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Instead light waves interact with one another through a process called "interference".

When two wave meet they superpose - meaning that they merge - and at that point in space the waves add their relative displacements together.

To make this clear, let's envisage two waves, A and B, which are both the same size, wiggling up and down as they travel and they meet at a certain point.

First, lets imagine that A is going "up" at this point, and wave B is also going up.

A added to B makes a much bigger wave with a displacement equal to A + B; so if A and B were both the same height, we'd get a wave at that point that was twice the height of A and B individually. Here you'd see a brighter spot.

Now let's imagine that A is still going up but B is going down. This time the two cancel out because A is equal but opposite to B, so at the point where they meet there is no net displacement and you'd see a dark spot.

But, these observations refer only to the point where the two waves meet. After they have interacted in this way, the waves continue in their original directions, un-changed.