When we have 2 sinusoidal waves with slightly different wavelengths and the same amplitude, the resulting wave has a changing amplitude (amplitude = intensity) which varies from 2*Amplitude to 0

Suppose we have two different sources of visible light (slightly different wavelength: different color) producing an electromagnetic wave in 3D.

This will result from point to point a wave similar to the sum from the picture.

Now let's say we have a 2d surface (let's say a paper) where this (superposition of waves) can be projected (and finally a human can perceive it ). What will appear in this surface?

# [Physics] Sum of 2 Light sources

electromagnetic-radiationopticssuperposition

## Best Answer

The full formula for the sum of two waves of different frequencies $f_1$ and $f_2$ is:

$$\cos(2\pi f_1t)+\cos(2\pi f_2t)=2\cos\left(2\pi\frac{f_1+f_2}{2}t\right)\cos\left(2\pi\frac{f_1-f_2}{2}t\right)$$

In other words, the combined wave can be thought of as oscillating with a frequency that is the average of the two frequencies that produced it, while its

intensityoscillates with a frequency equal to half of thedifferenceof the two frequencies that produced it.In order to perceive this oscillation, it must be slow enough that your eye can process it. Your eye has an effective "frame rate" of around 60 images per second (this is why movies work - despite the fact that they're displaying a flickering image, they're switching frames faster than your eye's

flicker fusion threshold, so it appears to be a continuous image). Any brightness oscillation that is faster than around 60 Hz will not be perceptible by the eye.So, if you somehow had two sources that produced light at frequencies that were

exactly, say, 4 Hz apart, and you set up this arrangement, then you would see a point on the paper brighten and darken 2 times per second. The phase of the brightness oscillation would depend on the path length that the light took to reach your eye, so different points on the paper would appear differently bright at any given instant. It would definitely be a pretty cool pattern, if you could get it to work.The problem is that for any

reallight source, the frequencies that it produces are spread out. For example, a helium-neon laser doesn't just produce light at just one frequency; it has abandwidthof around 1.5 GHz (which is still extremely small, considering its frequency is in the THz range), so it produces significant amounts of light with frequencies that are up to 1.5 GHz away from the center frequency. Two frequencies that are 60 Hz apart (the maximum possible difference perceptible to the human eye), but which can vary by up to 1.5 GHz, are completely indistinguishable. The 60-Hz difference is totally insignificant compared to the variation. In short, we needmuchbetter light sources in order for this to actually be practical.