Looking for proof of Superpositional Energy Conservation

[greswd]

Let's have waves with their carried power proportional to the square of their amplitude(s).

The waves obey the principle of superposition.

Before superposition, we can calculate the power output based on the amplitudes.
After superposition, there will be new values for the amplitudes, but the power output is expected to remain the same, by the Law of CoE.

I'm looking for a mathematical proof which shows that the superposition of such waves is consistent the Law of CoE.

 

[vanhees71]

The question you are asking needs a dynamical model for the waves. So you must tell us which waves you are talking about. Usually you start from a Lagrangian. Then Noether's theorem tells you how the energy-momentum tensor looks and provides the energy-balance equation.

E.g. take elecromagnetic waves in a vacuum. In Heaviside-Lorentz units the energy density is
$$\epsilon=\frac{1}{2}(\vec{E}^2+\vec{B}^2).$$
Together with the Poynting vector (energy flow density),
$$\vec{S}=c \vec{E} \times \vec{B}$$
you get the energy balance equation
$$\partial_t \epsilon + \vec{\nabla} \cdot \vec{S}=-\vec{j} \cdot \vec{E}.$$
On the left-hand side you have the temporal change of the energy density and the flow of energy out of the surface of the corresponding volume element, and on the right-hand side the energy lost by the field due to mechanical work on the charges. The total energy (field energy + mechanical energy of the charges) is conserved.

 

[greswd]

The question you are asking needs a dynamical model for the waves. So you must tell us which waves you are talking about. Usually you start from a Lagrangian. Then Noether's theorem tells you how the energy-momentum tensor looks and provides the energy-balance equation.

E.g. take elecromagnetic waves in a vacuum. In Heaviside-Lorentz units the energy density is
$$\epsilon=\frac{1}{2}(\vec{E}^2+\vec{B}^2).$$
Together with the Poynting vector (energy flow density),
$$\vec{S}=c \vec{E} \times \vec{B}$$
you get the energy balance equation
$$\partial_t \epsilon + \vec{\nabla} \cdot \vec{S}=-\vec{j} \cdot \vec{E}.$$
On the left-hand side you have the temporal change of the energy density and the flow of energy out of the surface of the corresponding volume element, and on the right-hand side the energy lost by the field due to mechanical work on the charges. The total energy (field energy + mechanical energy of the charges) is conserved.

Let's ignore charges for this problem.
Consider
$$(\vec{E_1} \times \vec{B_1})+(\vec{E_2} \times \vec{B_2})$$
versus
$$(\vec{E_1}+\vec{E_2}) \times (\vec{B_1}+\vec{B_2})$$That's my naive interpretation of the Poynting vector before and after superposition, which is most probably wrong.

So I'm trying to figure out how energy is still conserved after superposition.

 

[Dale]

I'm looking for a mathematical proof which shows that the superposition of such waves is consistent the Law of CoE.

Maxwell's equations are linear, so they obey superposition. And from Maxwell's equations we obtain Poynting's theorem, which shows energy conservation.

What more mathematical proof can you have?

 

[Dale]

Consider
$$(\vec{E_1} \times \vec{B_1})+(\vec{E_2} \times \vec{B_2})$$

This is not a relevant quantity. Poynting's theorem holds for Maxwell's equations, but this quantity is not related to it.

 

[greswd]

This is not a relevant quantity. Poynting's theorem holds for Maxwell's equations, but this quantity is not related to it.

what if we only have information about two Poynting vector fields, with no information regarding the E and B vector fields?

how should we superpose those two Poynting vector fields?

 

[vanhees71]

In the Poynting vector and the energy density you have the electromagnetic field components. Why should you split them somehow in two parts? It's always the full electromagnetic field from all sources entering. Maxwell's equations are linear and thus the superposition principle holds, i.e., the electromagnetic field from different sources add simply up, but that cannot hold true for the energy density and Poynting vector since these are products of field components.

 

[Dale]

what if we only have information about two Poynting vector fields, with no information regarding the E and B vector fields?

how should we superpose those two Poynting vector fields?

You cannot. If you don't have enough information then you don't have enough information

 

[Charles Link]

I think the OP is asking a very good question. With multiple occupancy of the Boson cavity modes (at the same frequency, e.g. as in a laser), if you assume the photon phases are all the same, it leads to a complete contradiction. One explanation that provides for superposition along with energy conservation is that the phases of individual photons go in randomly. (realizing of course that it is somewhat unsound to consider the phase of a single photon.) The phasor diagram becomes a 2-D random walk with the resultant E proportional to $\sqrt{N}$ (for large $N$) where $N$ is the number of photons. Since energy is proportional to E^2, it is then proportional to photon number. $\\$ The above is for plane wave sources that lay right on top of each other. In the case of mutually coherent point sources at separate spatial locations, the intensity patterns do not superimpose, (and they are not required to=the intensity is second order in the linear parameters (E,B) of the Maxwell equations=thereby they are not required to obey superposition and linear properties), but instead interference patterns necessarily result that conserve total energy. $\\$ And finally, in the case of mutually incoherent sources, the intensity patterns do superimpose in a linear fashion.

 

[Charles Link]

In the Poynting vector and the energy density you have the electromagnetic field components. Why should you split them somehow in two parts? It's always the full electromagnetic field from all sources entering. Maxwell's equations are linear and thus the superposition principle holds, i.e., the electromagnetic field from different sources add simply up, but that cannot hold true for the energy density and Poynting vector since these are products of field components.

I agree. I made a similar statement in post #9 that the intensity functions of the sources are not required to obey linear principles. Perhaps this comes as somewhat of a surprise to some, especially when they are working with equations (Maxwell's) that are completely linear. The system is said to be "completely linear" even though some features (e.g. intensity patterns) can appear that do not behave linearly. Perhaps it is a "misnomer" to call it a "completely linear" system because the energy equations of the system are not linear (but are in fact quadratic in the E and B fields).

 

[Dale]

think the OP is asking a very good question. With multiple occupancy of the Boson cavity modes

The OP is asking a purely classical question. I see no value in bringing quantum stuff into the discussion.

 

[Dale]

The system is said to be "completely linear" even though some features (e.g. intensity patterns) can appear that do not behave linearly. Perhaps it is a "misnomer" to call it a "completely linear" system because the energy equations of the system are not linear (but are in fact quadratic in the E and B fields).

I disagree completely. "Linear" has a well defined meaning, and the fields meet that definition, therefore they are linear.

You can always take any linear equation, do some mathematical operation on it and get something nonlinear. So if that were the requirement, then we could never call anything "linear". We might as well not even use the word.

 

[Charles Link]

The OP is asking a purely classical question. I see no value in bringing quantum stuff into the discussion.

The problem can be considered in a very classical context. Considering the energy requirements, the question arises, is it possible to overlay two identical macroscopic electromagnetic plane waves? It would appear the OP is also asking this question, and that question has a slightly more mathematically complex but completely classical solution. The apparatus (using optical interferometers) exists to perform such a superposition of two plane waves with complete conservation of energy, and the explanation of the physics involves the fresnel coefficients. The solution I gave when the question involves many, but individual photons, I think also deserves an answer, which I gave.

I disagree completely. "Linear" has a well defined meaning, and the fields meet that definition, therefore they are linear.

You can always take any linear equation, do some mathematical operation on it and get something nonlinear. So if that were the requirement, then we could never call anything "linear". We might as well not even use the word.

I was just pointing out a subtlety. The word "linear" will continue to be used for such systems, and I, of course, also use the word.

 

[Charles Link]

Just an additional item of interest. (The discussion hasn't had any additional responses, but if anyone is still following it they may find it of interest.) When my classmates and I were first presented with the Fraunhofer diffraction derivation of the two slit and multi-slit problem, it was presented as a sequence of mathematical steps to arrive at the intensity pattern. It wasn't taught as being a linear process, and the energy part of it is not a linear process. We learned the derivations in detail, but it wasn't until later that this whole process was shown to come from Maxwell's equations, with the E-fields and B-fields being the linear parameters of these linear (Maxwell's) equations. The interference patterns, which exhibit some properties that are not of a linear nature, is a result of the energy expressions being second order in the linear parameters rather than first order.

 

[greswd]

Charles, I'm rather fearful that you might derail this thread. Your verbosity makes it harder for others, sorry to say.

 

[Charles Link]

Charles, I'm rather fearful that you might derail this thread. Your verbosity makes it harder for others, sorry to say.

To each their own. I can't please everybody. I will refrain from further comment on your thread.

 

[greswd]

To each their own. I can't please everybody. I will refrain from further comment on your thread.

Thank you for your understanding.

 

[greswd]

What more mathematical proof can you have?

chill bro, I need some time to wrap my head around it.I was thinking of the double-slit experiment.

In this diagram you can see the nodal and anti-nodal lines, the bright and dark fringes of an interference pattern:

The nodal lines are hyperbolas.

If we consider distances far, far away from the sources, the hyperbolas can be approximated as radiating from one specific point.I created a Desmos worksheet to demonstrate a plot of the Intensity against angle:
https://www.desmos.com/calculator/jv5sk0mucx

It follows this formula:

Now consider a circular screen which is extremely large so we can do the radial approximation. We also don't have to worry about the inverse-square law because the circular screen makes all the distances about the same. I assume the energy output is approximated by integrating the intensity across the arc distance of the circular screen.

If you try integrating the formula above from π/2 to 0, you get this:

https://www.wolframalpha.com/input/?i=int+(cos((a*tanx)/sqrt(1+(tanx)^2)))^2+from+0+to+pi/2

Based on my naive assumptions about energy conservation, I assume that energy is transferred from dark fringes to bright fringes.

I thus expect the integral to be simply π/4. Because looking at my Desmos plot, I expect the area under the green curve to be equal to the area under the blue horizontal line if energy is conserved.However, from Wolfram Alpha, the integral is not simply π/4, but has a Bessel function component. This is what I'm confused about.

d represents the slit spacing and W_L represents the wavelength. Apologies for any grammatical errors, I'm quite tired as I'm writing this. Feel free to ask me any questions about the Desmos worksheet.

 

[Dale]

Based on my naive assumptions about energy conservation, I assume that energy is transferred from dark fringes to bright fringes.

I thus expect the integral to be simply π/4.

OK, this seems less like a problem with EM superposition and more like a problem with EM energy conservation.

The conservation of energy in EM is given by Poynting's theorem. What does Poynting's theorem actually say? How would you apply it to a given field to check if energy is conserved?

 

[greswd]

The conservation of energy in EM is given by Poynting's theorem. What does Poynting's theorem actually say? How would you apply it to a given field to check if energy is conserved?

Hmm, Poynting's theorem talks about the work done on charges, which aren't present in the double-slit setup.

I thought of Poynting flux and dot products, but at large distances the Poynting or P-vectors are all close to being perpendicular to the screen. If we assume that all the E vectors are pointing in the same direction, at large distances the B vectors of waves from both slits are close to parallel at the point where they meet on the screen, the vector sum would be the sum of their magnitudes.

I'm still wondering why there's a Bessel component.

 

[greswd]

there is a slight mistake in the formula I posted. let me correct it.

 

[greswd]

here's the new Desmos link: https://www.desmos.com/calculator/8gcwydxb4t

In the top left corner, we can see that for a slit-spacing 9.5 times that of the wavelength, about 7.3% of the energy is "lost". For other ratios, there may be a "gain" instead.

 

[Dale]

Hmm, Poynting's theorem talks about the work done on charges, which aren't present in the double-s

OK, that is one of the terms in Poyntings theorem. We set that term to 0 in vacuum.

What about the other terms. What do they say and how would you check to see if they conserve energy?

 

[greswd]

OK, that is one of the terms in Poyntings theorem. We set that term to 0 in vacuum.

What about the other terms. What do they say and how would you check to see if they conserve energy?

ok, I can see how it relates to the rate of change of energy density.

but idk how I should modify the model in #22.

 

[Dale]

ok, I can see how it relates to the rate of change of energy density.

but idk how I should modify the model in #22.

The model in 22 does not have enough information. You need to know the fields, not just the intensity. Then you can evaluate Poynting's theorem and confirm energy conservation.

 

[greswd]

Sorry for my late replies, I'm pretty busy.

The model in 22 does not have enough information. You need to know the fields, not just the intensity. Then you can evaluate Poynting's theorem and confirm energy conservation.

As I mentioned earlier:

I thought of Poynting flux and dot products, but at large distances the Poynting or P-vectors are all close to being perpendicular to the screen. If we assume that all the E vectors are pointing in the same direction, at large distances the B vectors of waves from both slits are close to parallel at the point where they meet on the screen, the vector sum would be the sum of their magnitudes.

Also, I realized that when the slit spacing 'd' becomes arbitrarily small, both sources converge towards behaving like a single source with double the amplitude and with double the power output, which sounds very paradoxical.

 

[Dale]

I realized that when the slit spacing 'd' becomes arbitrarily small, both sources converge towards behaving like a single source with double the amplitude and with double the power output, which sounds very paradoxical.

But the slit spacing cannot be arbitrarily small. In fact, it cannot be less than the slit width.

 

[greswd]

But the slit spacing cannot be arbitrarily small. In fact, it cannot be less than the slit width.

Is it okay to assume point sources? meaning arbitrarily small slit widths.

 

[Dale]

Is it okay to assume point sources? meaning arbitrarily small slit widths.

Sure, but then the energy is 0 so double 0 is still 0.

 

[greswd]

Sure, but then the energy is 0 so double 0 is still 0.

What about infinitesimally small slit widths and slit spacings? The energy is near zero but the ratio of superposed vs un-superposed energy remains the same.

If we really need to abandon infinitesimally small slit widths, we can just increase the wavelength used, as the far-field result largely depends on the slit-spacing to wavelength ratio.

For a slit-spacing and wavelength ratio of 1, the superposed area is 22% larger than the un-superposed one. That's a pretty large discrepancy when we're trying to make the case that there should be no discrepancy at all.