Why Are Both KE and PE Maximal at Equilibrium in a String Wave?

[Saw]

I am in the car now ( passenger!)

Thanks for your interest. I am in no hurry for answers ;)! It is just out of curiosity that I am asking.

I have no textbook but you need to read what it says, carefully.

I will quote again, together with the diagram the text refers to:

Kinetic Energy


A string element of mass dm, oscillating transversely in simple harmonic motion as the wave passes through it, has kinetic energy associated with its transverse velocity . When the element is rushing through its y = 0 position (element b in Fig. 16-9), its transverse velocity —and thus its kinetic energy —is a maximum. When the element is at its extreme position y =ym (as is element a), its transverse velocity —and thus its kinetic energy —is zero.


Elastic Potential Energy


To send a sinusoidal wave along a previously straight string, the wave must necessarily stretch the string. As a string element of length dx oscillates transversely, its length must increase and decrease in a periodic way if the string element is to fit the sinusoidal wave form. Elastic potential energy is associated with these length changes, just as for a spring. When the string element is at its y = ym position (element a in Fig. 16-9), its length has its normal undisturbed value dx, so its elastic potential energy is zero. However, when the element is rushing through its y = 0 position, it has maximum stretch and thus maximum elastic potential energy.

I would say that the energy density ( per m) is uniform but that isn't the Potential.

Actually the text is saying that the energy density (per m) is not uniform. For example, if you make a snapshot of the wave at a given moment, the element at y = 0 has both KE and PE densities = maximum, whilst the element at y = maximum has both KE and PE densities = 0.

If you switch now to the global perspective and integrate, for example, along a 1/4th cycle, from y = 0 to y = m, you may say that total E is whatever and KE is 1/2 whatever and PE is 1/2 whatever, but you cannot say that one of them (for instance, PE) is bigger than the other. If they are always equal, if they constantly travel in phase, there is no way to distinguish between them… either at one level or the other! Either at the level of energy density at a point or of total energy along the cycle or part of it!

 

[Jilang]

By the way: this applies also to EM waves. The E and B fields are also said to travel in phase. How can that be and, what is more, how can they create each other this way? How can something that is zero create anything?

There was a thread about this that you might be interested in.
https://www.physicsforums.com/showthread.php?t=36797

 

[AlephZero]

- It strikes me that what you are elaborating would apply to any (?) wave,

Not quite to "any" wave, but certainly to any linear wave, whether it occurs in mechanical vibration, electricity, optics, acoustics, fluid flow, or whatever.

whereas the textbooks' description of what happens in a string wave is presented as a peculiarity, vis-à-vis other types of waves, even vis-à-vis string standing waves.

If you want to look at kinetic and potential energy, energy flows, etc, this is easier with a definite example of a wave in some medium, but the the results you get will be true for every type of linear wave motion.

Actually, transverse waves on a stretched string are not the easiest example to take, because finding the potential energy is a bit tricky, and "common sense" can easily be wrong.

Longitudinal waves in a rod are simpler to analyze, but the results are not "obvious". If you have a standing wave, then for the whole wave the total energy is constant and is transferred back and forth between kinetic and potential energy. But if you look at a small element of the rod at different positions, that is not what happens. At the nodes, the kinetic energy is always 0, but the potential energy varies from 0 to a maximum value, at twice the vibration frequency. At the antinodes, the opposite is true: the PE is always 0, but the KE changes from 0 to a maximum, twice per cycle. The reason is that energy is traveling along the rod, in both directions simultaneously (a standing wave is the sum of two traveling waves) and at the nodes and antinodes you get "destructive interference" in either the KE or PE.

Transverse waves on a stretched string are exactly the same, but the tricky part is finding the potential energy correctly from first principles.

At a more advanced level, you can avoid the specifics of finding the KE and PE for each separate example of a wave, and get them both direct form the partial differential equation of the wave motion - and if you do it that way, the results obviously apply to every type of linear wave.

And of course the fact that waves are "always the same" in so many different physical situations, is why they are an important topic to study!

 

[sophiecentaur]

When you lift a block up in the air, you just say that the block has gravitational energy by virtue of its height. No one talks about the GPE being in the space below or how it's 'shared out' amongst the space. If you use a set of springs to simulate the same forces in the absence of gravity, would that mean we would define the PE differently? Surely it would still be described as the work done in getting the block to that position. If the 'normal' definition of PE is used then the relationship between KE and PE over cycles of a standing or a progressive wave seems to be well behaved and as I would expect. I suppose it isn't surprising that, if you use a different way of defining / calculating PE, you may get a different answer.

Something that bothers me is that two different strings, with the same tension and density (and therefore with identical wave speeds) can have different moduli and, hence, different stored energy for the same tension. If the wave PE energy is said to be related to the tension then I reckon it would have be different in the two cases. But the KE would be the same (frequency, velocities, amplitudes could all be the same). Now I thought that the energy of the wave is the same over the cycle and is exchanged from PE to KE and back. This implies that the max PE is the same as the max KE (as with plain oscillators).
The above is not self consistent. What can be said to resolve this? I am confused.

Is this all, in the end, one of those traps which appear when you try to categorise too much and say 'what is really happening'?

 

[Jilang]

I guess the difference is that when you lift a block under gravity you are just doing work on the block, but when you move the block attached to springs in the absence of gravity you are doing work on the springs and actually no work is done at all on the block at all.

For the two different strings I would have thought that different moduli would lead to different tensions.

 

[Saw]

Jilang: the thread you mentioned about E and B is very interesting and some comments may apply here. There other branches that have opened up through others’ comments, but before touching them may I ask you to stop for a moment and look at the following recapitulation on the main subject?

If you have a standing wave, then for the whole wave the total energy is constant and is transferred back and forth between kinetic and potential energy. But if you look at a small element of the rod at different positions, that is not what happens.

Assuming that the distinction also applies to traveling waves, I understand that it is important to distinguish between the two approaches: whole wave and segment. The segment approach gives out “energy density” at a specific location, which would be –in free language- a sort of “energy pressure” = E / A or in this case (strings) E/L. But only the whole string approach gives you the necessary information to predict what is going to happen.

This links with other comments by Sophiecentaur.

Thus in a spring-mass system if we take a snapshot of the instant when the spring is fully compressed, the segment closer to the end may have more of this “E pressure” but the other coils also have some and it is looking at the total E stored in the whole spring what gives you the ability to predict how the system is going to behave.

In particular, that is the way we should define PE, as the work done to get the whole string to a given state:

If the 'normal' definition of PE is used then the relationship between KE and PE over cycles of a standing or a progressive wave seems to be well behaved and as I would expect. I suppose it isn't surprising that, if you use a different way of defining / calculating PE, you may get a different answer.

So I gather that the text I quoted from Halliday is taking an unconventional approach, in that it is focusing on a specific segment. No wonder then that it puzzles us if we want to extract from it an account about what is going to happen.

And if we now want to take the whole spring approach to see what happens in a traveling string transverse wave, would this comment I made earlier be correct? Or is it, as AlephZero suggests, a more complicated question?

I am thinking that simply the string is communicated a velocity v (maximal KE, PE still zero) and this motion stretches the string, although this effort progressively consumes the v, until the maximum amplitude is reached at the crest (KE zero, maximal PE) and then the cycle continues with compression entailing that the PE is consumed to the benefit of KE, until upon return at equilibrium we get again maximum KE and PE zero…

 

[Jilang]

Yes, you really need to consider the whole string because the segments are joined to each other and are not moving independently of each other. I read through lots of sites over the last couple of days and most take the Halliday approach so I wouldn't say it was unconventional. (In fact I only found one that didn't). Your last quote looks fine to me, KE transferring to PE and back again for the string as a whole.

One last thought, following on from the mass attached to springs that Sophie mentioned. Consider threading a bead onto the string fixed at both ends and stretching the string by pulling on the bead. The work is not done on the bead, but on the string as a whole and the PE is stored in the string all he way along it.

 

[sophiecentaur]

I guess the difference is that when you lift a block under gravity you are just doing work on the block, but when you move the block attached to springs in the absence of gravity you are doing work on the springs and actually no work is done at all on the block at all.

For the two different strings I would have thought that different moduli would lead to different tensions.

Problem is that the formula for speed of the wave (and hence the wavelength) contains just tension and linear density. You can obviously put whatever tension you like on the string by turning a screw.

I am coming to terms with this stuff, a bit at a time.

PE seems to be a very context-specific idea and we need to avoid imposing the comforting notion that just classifying something will give us a full explanation. In a Potential Field, there is no Energy at all, until you involve another mass / charge or whatever. I now see and accept why maximum 'tension' energy density on the string has to be near the equilibrium position, in a progressive wave and on a standing wave. In a standing wave the total PE is a maximum when the amplitude of the standing wave is a maximum but, paradoxically, if you like, the most PE is 'located' at the nodes

 

[sophiecentaur]

Yes, you really need to consider the whole string because the segments are joined to each other and are not moving independently of each other. I read through lots of sites over the last couple of days and most take the Halliday approach so I wouldn't say it was unconventional. (In fact I only found one that didn't). Your last quote looks fine to me, KE transferring to PE and back again for the string as a whole.

One last thought, following on from the mass attached to springs that Sophie mentioned. Consider threading a bead onto the string fixed at both ends and stretching the string by pulling on the bead. The work is not done on the bead, but on the string as a whole and the PE is stored in the string all he way along it.

Looks like this is turning out to be a successful thread - aiming at a common improvement in understanding without anyone throwing their toys out of the pram. :thumbs:
PF strikes again.

 

[Jilang]

Looks like this is turning out to be a successful thread - aiming at a common improvement in understanding without anyone throwing their toys out of the pram. :thumbs:
PF strikes again.

I'll second that. Thanks for starting this thread Saw. It's been interesting and educational.

 

[Jilang]

Problem is that the formula for speed of the wave (and hence the wavelength) contains just tension and linear density. You can obviously put whatever tension you like on the string by turning a screw.

I am coming to terms with this stuff, a bit at a time.

Me too. It looks like my problem is that we are taught that the tension is equal along the string. I always thought that this was because the tension in the string travels much faster than the wave and so that is why as a first approximation the tension is assumed to be the same everywhere. But if that were the case the PE along the real contour of the string would be the same everywhere (and to a first approximation the same along the x axis). Would it be right to assume that equal tension would require infinite stiffness so there would be zero amplitude in the first place?

 

[sophiecentaur]

Me too. It looks like my problem is that we are taught that the tension is equal along the string. I always thought that this was because the tension in the string travels much faster than the wave and so that is why as a first approximation the tension is assumed to be the same everywhere. But if that were the case the PE along the real contour of the string would be the same everywhere (and to a first approximation the same along the x axis). Would it be right to assume that equal tension would require infinite stiffness so there would be zero amplitude in the first place?

In practical cases,the static tension waves will be high and the incremental tension will be small (for small amplitude). I imagine this could imply that 'the medium' isn't actually linear with string waves. We need an experiment to demonstrate this and I'd bet someone has done one. The way to find what could be small non linearities would be to look for intermodulation products between two frequencies with no common factors. That's much more sensitive than just looking for harmonics, which can arise from many distracting mechanisms.
[Edit: On re-reading this, it strikes me that many mechanical waves are only SHM based for small perturbations so what I've said is just a bit Bloomin' Obvious.]

 

[Saw]

Before closing... a little thinking aloud:

We have concluded that, in a traveling string wave, there is an interchange between the KE and PE of the string as a whole, just like in any other oscillating or waving phenomenon, and that it is this sloshing what explains the Energy transport.

Then I wonder what to make with the common expression that in a traveling string wave “KE and PE go in phase” or “rise and fall together”.

See here for example, in the article found by Jilang (http://faculty.ifmo.ru/butikov/WaveEnergyPS.pdf):

For a purely transverse traveling wave of an arbitrary shape ψ(x, t ) = f (x -vt ), equation (1) shows that the linear densities of kinetic and potential energies are equal to one another at a spatial point x at a time instant t ; they rise and fall together.

Is this expression still valid within the domain of the segment approach?

I think that it would still be true (another side of the truth meriting “even honors”) to the extent that we could state that the particular segment which is transporting the energy at a given instant always stores KE = PE.

For example, the wave has reached y = A, the point of maximum amplitude. Here it is true that KE = 0 (it has been exhausted) and the PE of the segment in question (which looks flat) is also = 0. The wave will keep moving because there is PE of the string outside that segment and precisely at its peak value, but that is another story.

So far, so good. But now the wave starts descending and reaches y = 0, equilibrium position. The segment that is then visited by the wave will have maximum KE, but it will be also flat and its PE (like in the rest of the string, which is flat as well) will be zero. Hence now it is not true that “local” or “segment” KE and PE have risen together.

A different thing is that, at an earlier moment, when the wave was at y = A, this same segment was stretched, with maximum slope. But not now, at the relevant time, when the wave visits it.

Well, it is true that I am thinking of a wave with a single pulse. If we now imagine a train of waves, the segment in question may be stretched out by the next pulse…

 

[sophiecentaur]

The simplest wave solution of Maxwell's equations gives H and E fields varying in phase with each other. This always upset me a bit because it seemed to be out of step with how mechanical waves were said to behave - i.e the kinetic and potential energy varying in quadrature. Now, it seems, the same thing seems to be true of mechanical waves. This is a relief to me because there is now some consistency. This just shows the flaw in intuitive reasoning and attempts to over-classify things.
In the case of individual harmonic oscillators like springs, rods and EM antennae, the KE and PE are certainly in quadrature but, once the energy starts to flow, the two are in phase.
The only unsettling bit is to explain how and where the phase shift occurs as the wave is launched. The antenna calculations seem to predict both near field (quadrature V and I related) and far field (E and H in phase) energy, with the near field variations being localised round the antenna as they cannot carry energy away. What's the explanation for a wave launched along a string by a vibrating 'hand', I wonder? There is, presumably some evanescent wave near the end of the string.

 

[Saw]

how mechanical waves were said to behave - i.e the kinetic and potential energy varying in quadrature. Now, it seems, the same thing seems to be true of mechanical waves.

But quadrature means a phase shift of pi/2 = 90^o, doesn't it? However, in my understanding the components of the total ME of the string as a whole (KE and PE) vary with phase shift = pi = 180^o. The same applies for the sloshing between KE and PE in SHM. That is why when PE = max, then KE = 0 and vice versa. Or am I missing anything?

 

[sophiecentaur]

But quadrature means a phase shift of pi/2 = 90^o, doesn't it? However, in my understanding the components of the total ME of the string as a whole (KE and PE) vary with phase shift = pi = 180^o. The same applies for the sloshing between KE and PE in SHM. That is why when PE = max, then KE = 0 and vice versa. Or am I missing anything?

There are two maxes in magnitude (a+ and a-) and two zero crossings per complete cycle of a wave / oscillation. When I say quadrature, I mean that the max of one coincides with the zero crossing of the other. That is a ∏/2 difference in phase (2∏ in a complete cycle).

 

[Saw]

Understood. I had relied on a faulty drawing...

In the case of individual harmonic oscillators like springs, rods and EM antennae, the KE and PE are certainly in quadrature but, once the energy starts to flow, the two are in phase.

I am not sure if you are you seeing this as a distinction based on time. i.e. one approach follows the other in time? For example: PE and KE out of phase by pi/2 in a spring-mass system which is connected to a string and then when the E starts to flow through the string, you get the two in phase.

I am seeing it as two simultaneous perspectives. For example, at y = A "the E carried by the wave" (both PE and KE) is zero, but the wave progresses because the PE of the whole string is maximum.

Point y = 0 is a little more problematic, because here the PE of the string as a whole should be zero, but this element is actually fully stretched out, so it sounds difficult that global PE is zero. I was suggesting that in a single pulse the PE, also of this element, should be zero, and what streches it is only the fact that it is followed by a new pulse. Thus what would oscillate out of phase by pi/2 would only be the "energy of the whole string for a single pulse" or maybe "for a wavelength"... I don't know... A slippery concept, anyhow...

 

[sophiecentaur]

The problem is that we want to 'know' where the energy actually is, in a wave. That is our mistake. Somehow, for a simple oscillator, we are we are happy to have KE and PE - the location of the PE is just as vague as in a wave but we don't seem to care particularly. We can 'clearly' see when there's movement or not.

With progressive waves, the actual location of the PE is not always clear but on strings, the energy stored as tension is greatest around the zero crossing - as is the maximum of the KE. But both forms are actually spread over the whole of the wave (except for the KE at the stationary peaks). We should just stop worrying further I think.

Your last comment is about a stationary wave, I take it. When the string is moving through the equilibrium position, the PE is not a true zero because there is existing tension in the string but that isn't wave energy. Also, you cannot have a meaningful Phase for a single pulse. A single pulse must consist of an infinity of components which come together during the time of the pulse and the phase is different for each.

I think we should all have a time to reflect and do some useful personal sketches. There can't be much more useful, said about the topic except to delve deeper into the derivation of the wave equation from the basic equations of motion in each case. I can't trust my Maths enough for that, these days.

 

[Saw]

I was not referring to a standing wave, but to a traveling wave, but I do not know why I said that at y = 0 the string is flat and PE should be zero. Obviously, with a traveling wave the string is never flat and so the PE of the string as a whole would never be zero.

But then I realize that my understanding about the teaching of the thread was wrong. Is the one below a better understading?

I always refer to a traveling wave:

- Distribution of the energy of *the string as a whole* between PE and KE --> no interchange or sloshing, but constant 50/50.
- Distribution of the energy of a *segment* between PE and KE: what Halliday says, they always rise and fall together, being both maximal at segment around y = 0 and zero around y = A.

 

[sophiecentaur]

I was not referring to a standing wave, but to a traveling wave, but I do not know why I said that at y = 0 the string is flat and PE should be zero. Obviously, with a traveling wave the string is never flat and so the PE of the string as a whole would never be zero.

But then I realize that my understanding about the teaching of the thread was wrong. Is the one below a better understading?

I always refer to a traveling wave:

- Distribution of the energy of *the string as a whole* between PE and KE --> no interchange or sloshing, but constant 50/50.
- Distribution of the energy of a *segment* between PE and KE: what Halliday says, they always rise and fall together, being both maximal at segment around y = 0 and zero around y = A.

There's an argument that says that could be right. If it isn't 100% sloshing each way then what actual proportion would it need to be? This link shows it's shared equally for a progressive wave.

 

[Saw]

But... what the link says is that the average KE / metre = average PE / metre. Consider a string of a given length and that means that the total KE and PE of the string are equal. But that doesn't preclude sloshing. If we want to reintroduce interchange between KE and PE, we just have to stipulate that if anything happens somewhere in the string, the opposite takes place, simultaneously, somewhere else. For example, if somewhere KE and PE jointly fall, it is because they jointly rise somehere else. In other words, a double sloshing but sloshing after all.

 

[sophiecentaur]

It is true that the energy density may not be uniform along the line at any instant. It is certainly true for EM waves, where the energy arrives with maxima, every half cycle.

 

[Saw]

And the requirement of your link that average densities for KE and PE are equal puts a constraint to that: if in some place there is a gain of one of them, there must be a loss somewhere else.

The requirement of Halliday and Butikov puts a second constraint for specific segments = also local densities of KE and PE should be equal:

For a purely transverse traveling wave of an arbitrary shape ψ(x, t ) = f (x -vt ), equation (1) shows that the linear densities of kinetic and potential energies are equal to one another at a spatial point x at a time instant t ; they rise and fall together.

I don't know what to do with this second requirement. It sounds strange but it has much support. But if we accept it, then -in order to harmonise it with the first- we should infer that if PE and KE fall somewhere together at a given instant, they must be rising somewhere else (also together) simultaneously.

 

[sophiecentaur]

And the requirement of your link that average densities for KE and PE are equal puts a constraint to that: if in some place there is a gain of one of them, there must be a loss somewhere else.

The requirement of Halliday and Butikov puts a second constraint for specific segments = also local densities of KE and PE should be equal:



I don't know what to do with this second requirement. It sounds strange but it has much support. But if we accept it, then -in order to harmonise it with the first- we should infer that if PE and KE fall somewhere together at a given instant, they must be rising somewhere else (also together) simultaneously.

Yes (you know this); as the peak travels forward, the energy level rises in the region in front and falls in the region behind - just as the displacement does. The whole point about waves is that the energy is carried. forward - as the boat bobs up and down.

 

[Saw]

Yes (you know this); as the peak travels forward, the energy level rises in the region in front and falls in the region behind - just as the displacement does. The whole point about waves is that the energy is carried. forward - as the boat bobs up and down.

I was mentally developing something similar but I am not sure it is the same. Would you say that the E rises in front and falls behind? It is obvious that E is moving forward and that there is no E where the wave has passed by. But here we are talking about E "inside" for example a wavelength. Thus I would rather say that the wave pulls up the particles in front and pushes down those behind the peak, but the E remains the same at both sides. What is more, in a string wave, unlike in pure SHM, each of KE and PE remain constantly the same along one wavelength because what is happening at one side is set off by exactly the opposite happening simultaneously at the other side of the peak.

 

[sophiecentaur]

Yes but what is it that's actually 'moving along'? Nothing but the Energy. What's strange is that the energy goes one way and not the other - bearing in mind that the string (in a particular case) is just going up and down. Think that one over for a bit. lol

 

[Jilang]

Yes but what is it that's actually 'moving along'? Nothing but the Energy. What's strange is that the energy goes one way and not the other - bearing in mind that the string (in a particular case) is just going up and down. Think that one over for a bit. lol

Yes, when you put it like that it certainly sound weird! Is there a natural propensity for energy to spread outwards? I'm imagining plucking a string and the wave traveling outwards in both directions. After that what keeps it going? Does the energy have an associated momentum outwards that must be conserved? I thought strings were pretty straightforward in 2013. Lol.

 

[Saw]

It is a good point. I also think that the disturbance tends to move in both directions. When you pluck a string, it actually does. When a SHM device connected to a strig starts doing its job, again the disturbance attempts to move in both directions, but the fixed end where the device is prevents it from making progress in that direction (?).

 

[Jilang]

Ah, I've figured it out I think. The backward waves originating at different times must all cancel each other!

 

[sophiecentaur]

Ah, I've figured it out I think. The backward waves originating at different times must all cancel each other!

That sounds like good thinking. Do a Google search on Huygen's principle for predicting the progress of a wavefront of light. That's the more general case of what you just wrote and explains why the beam just carries on and only spreads out at the edges (i.e. it's a graphical way to explain diffraction).

 

[Jilang]

Thanks Sophie, That seems to be sort of thing indeed! It is a bit disconcerting though in as to how much Huygens Principle looks like it could be the earliest description of Quantum propagation. If this indeed the explanation of why the wave only goes one way it would appear that all classical waves are governed by quantum behaviour!

From what I can gather he assumed de facto that the waves only spread out forwards though. Later work seems to suggest that the backwards cancellation only works properly in an odd number of dimensions. So string yes, water no (!), space yes. Fascinating stuff...

 

[sophiecentaur]

Thanks Sophie, That seems to be sort of thing indeed! It is a bit disconcerting though in as to how much Huygens Principle looks like it could be the earliest description of Quantum propagation. If this indeed the explanation of why the wave only goes one way it would appear that all classical waves are governed by quantum behaviour!

From what I can gather he assumed de facto that the waves only spread out forwards though. Later work seems to suggest that the backwards cancellation only works properly in an odd number of dimensions. So string yes, water no (!), space yes. Fascinating stuff...

That makes it difficult to plot with a paper and pencil then! Just as well I never bothered to try in detail.