# Light as a wave, frequency question

1. Apr 19, 2010

### fluidistic

Considering a 1 dimensional problem:
If I'm not wrong, in case of having 2 monochromatic plane waves such that $$E_1=E_{01} \cos (k_1 x - \omega _1 t)$$ and $$E_2=E_{01} \cos (k_2 x - \omega _2 t)$$ (so the same intensity but different wavelength), if we sum them up we reach another wave: $$E=2E_{01} \cos \left ( \frac{(k_1+k_2)x - (\omega _1 + \omega _2)t}{2} \right ) \cdot \cos \left ( \frac{(k_1-k_2)x - (\omega _1 -\omega _2)t}{2} \right )$$. I've asked at my university if I choose the waves as being in the visible spectra if the resulting wave would be in the visible spectra. I've also asked that in case of an affirmative answer, why when we sum up all the waves (infinitely of them, but a big number is a good approximation I guess) forming the visible spectra, we get a "white color" while white hasn't even a defined wavelength.
Now I realize that if you mix paints like yellow and blue, your eyes will see "green" but I believe that any instrument could show that the light is composed of yellow and green. While I do not think any instrument could tell whether the light coming from 2 very very close Red+Green lasers is composed or not. I might be plain wrong, I'm just guessing here.
Anyway, my biggest worry is to know whether the resulting wave has a proper wavelength, thus "color". I've been told that no. But I can't grasp it mentally, I mean it's the sum of 2 waves that have a proper wavelength. So although the resulting wave might look ugly, it must repeat itself over time... right? And so having a wavelength?

Another question: Can sunlight on Earth be considered as a coherent light? I think we can consider the rays as being parallel.

Last edited: Apr 19, 2010
2. Apr 19, 2010

### Antiphon

Sunlight is not coherent. Nothing to do with parallelism (and the sun subtends an angle of about 1/2 degree in the sky).

The resulting wave has a color. Color is a subjective perception and isn't something that can be prescisely defined like energy. There are standards though. Look up CIE1934. There are formulas that take a spectrum and tell you exactly what color it looks like to the average eye.

3. Apr 19, 2010

### fluidistic

Ok... but can a non parallel light be coherent?
Ok thanks for the reference. I know that color is a subjective perception. I'm wondering if the resulting wave has a well defined wavelength. It seems like yes, like I was thinking although I've been told that no?

4. Apr 19, 2010

### Antiphon

Coherent light can be parallel or not. Parallel light can be coherent or not. They are not related concepts.

The wave doesn't have a single unique wavelength. It has a definite color. Many different spectra map onto the same perceived color.

5. Apr 19, 2010

### fluidistic

Ok thanks a lot, you're clearing my doubts.
What interested me wasn't the perceived color, rather the wavelength from the resulting wave.
Ok so it doesn't have a single unique wavelength. I guess I can't rewrite the expression of the resulting E field of the EM field (light) under an expression of the form $$E=E_0 \cos (kx + \omega t)$$ hence the non existence of a unique wavelength.
So if I understand well, there exist light "rays" or photons or whatever light is, that does not have a wavelength that one could situate in the electromagnetic spectrum. Am I right on this?

6. Apr 19, 2010

### Antiphon

Almost. They are all in the spectrum but not with an exact precise frequency. It can be very very narrow but there's always a width to the frequency at least and sometimes a very complex shape for the spectrum.

7. Apr 20, 2010

### Andy Resnick

There seems to be confusion between two limiting cases of coherence: spatial and temporal. Temporal coherence involves the frequency spread, and for sunlight it's very short. The spatial coherence depends on the size of the source, and for sunlight at earth, the coherence area is about 3.5*10^-3 mm^2. By contrast, the coherence area from a (more distant) star is about 6 m^2. Plane waves have an infinite coherence area.

The coherence area is a measure of how far apart the two slits in a double-slit experiment can be and still produce interference fringes, and also relates to the size of the speckle pattern.