Why is the power of a particle on a wave zero in a stationary wave?

[Gourab_chill]

Homework Statement

I'm confused on the concept of power carried by particles in a wave.

Relevant Equations

--i don't know any equations which can help here--

I've marked the right answers.
They mainly indicate at power carried by the particles being zero, and here is my doubt- why should it be zero? Shouldn't it have some definite value?
I do understand that the kinetic energy is max at the y=0 and potential energy is max at y=A, but I don't know whether that is needed for this question.
Why should the power be zero?

 

[haruspex]

I am at a loss to know what is meant by the power "across" two points. Do you have any course notes that might define this?

 

[etotheipi]

I am at a loss to know what is meant by the power "across" two points. Do you have any course notes that might define this?

I agree, it is terrible wording. We can speak of the total energy of the wave between any two points, and if we know the velocity of the wave we can use that to deduce the power transmitted across a certain point (e.g. on the $x$ axis).

But it makes no sense to speak of the power "across two points", or the power "possessed by the rope".

 

[Gourab_chill]

I am at a loss to know what is meant by the power "across" two points. Do you have any course notes that might define this?

I don't know I'm confused myself. It probably is asking about the difference in power at the points (my concepts regarding this is a bit shallow); I don't know this makes any sense either.

But, in simple terms the rope or string does carry some power ?

 

[etotheipi]

But, in simple terms the rope or string does carry some power ?

It doesn't make sense to say something "has" power. Power is a rate of transfer of some form of energy across a boundary (e.g. the power of a force that is working on a particle, or the power incident on a surface).

However, you can say that a section of the rope has energy. If the wave is "moving" (i.e. not a stationary wave), energy is being transferred past a point on the $x$ axis.

You may calculate this by calculating the total energy (kinetic + potential) of a single wavelength of the rope (notice that it will be $\propto A^2$) by integrating up, and then you can divide this by the time period.

In electromagnetism, the power flux (power per unit area) is the Poynting vector $\mathbf{S}$; this is a similar concept, and the energy through a surface per unit time is $P = \iint \mathbf{S} \cdot d\mathbf{A}$.

 

[Gourab_chill]

It doesn't make sense to say something "has" power. Power is a rate of transfer of some form of energy across a boundary (e.g. the power of a force that is working on a particle, or the power incident on a surface).

However, you can say that a section of the rope has energy. If the wave is "moving" (i.e. not a stationary wave), energy is being transferred past a point on the $x$ axis.

You may calculate this by calculating the total energy (kinetic + potential) of a single wavelength of the rope (notice that it will be $\propto A^2$) by integrating up, and then you can divide this by the time period.

In electromagnetism, the power flux (power per unit area) is the Poynting vector $\mathbf{S}$; this is a similar concept, and the energy through a surface per unit time is $P = \iint \mathbf{S} \cdot d\mathbf{A}$.

Since this wave is moving or its particles are, can I say there is a constant power across this string and is constant at any region(or point?) over the string as the energy transmitted is conserved?

 

[etotheipi]

Since this wave is moving or its particles are, can I say there is a constant power across this string and is constant at any region(or point?) over the string as the energy transmitted is conserved?

You must speak of power through a particular surface, or in this case through a particular point.

Of course, if you consider two points on the axis that enclose a region of rope, then $\frac{dE}{dt}$ of this region must be zero and as such the net power into this region must be zero. This is an expression of the continuity equation, since power is energy flux: the integral of energy flux density through the closed surface!

 

[Gourab_chill]

You must speak of power through a particular surface, or in this case through a particular point.

Of course, if you consider two points on the axis that enclose a region of rope, then $\frac{dE}{dt}$ of this region must be zero and as such the net power into this region must be zero. This is an expression of the continuity equation, since power is energy flux: the integral of energy flux density through the closed surface!

Okay, so the energy across two points never changes- rate of change of energy is zero - power is zero. Thanks a lot for explaining:)

 

[etotheipi]

Okay, so the energy across two points never changes- rate of change of energy is zero - power is zero. Thanks a lot for explaining:)

Essentially yes, but be careful with wording. I would call it the "energy of the wave between two points" on the $x$ axis, and speak of the "net power" into that region.

You will see that this is a common theme amongst different applications of the continuity equation. For instance, electric current is the rate of flow of charge past a point (charge flux), analogous to power in this example (energy flux). The current is the surface integral of the current density, and the power the surface integral of the energy flux density.

You will notice that we speak of the charge enclosed between two points of a circuit and the current into/out of this region.

Likewise, here we speak of the energy enclosed between two points of the axis, and the power into/out of this region.

 

[DrClaude]

Okay, so the energy across two points never changes- rate of change of energy is zero - power is zero. Thanks a lot for explaining:)

Then answer (B) would also be correct. I don't understand the question, but I have a feeling there is more to it than this.

 

[Gourab_chill]

You will notice that we speak of the charge enclosed between two points of a circuit and the current into/out of this region.

Likewise, here we speak of the energy enclosed between two points of the axis, and the power into/out of this region.

Yes, the analogy is quite clear!

@DrClaude yes I think that might also be the right answer.

 

[Delta2]

If the wave is stationary then the power across any point is zero. Stationary waves don't transfer energy and momentum. So B is also correct yes.

 

[Gourab_chill]

If the wave is stationary then the power across any point is zero. Stationary waves don't transfer energy and momentum. So B is also correct yes.

This is not a standing wave, but I'm confused as you said "if the wave is stationary";there are two types of waves- moving waves and stationary waves, right?

 

[Delta2]

This is not a standing wave, but I'm confused as you said "if the wave is stationary";there are two types of waves- moving waves and stationary waves, right?

yes and i thought that standing and stationary waves are the same thing. Why do you think are different?

 

[Gourab_chill]

yes and i thought that standing and stationary waves are the same thing. Why do you think are different?

But this is not a standing wave, but shouldn't the power be zero across two points? You said power should be zero assuming it as a standing wave

 

[Delta2]

It depends how you interpret the term "Power across two points". The way you interpret it it is correct that the power across two points is zero even for a non standing (progressive) wave.
The way i interpret it is different though: I interpret it as Power across point 1 and Power across point 2 and these two powers are not zero for a non standing wave.

 

[Gourab_chill]

It depends how you interpret the term "Power across two points". The way you interpret it it is correct that the power across two points is zero even for a non standing (progressive) wave.
The way i interpret it is different though: I interpret it as Power across point 1 and Power across point 2 and these two powers are not zero for a non standing wave.

Yes, at a point the power is zero but over a region the power is zero as energy flux is zero, right?

 

[Delta2]

The way i understand it is that the power across a point is not zero but the power in the region across two points is zero because the energy that enters from one end point of the region equals the energy that leaves from the other end point of the region

 

[Gourab_chill]

The way i understand it is that the power across a point is not zero but the power in the region across two points is zero because the energy that enters from one end point of the region equals the energy that leaves from the other end point of the region

Yes, I agree with this

 

[haruspex]

The way i understand it is that ... the power in the region across two points is zero because the energy that enters from one end point of the region equals the energy that leaves from the other end point of the region

Wouldn't that be true for all pairs of points in all lossless waves?
In post #12 you took the view it was to do with the transfer of energy between points. But that would be zero for all pairs of points in a standing wave and nonzero for all pairs in a progressive wave.

 

[haruspex]

But this is not a standing wave

All options refer to case III, and case III is described in the question as as a stationary wave. If you think that does not mean a standing wave, please explain the difference.