# Does destructive interference contradict the law of conservation of energy?

• ifanyonecan

#### ifanyonecan

When two waves that are 180 degrees out of phase interfere, the energy from both cancels out. Why is it that, for example, a sound wave, which is a form of energy, can disappear when combined with another when the Law of Conservation of Energy states that energy cannot be destroyed?

I posed this thought in my Physics class the other day, and the instructor said, "The Law of Conservation of Energy works great when you're working with concrete trucks and things, but when you get into more detailed physics it doesn't really apply." I think this is bogus, since it is a law, which means it is accepted to work all the time with no exceptions. Unfortunately, I don't have a sufficient argument to prove my point.

Any help is greatly appreciated.

Chris

It doesn't contradict it because there is still kinetic energy, or there will be constructive interference somewhere else.

It doesn't. As energy is not lost. Two colliding sound waves (longitudinal waves) do cancel each other locally but no energy is lost. The colliding air molecules generate heat or just change the direction of their wave ( that means that you get two waves in opposite directions, which would look like the first two just passed through each other). It's a bit more complicated than that actually and as for EM waves the same applies.

A simple experiment is sitting between two speakers that play the same music. Right in the middle of them the sound waves cancel out. you can percieve that as a zone of low intensity (cover one ear for maximum effect). Even though those speakers have a whole plane between them that cancells out, they still add up at a big distance from them so basically energy never got lost. Which means that the energy from one can pass the middle cancel zone.

You don't have a violation of conservation of energy in destructive interference any more than you have it in constructive interference. In fact, if you add up the energy "gained" in constructive interference it exactly compensates for the energy "lost" in destructive interference.

When two waves that are 180 degrees out of phase interfere, the energy from both cancels out. Why is it that, for example, a sound wave, which is a form of energy, can disappear when combined with another when the Law of Conservation of Energy states that energy cannot be destroyed?

I posed this thought in my Physics class the other day, and the instructor said, "The Law of Conservation of Energy works great when you're working with concrete trucks and things, but when you get into more detailed physics it doesn't really apply." I think this is bogus, since it is a law, which means it is accepted to work all the time with no exceptions. Unfortunately, I don't have a sufficient argument to prove my point.

Any help is greatly appreciated.

Chris

W. N. Mathews "Superposition and energy conservation for small amplitude mechanical waves", Am. J. Phys. v.54, p.233 (1986).

Zz.

If you have two one-dimensional waves like A*cos(omega*t-k*x) and -A*cos(omega*t-k*x), they cancel everywhere (zero). But let us not forget that a wave is generated by a source. If your source does not generate waves, you can write zero in many ways (A is arbitrary). If you speak of two separated sources, then there will be space points where the corresponding interference is constructive, so there is no zero everywhere. And the energy is not local but volume integral variable. That is why it may "flow" from one region to another being conserved in the numerical value.

Bob.

You don't have a violation of conservation of energy in destructive interference any more than you have it in constructive interference. In fact, if you add up the energy "gained" in constructive interference it exactly compensates for the energy "lost" in destructive interference.

You gain nothing in constructive interference ,it is the sum of the energy of the two waves.
so constructive interference can not compensates for the energy "lost" in destructive interference.

It doesn't contradict it because there is still kinetic energy, or there will be constructive interference somewhere else.

You gain nothing in constructive interference ,it is the sum of the energy of the two waves.
so constructive interference can not compensates for the energy "lost" in destructive interference.

It doesn't. As energy is not lost. Two colliding sound waves (longitudinal waves) do cancel each other locally but no energy is lost. The colliding air molecules generate heat or just change the direction of their wave ( that means that you get two waves in opposite directions, which would look like the first two just passed through each other). It's a bit more complicated than that actually and as for EM waves the same applies.

A simple experiment is sitting between two speakers that play the same music. Right in the middle of them the sound waves cancel out. you can percieve that as a zone of low intensity (cover one ear for maximum effect). Even though those speakers have a whole plane between them that cancells out, they still add up at a big distance from them so basically energy never got lost. Which means that the energy from one can pass the middle cancel zone.

What about conservation of energy in constructive interference of electromagnetic waves?

You gain nothing in constructive interference ,it is the sum of the energy of the two waves.
so constructive interference can not compensates for the energy "lost" in destructive interference.
This is slightly incorrect. The wave amplitudes sum to create constructive and destructive interference.

If we have a pair of coherent waves (each amplitude 1) which are interfering with each other we will get a spatially distributed pattern of nodes and anti-nodes. At the nodes, the amplitude will be 0, so the energy density will be 0, and at the antinodes the amplitude will be 2, so the energy density will be 4 times the energy density of one wave. Integrating over space you get that the energy is 2 times the energy of one wave, as required by the conservation of energy.

What about conservation of energy in constructive interference of electromagnetic waves?
EM waves are no different from any other waves in terms of constructive and destructive interference. In the absence of any interaction with matter, destructive interference in one location will always be balanced by constructive interference in another location.

Here is my favorite page on energy conservation in EM waves:
http://farside.ph.utexas.edu/teaching/em/lectures/node89.html

Yes thanks ... this implies that there is no local conservation for the energy in electromagnetic field but this leads me to an important question related to General Relativity
so I am going to open new thread there.

Yes thanks ... this implies that there is no local conservation for the energy in electromagnetic field.
This is not correct at all. See the link I provided, the conservation law can be written locally also.

Simply because something has a nonlocal explanation does not imply that it does not have a local explanation also. The local explanation is not as easy to follow, but is given in full detail and generality in the link.

By detail, maybe he means locally. Because sometimes if there is destructive interference locally there would be constructive ones that have energy of x4.

Or probably quantum mechanics, energy is not conserved due to an uncertainty, but it is off topic.

Indeed, destructive interference is NOT a total effect, rather a nodal effect.
As such, the waves actually continue with the OPPOSITE nodal effect, satisfying conservation of energy, and all is well...

Special note: Shock-wave propagation is much different, and do NOT obey the same rules, as there is no inherent oscillation.

Last edited:
This is not correct at all. See the link I provided, the conservation law can be written locally also.

Simply because something has a nonlocal explanation does not imply that it does not have a local explanation also. The local explanation is not as easy to follow, but is given in full detail and generality in the link.

I am not convinced. That link explains energy conservation for an electromagnetic wave, but it does not discuss the issue when two waves interfere. The following is a specific example when there seems to be a problem.

Suppose you have two plane square waves traveling in the $\textit{z}$ and $\textit{-z}$ directions, respectively.
For the first wave $\textbf{E}_{A}\textit{(x,y,z,t)} = (\textit{E}_0 \textit{f(z}-\textit{c t)}, 0, 0 )$ and $\textbf{B}_{A}\textit{(x,y,z,t)} = ( 0, \textit{B}_0 \textit{f(z}-\textit{c t)}, 0 )$,
where $\textit{f(v) = 1}$ if $\textit{-L<v<L}$ and $\textit{f(v) = 0}$ otherwise.
For the second wave $\textbf{E}_{B}\textit{(x,y,z,t)} = ( -\textit{E}_0 \textit{f(z}+\textit{c t)}, 0, 0 )$ and $\textbf{B}_{B}\textit{(x,y,z,t)} = ( 0, -\textit{B}_0 \textit{f(z}+\textit{c t)}, 0 )$.

In this case, there is complete destructive interference everywhere in space at $t=0$, since $\textbf{E} = \textbf{E}_{A}+\textbf{E}_{B} = (0, 0, 0 )$ and $\textbf{B} = \textbf{B}_{A}+\textbf{B}_{B} = (0, 0, 0 )$. Thus, the energy density vanishes at this instant everywhere as well, $U(x,y,z,0) = 0$. However, it is nonzero at other times. In particular, $\textit{U(x,y,z,t)} = \textit{U}_0 \textit{f(z}-\textit{ct)} + \textit{U}_0 \textit{f(x}+\textit{c t)}$ if $|\textit{t}| > \textit{L}/c$, where $\textit{U}_0 = \frac{\epsilon_0 E_0^2}{2} + \frac{B_0^2}{2 \mu_0}$. The total energy in the electromagnetic field in a box enclosing the two waves is $\int \textit{U} \;\textit{dV} = 0$ at $t=0$, while it is $\int \textit{U} \;\textit{dV} = \textit{ 2 L A U}_0$ if $|\textit{t}| > \textit{L}/c$ when the two waves are well separated. Here $\textit{A}$ is the area of the box in the $\textit{(x,y)}$ plane. (Note that the energy flux $\textbf{u}$ (eq. 1034 in the link) drops out if you integrate over the space containing the waves.)

Therefore energy conservation seems to be violated during interference in this particular example. The problem seems to be that the two waves add linearly when they interfere, while the energy is a nonlinear function of the amplitudes. ZealScience, note that in this example there is an instant when there is destructive interference everywhere, so that there are no regions where a positive interference would compensate. Pallidin, isn't the wave equation satisfied for shock fronts?

Can someone explain what I am missing?

I am not convinced. That link explains energy conservation for an electromagnetic wave, but it does not discuss the issue when two waves interfere. The following is a specific example when there seems to be a problem.
...
Can someone explain what I am missing?
The link explains conservation of energy for Maxwells equation. Any time your system follows Maxwells equations it conserves energy. The number and details of the waves are not relevant.

In your particular wave B does not satisfy Maxwells equations.

In the sense of say the interference of electrons with themselves in the double slit experiment, is it correct to say that the "interference" merely determines the probability of where the electron will end up? Obviously the electrons aren't destroyed due to destructive interference, its just the positions of many electrons added together that forms the pattern. (I'm about 95% sure that this is correct, but I just wanted to make sure)