# Vibrations account for interference

Physics Narrative for 14-16

## Coherence is difficult to arrange

Now let's work out how the trip times and frequencies combine to predict the different fixed patterns of displacements. These depend on the geometry of the situation.

In each case here the vibrations at the sources are in step. This is a rather special situation, and it's often not easy to arrange. Remember that light, just as an example, has a frequency in the terahertz range. So keeping vibrations in step at this kind of frequency is a pretty stringent condition – not easy to realise in practice. Where there is a single source that provides both beams (for both cases where there are reflections, or for the pair of slits), it's a bit easier to see how this might be arranged.

The essential idea is rather simple. Each path is of a fixed distance, from the source to the detector. This introduces a fixed delay between the displacement at the source and the displacement at the detector. This delay is just the trip time, calculated, as ever, from the distance and speed of propagation of the waves. For two sources and one detector there will be a pair of trip times.

The difference between the the two trip times determines whether the two contributions, one from each source, are in step or out of step.

## How interference comes about

Interference occurs when coherent beams both arrive at a point. The vibrations in the beams may be in or out of step.

If the vibrations of the two are in step then you'll see constructive superposition. The contribution from one beam adds to the contribution from the other beam to give a large resultant amplitude. If both beams have the same amplitude – so they are the same intensity or brightness – then the resultant amplitude will be twice the amplitude of either beam because the amplitudes simply add.

If the vibrations of the two are completely out of step, then the two contributing amplitudes will add to give a resultant amplitude of zero. This will lead to a point where there is no illumination.

As you scan across the possibilities, varying the trip times of the two beams systematically, so the two contributions move from being in step, through being partially in step, to being completely out of step. The two contributions combine in exactly the same way: they add. But the resultants are different. These resultants predict different brightnesses, or different loudnesses, or, more generally, just different intensities.

Here there are two distinct, and apparently physical, beams, both of which are modelled by paths. Yet, if you remember the work on paths from episode 01, paths can also be used to explain why certain rays are drawn. As you delve more and more deeply into the nature of radiating, you'll get more and more entangled with paths.