other than the power becomes 4 times the power of one wave..

To superimpose two waves on each other you will need to use 4 times the power of one wave.

[5614]

[edit] I thought I had a solution, it seems I've just re-proved the violation of the conservation of energy, which sucks because it's crap, I put all of this in grey. I have got some more ideas, including what I think is the real answer in black below. So you can skip the grey if you want!

 

We know the power of the wave is proportional to the amplitude squared. For the sake of this explanation I'm going to assume that the proportionality constant is 1, that is, power equals amplitude squared. Follow this through the and constant would cancel anyway, it's just easier omitting it for now.

 

Initially lets deal with the amplitudes within the system. So you have your Y-formation with two waves, each of amplitude 2, along the two branches of the Y.

When these waves reach the middle of the Y formation half of the wave bounces back towards the source and the other half combines to form the tail of the Y.

The blue shows what happens after the wave reaches the mid point of the Y. Half of the two combining waves reflect back, and the other half of both of them combine. As each input wave has an amplitude of 2, when the wave reached the mid point of the Y half the wave bounces back with amplitude of 1, and the part of the wave continuing along the tail has amplitude 1. However there are two waves constructively superimposing, giving an actual amplitude of 2 along the tail.

 

OK, so we have two input waves, each of amplitude 2. Amplitude squared is the power input, 2²=4, and there's two inputs, so total power input is 2*4=8.

 

And now lets look at the output, represented by the blue lines in the diagram. We have an amplitude of 1 at source 'a' and 'b' and an amplitude of 2 at 'c'. 1²+1²+2²=6

 

6 (out) doesn't equal 8 (in)... That's not meant to happen.

 

======================

Useful part of the reply below

======================

 

The whole Y-formation setup seems to be over complicating the issue, mainly because how the wave behaves on within the Y setup is not easy to predict. The wave does not totally superimpose, and it's not a simple 50/50 split at the mid point. The behaviour of the waves should be able to be predicted, but I do not know how. Maybe the energy splits and not the amplitude, or maybe, taking into account factors such as tension and other resistive forces, you end up with a complex answer which we won't guess just by looking at it.

 

So I went back to look at the question, and to calculate the answer without the Y experiment.

 

say that you have two identical waves traveling along a string, what is the average power of the two waves.

 

OK, so we have two waves, each of amplitude 1 travelling along a string (black). For simplicity of thinking lets assume the waves constructively interfere. Producing a wave of amplitude 1+1=2 (blue).

The average power of the two waves could be considered as the power when the two waves are superimposed. If power equals amplitude squared (I'm ignoring the proportionality constant for simplicity), then the power of one wave is 1²=1 and the power of the superimposed wave is 2²=4.

 

As 4 is 4*1 the average power, or the power of the two waves superimposed, is 4 times the power of one wave.

 

My question is this: if these two waves were to destructively interfere then where would the energy go?

[m4rc]

Experimental Results

 

I have just done a quick experiment to test what the amplitude will be. The experiment consists of attaching three slinkies together, with one going to the left and two going to the right (almost parallel to each other ie this orientation: --======= ). I then made identical pulses along the two slinkies and filmed the wave as it reached the single slinky.

 

Results: The amplitude transmitted to the single slinky is the same as that of each of the individual waves that made it up. So if two waves each with an amplitude A combine at the single slinky, the resulting amplitude is A and NOT 2A as expected by the superposition priciple.

 

Explanation If you are using the superposition priciple, you are assuming that the medium is not changed as the waves meet. However in this case, the tension in the single slinky is twice that in each slinky of the pair. The energy of the wave is proportional to the tension in the rope and the square of the amplitude. So if the available energy doubles and the tension doubles, the amplitude must be unchanged.

 

I have to admit that there was a fair bit of friction in this setup which may have reduced the resulting amplitude a bit.

[CPL.Luke]

thanks m4rc by any chance can you post the film?

 

hmm so now we can say that two mechanical waves cannot be superimposed on eachother in a string,

 

but what about for soundwaves and electromagnetic waves which could easily be superimposed on eachother?

[5614]

hmm so now we can say that two mechanical waves cannot be superimposed on eachother in a string

You can get two waves to superimpose on a string easily... Get two people to hold opposite ends of a piece of string and wave it up and down. If they do it well then you will get the two waves superimposing... standard demonstration for nodes, antinodes & standing waves.

 

(Alternatively you could get a vibrator and fix the other end of the string to something stationary.)

[CPL.Luke]

I meant when superimpsing two waves traveling in the same direction, sorry I should've been clearrer

[5614]

Well, I assume you're talking about two waves completely constructively superimposing... if they were going along a single string then for the 2nd wave to catch up with the 1st it would have to be travelling faster. Consequently the two waves would not constructively superimpose, as one wave overtook the other the superposition would vary, ie. it would not always be constructive.

 

Which brings us back the Y-setup.

 

Explanation If you are using the superposition priciple, you are assuming that the medium is not changed as the waves meet. However in this case, the tension in the single slinky is twice that in each slinky of the pair. The energy of the wave is proportional to the tension in the rope and the square of the amplitude. So if the available energy doubles and the tension doubles, the amplitude must be unchanged[/u'].

Ignoring all proportionality constants:

energy = tension = (amplitude)²

 

So if the input energy per slinky is 1J and you have two slinkies then total input is 2J. Also the tension in each slinky is 1N.

 

When the pair of slinkies join to the single slinky, as m4rc said; "the tension in the single slinky is twice that in each slinky of the pair". The tension doubling seems a logical step, as it's twice the force, 1N from each slinky provides a net force of 2N.

 

Therefore the tension in the single slinky is 2N. This would make the energy out 2J.

 

All good so far, however now we come onto the amplitudes.

 

Originally there were 2 waves of energy 1J and 1J. By square rooting this energy we get the amplitude of the two original waves, which are 1 and 1.

 

However at the end of the experiment we have one wave of energy 2J. By square rooting this energy we get the amplitude of the single and final wave, which is [math]\sqrt{2}[/math].

 

Having an amplitude of [math]\sqrt{2}[/math] had crossed my mind before, but it was only rereading m4rc's:

energy [math]\propto[/math] tension [math]\propto[/math] (amplitude)²

that made me think along those lines further. What do you think of this?

[CPL.Luke]

actually the energy is power is proportional to the square root of the tension.

 

 

the full formula for power is,

 

1/2 u v w^2 A^2

 

where u is the linear density, and w is the angular frequency

 

v is the velocity of the we and is equal to the square root of tension/ linear density.

 

and it is possible to make the tensions exactly equal, for instance if they all meet at 120 degrees.

 

 

For the Y setup I'd buy that the amplitude of the wave that procedes down the third string is equal to the original two waves, and half the power travels back up the two strings, or that the two waves hit the center point and create a thrid wave that has root 2 the amplitude of each component wave.

 

however for an instance where you have to electromagnetic waves that interfere constructively, what hapens then?

 

note the formula for the power of an electromagnetic wave is E^2/c

[m4rc]

Here are the movies that I analysed to conclude that the amplitude does not change. I only did a quick analysis, Shortly after posting my last comments I found the relation between tension and energy but the equation also includes a velocity term I suspect may also depend on the tension. I intended on looking into it but other projects have been keeping me busy.

 

http://www.geocities.com/marc.spooner/spring1.avi

http://www.geocities.com/marc.spooner/spring2.avi

For a sence of scale: the slinky has a diameter of about 6 cm.