# Fields cancellation and energy accounting

When fields cancel in superposition. Where does the energy go? Do they revert to their potential form? Please give an example.

Two fields can interfere destructively only over a finite region. Consider a sum of two counter-propagating wave solutions. $$\psi(x,t)=\psi_{1}(x,t)+\psi_{2}(x,t)=A\cos(kx-\omega t)+A\cos(kx+\omega t)=2A\cos(kx)\cos(\omega t)$$ There are nodes (regions of complete destructive interference) at $$x=\frac{2\pi}{k}\left(n+\frac{1}{2}\right)$$; the amplitude of the temporal oscillations at these nodal points is zero. However, there are antinodes, where there is complete constructive interference at $$x=\frac{2\pi}{k}n$$.

Therefore, the interference relocated the energy away from the regions of destructive interference to the regions of constructive interference. The total energy, integrated over all space, remains the sum of that of the two interfering fields.

The only way to have complete cancelation over all space is to sum two waves that have equal wavelengths, directions, and amplitudes, but differ in phase by $$\pi$$. But that means a sum like $$A\cos(kx-\omega t)+A\cos(kx-\omega t+\pi)=A\cos(kx-\omega t)-A\cos(kx-\omega t)=0.$$ Note that this wave cancels everywhere in space and time, including at wherever it is supposed to be emitted. So there was really no nonzero waves present, even at the time of emission; and that is the only way that no energy can be involved.

• But what if the two waves only have destructive interferences and no constructive interferences? – Jtl Dec 25 '18 at 23:52
• @Jtl That can't happen. – Buzz Dec 26 '18 at 0:08
• Look ag right side of google.com/…: there is no constructive interferences hence output is only a straight line. – Jtl Dec 26 '18 at 0:46
• @Jtl See my update. – Buzz Dec 26 '18 at 3:39
• @Jtl Buzz is correct here. You cannot have complete destructive interference. The pictures you linked to are not complete solutions of Maxwell’s equations – Dale Dec 26 '18 at 4:19

Let us start with the example. In a noise cancellation the external sounds vibrations are canceled out by vibrations with amplitudes of opposite signs. The magic behind is the dissipation of energy. An old good hammer, hammered on a piece of steel, will be pushed back. Doing this many times, you could feel, that the steel gets hot, so not all of the kinetic energy of the hammer is reversed. Using a hammer on the ISS is not a good instrument because of the pushback. That’s why their hammer is hollow and filled with marbles. The dissipation of the kinetic energy is much more better.

For water waves nearly a perfect cancelation of coherent waves is possible. All the energy is going into heat. Think this like two soft squash balls, all the kinetic energy is converted to heat, means the increase of the vibrations of the involved molecules of the balls.

If 2 waves travel cross a pond from opposite sides, and let's say they are opposite in phase, in the middle of the pond when they meet they cancel, BUT this is only temporary. After they pass each other, they are visible again and continue along to the opposites shores!. Waves cancelling is one of the most poorly explained phenomenons in physics, because they really don't literally cancel, they just temporarily intertact with each other. The same is true for light waves. In the middle of the pond all the wave energy is stored as tension between the water molecules, this energy is very real but not visible unless you are very very tiny!

• How about magnetic field? When opposite currents exist in 2 wires besides each other. The field cancel (basis of GFCI). So they dont really cancel and just hiding themselves? – Jtl Dec 26 '18 at 3:52
• How about magnetic field? When opposite currents exist in 2 wires besides each other. The field cancel (basis of GFCI). So they dont really cancel and just hiding themselves? – Jtl Dec 26 '18 at 3:52
• A non-moving field (static field) is an interesting question, it's not really a wave. You could say it is similar to 2 people pushing on either side of a car, the net force is cancelling but you don't want to get in between them! For static EM fields scientists have invented the term virtual photons to explain things pushing on each other (no energy is transferred during this process only force), these are different than regular light wave photons. Virtual photons is a very advanced area of study and I'm not sure of other practical application of this concept but to explain static EM fields. – PhysicsDave Dec 26 '18 at 4:02
• In fact the 2 wires want to push apart. – PhysicsDave Dec 26 '18 at 4:04
• In the case of magnetic field. There seems to be two issues involved. How energy accounting works in magnetic waves cancellation and how you can store energy in magnetic field via inductors. What is these two different concepts called in physics? – Jtl Dec 26 '18 at 4:15