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

[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.