What adding does to the energy

Waves add: where two arrive at once, the disturbance is the sum of what each would have produced. That is the whole of superposition, and it is a statement about amplitudes.

What anything measures is not an amplitude. A detector reads a power, which goes as the square of the amplitude — and the square of a sum is not the sum of the squares. There is a third term, and it is what interference is.

The immediate objection is the one everybody has. At the bottom of that curve the two waves together deliver less than either would alone — and for equal amplitudes, nothing at all. Both waves are still there, both still carry energy, and the detector reads zero. Where has it gone?

Before answering, it is worth being exact about what the question is asking, because the loose version of it contains an error that the precise version does not.

The loose version is “two waves cancel, so their energy has vanished”. The precise version is that at a particular place and a particular phase relationship, the intensity is less than the sum of the two intensities. Those are different claims. The first is about energy, which is a quantity summed over a region; the second is about intensity, which is a density at a point. A density can be smaller somewhere without any quantity being smaller anywhere.

So the question worth asking is: integrated over what? And the answer turns out to depend on how big the region has to be to contain the sources.

The answer for sources that cannot feel each other #

The question contains a mistake and the figure shows what it is. Energy conservation is a statement about a region, and the objection was made about a point. Integrate across the screen and the cross term integrates to zero, because it is a cosine of the path difference and the path difference sweeps through whole cycles as the position moves.

So the two sources deliver exactly what they would have delivered separately, and the pattern is a redistribution. The bright fringes carry the missing energy of the dark ones, order for order, and the accounting closes without anything being created or destroyed anywhere.

That is the correct answer in the ordinary case and it is worth being precise about what makes the case ordinary. The two sources are supposed to be far apart compared with a wavelength and to be driven independently — each pushing out its wave without regard to what the other is doing. Under that assumption the total emitted is fixed before the waves meet, and everything that happens afterwards is arrangement.

What the assumption is hiding #

It is not always true, and where it fails the answer changes completely.

Two loudspeakers side by side, wired in opposition, are much quieter than one — not because the sound is cancelling somewhere and reinforcing somewhere else, but because almost no sound is being produced. Push them together and the radiated power goes to zero, and it goes to zero everywhere, so there is no compensating bright region to point at.

The mechanism is at the sources. A source radiates by pushing against the medium, and how hard the medium pushes back depends on what the field there already is. A second source in antiphase nearby cancels the field the first is trying to push into, so the first meets almost no resistance — its cone moves, and moves air about, and does no net work on the far field. The pair is not converting its input into sound and losing it; it is not converting it.

The measurable consequence is at the amplifier rather than in the room, and it is the honest test of the claim. Two speakers wired in opposition and pressed together draw less power than one alone, at the same voltage. Move them apart by a wavelength and the current goes back up. Nothing about the field measurement distinguishes “the energy went elsewhere” from “the energy was never emitted”; the ammeter does.

What the field does between the sources and the screen #

It is worth being concrete about the redistribution, because “the energy goes to the bright fringes” sounds like a bookkeeping trick and is a statement about a flow.

Between two slits and a screen, the energy is not travelling in straight lines. The two waves combine everywhere in that region, not only at the screen, so the direction of energy flow at each point is set by the combined field — and it is not radial from either slit. Mapped out, the flow lines curve: they leave the slits, bend sideways in the region between, and arrive at the screen bunched into the bright fringes and absent from the dark ones.

Nothing arrives at a dark fringe because nothing was heading there by the time it got close. The energy that would have arrived at that point in the absence of the other slit was deflected, gradually, over the whole distance from the slits, by the presence of a field it was adding to.

That picture also explains why the dark fringes are dark all the way to the screen and not only on it. Put a detector halfway and the fringe pattern is there too — wider, because the fringes have not yet separated as much, but present. The redistribution is continuous, and the flow lines are what it looks like.

There is a corresponding statement about a beam that is worth carrying, because it recurs. Where a wave’s intensity varies across a beam, the energy is not travelling straight. That is true of a focusing beam, of a wave spreading from a source, and of every interference pattern, and it is why an argument that follows rays and adds intensities at the end gets the pattern right and the mechanism wrong.

The same statement in three settings #

The distinction between the two answers recurs, and the useful summary is a question rather than a rule: can the sources feel each other’s field?

Where they cannot — two lamps, two independent radio transmitters, two slits illuminated by one distant source — the total is fixed and interference arranges it. Two lamps do not interfere at all on any timescale a detector resolves, and even the coherent case only moves light about.

Where they can — two antennas a fraction of a wavelength apart, two speakers in a cabinet, two atoms within a wavelength of each other — the total is not fixed, and the interference reaches back into the emission. An antenna’s radiation resistance changes when another antenna is brought near, by an amount that is exactly the interference term integrated over all directions, and antenna designers compute it under the name of mutual impedance.

The quantum version of the same statement is superradiance: a group of excited atoms within a wavelength of one another radiates not $N$ times faster than one but $N^2$ times, because the emission is coherent and the sources are coupled. Its mirror image, subradiance, is a set of atoms whose collective state cannot radiate at all and which therefore does not decay. In each case the lifetime — the emission rate — has changed, which is the same statement as the loudspeakers’: what changed was not where the energy went but how much was emitted.

A third case: where the energy goes into the medium #

There is an arrangement in between the two, and it is the one most often met in practice: sources that cannot feel each other, in a system that reflects.

Consider a wave sent down a line towards a load that reflects part of it. The forward and backward waves interfere, giving a pattern of maxima and minima along the line, and the question of where the energy goes at a minimum has a third answer: it is passing through. The net power crossing every point is the forward wave’s minus the backward wave’s, which contains no cross term at all and is therefore the same everywhere, while the amplitude swings by the standing-wave ratio.

That case is instructive because it looks exactly like the first — an interference pattern with maxima and minima — and the accounting is different again. Here there is no screen to integrate over, no set of fringes whose brightnesses average out, and the resolution is that the flow is uniform while the amplitude is not. Two quantities that are proportional to each other in a travelling wave stop being so as soon as there are two waves.

The general lesson is worth stating once, because it is what makes interference arguments error-prone. In a single travelling wave, amplitude squared, energy density and energy flux are all proportional and the distinction between them never has to be made. In any superposition they come apart, and every apparent paradox about interference and energy is a place where one of them has been used where another was meant.

Where the cross term goes when the waves are not steady #

Beats are the same arithmetic with the phase difference sweeping in time rather than in position, and putting them beside the fringes makes the general statement clearer than either alone.

The cross term is $2a_1a_2\cos\delta$, and everything about interference is a statement about what $\delta$ does. If it is fixed, the two waves are coherent and there are fringes. If it sweeps through position, the fringes are in space. If it sweeps through time, they are beats. If it is random and changes faster than the detector, the cross term averages to nothing during every measurement and there is no interference to observe — which is the whole of why two ordinary sources do not interfere.

In every one of those cases the average of the cross term is zero, which is why the energy question keeps having the same answer. The exception is the one above: where the sources are coupled, the cross term does not average to zero over all directions, because the geometry weights it — and a weighted average of a cosine need not vanish.

The array, where this is engineering #

Two coupled sources is the simplest case of an antenna array, and it is worth spending a paragraph on because the whole subject is the deliberate use of the effect this essay is about.

An array of radiators fed with chosen amplitudes and phases has a directional pattern that is the interference of their contributions, and the design problem is to put the lobes where the signal is wanted and the nulls where it is not. That much is redistribution and is the first answer above: the total radiated is roughly the sum of the elements’, and the pattern arranges it.

What makes the subject harder than that is the second answer. Each element’s radiation resistance is changed by its neighbours — an element in a close-packed array can find its own resistance driven near zero, or negative, so that it absorbs power from the array rather than delivering it. Feeding such an array with the currents the pattern requires then needs voltages that have nothing to do with the currents in isolation, and a design that ignores the coupling produces a pattern nothing like the intended one.

The extreme case is the superdirective array: elements much closer than half a wavelength, driven in a pattern of alternating phases, which in principle produces an arbitrarily narrow beam. It fails in practice for exactly the reason the loudspeakers went quiet. The radiation resistance collapses, so enormous currents are needed for any radiated power, and the ohmic losses in the conductors — which do not collapse — swamp everything. The limit is not a wave-optics limit at all; it is that the interference has changed what the sources are working against.

Where the model stops #

Superposition is a property of the equation, not of waves in general. It holds because the wave equation is linear, and it fails wherever the medium’s response depends on the amplitude — which is what makes a nonlinear medium generate frequencies nobody supplied and what makes a large-amplitude wave steepen until it breaks. None of the accounting in this essay survives that.

The energy argument assumes the medium is lossless. In an absorbing medium, moving energy from one place to another changes how much is absorbed, so a redistribution genuinely changes the total delivered — and an interference pattern in a lossy medium is a real change to the power budget rather than a rearrangement.

The coupled-source result is stated for two point sources in an unbounded medium. Put them in a room, a pipe or a cabinet and the reflected field is another contribution to what each source pushes against; the mutual loading is then a property of the enclosure as much as of the separation, which is most of what makes loudspeaker cabinets difficult.

And nothing here is about detection. The claim is about power, and a photon-counting measurement has its own statistics on top of it. The interference of single photons has exactly the pattern computed here and a completely different account of what is happening at any one arrival, which is the subject of another ladder.

How the question is best asked #

Given three different answers for three arrangements, it is worth having a procedure rather than a rule.

Start by identifying the region. If it can be drawn so that it contains the sources and its boundary is far away, then the total crossing the boundary is the total emitted, and any interference inside is redistribution. If the region cannot be drawn that way — because the sources are inside each other’s near field, or because the boundary would have to cut through a source — then the emission itself is in question and the field measurement alone will not settle it.

Then ask what is being held fixed. Two sources driven by fixed currents radiate what the interference says; two driven by fixed voltages do not, because the current adjusts. Almost every apparent paradox about interference and power turns out to be a case where the thing held fixed was not stated, and stating it resolves the paradox without any new physics.

What the pictures cannot show #

The fringe figure draws intensity against position on a screen and cannot show the flow. Between the source and the screen the energy is not travelling in straight lines from each source to each point: the two waves’ fields combine everywhere, and the resulting flow curves — light really does move sideways in the space behind two slits, from where the dark fringes will be to where the bright ones will be. Drawing that requires a map of the Poynting vector, and it is the picture that makes the redistribution concrete rather than merely accounted for.

The coupled-source figure draws total radiated power against separation and hides the pattern entirely. At any separation the pair has a directional pattern — lobes and nulls — and the total is that pattern integrated. Two sources half a wavelength apart in phase are not simply twice as loud; they are much louder broadside and much quieter end-on, and the total happens to be about twice.

Where the ladder goes next #

This ladder began with two waves meeting and simply adding. This rung asks what that does to the energy, and finds two different answers for two different arrangements. The rungs after it: mutual impedance in earnest, where the coupling between two radiators is computed and used to design arrays; the general interference of many sources, where the sum over random phases gives an intensity proportional to the number rather than to its square; and the failure of superposition itself, where the medium’s response depends on what is already in it.

The habit worth carrying away is about where to draw the box. A conservation argument about interference is only as good as the region it is made over, and the two questions worth asking of any such argument are whether the region contains a whole number of fringes and whether it contains the sources.