Interference

Constructive and destructive interference

If we add up two identical waves, the resulting amplitude doubles:

\[\sin (x) + \sin (x) = 2 \sin (x)\]

While mathematically obvious, have a look at what it means visually, with two sinusoids one above the other, as in figure 2.

The points that have the same x coordinate also have the same amplitude. Applying the sum causes the wave to stretch. Where the amplitude was 1, it is now 2, and where it was -1, it is now -2. This is called constructive interference. An example from everyday life is two speakers playing the same track. The volume of the music perceived is maximum when the waves produced by the speakers are in phase, interfering constructively.

When the waves have different phases, the result of the superposition changes, especially if they are in phase opposition or counter-phase when every point is added to one of the opposite value, e.g., 1 + (-1), as in the graph below.

The sum of these waves is zero due to destructive interference. Notice how, in this case, the phase causes the same result as the negative sign before the function.

\(\sin (x - \frac{\pi}{2}) + sin (x + \frac{\pi}{2}) = 0\)

\(\sin (x - \frac{\pi}{2}) - \sin (x - \frac{\pi}{2}) = 0\)

We defined the two ends of the line. In between, there are all the combinations of the two waves. The phase of the resulting wave is shifted to somewhere between the phases of the interfering waves, depending on their amplitudes, and the value of its amplitude will be between zero and twice the amplitude of the interfering waves.

Interference patterns

We talked about the interference between one-dimensional waves. The same phenomenon occurs when the propagation happens along two or more dimensions. In this case, two waves interfere and create an interference pattern.

When two stones are thrown into a lake, with one being thrown from a slightly different spot close to the one from which the first stone was thrown, a bi-dimensional wave forms on the surface of the water. In this scenario, the surface of the water gets corrugated but still exhibits a regularity, hence the name of this type of interference.

In the image, two circular waves propagate towards each other at an angle of \(\frac{\pi}{2}\). The wave fronts interfere almost orthogonally, giving the water a grid-shaped surface. The lines of the grid are points of destructive interference, while between them, there are points of constructive interference.