# Does $\lambda\nu = c$ hold for all the waves in the universe?

Are all waves in the universe the same as electromagnetic waves?

Basically, my question arises from an equation I found in my chemistry textbook:

$$\lambda \nu ~=~ c.$$

This states that the wavelength (distance from crest to crest) times the frequency (amount of times the wave passes the center point) equals the speed of light. Now, I know this applies in a vacuum and that the speed of light changes based on density. However, does this apply to all waves? If so, does this apply to a wave in water?

It seems as though it should. You have a high frequency of light that is incredibly high compared to water, but you have a incredibly small wavelength which causes that value to be lower. In the case of water, if you considered only the wavelength, the value would be too high, but if you also consider that the frequency is going to be orders of magnitude lower than light, you can see where I might arrive at this conclusion.

If we measure the crest of a wave of water and the frequency of that wave (I assume from the surface of the body of water) and consider the density of the water in our calculation (as some other variable that is normally used when calculating this), would the result also be the speed of light? In addition, if water was within a vacuum and we were to create a wave, how would this react? If we could create a wave in water in this vacuum, would our values reflect $c$ or a variation of $c$ based on the density of the water?

If $\lambda \nu ~=~ c$ is valid for all waves, what other attributes must be supplied within the equation to make the math work out to give the correct answer of $c$?

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This equation is not derived, it's straight forward, in sense it is something like $a\frac{b}{c}\frac{c}{b}=a$ and there is really no hidden physics inside, to understand this better I suggest you to see how one construct it. – TMS Oct 28 '12 at 20:10

## 1 Answer

Yes, this equations applies to all waves... with the caveat that you replace c by the speed of the wave you're studying! In a water wave, the product of the wavelength and the frequency will be the speed of the water wave, not of light. For sound waves in air, it will be the speed of sound, etc.

Because of this, the general form of the equation you provided is: $$\lambda \nu ~=~ v_{wave}.$$

Another interesting thing is that the speed of the wave need not be constant. The equation is always valid, but it might be possible that the wavelength depends on the frequency, in which case the speed will also depend on the frequency. This happens with water waves; you can notice not all waves travel at the same speed on the ocean. It also happens with light; light of different colours travel at different speeds through glass, which is what allows a prism to disperse white light into a rainbow.

When the wavelength depends on the frequency, we call it "dispersion". If it doesn't, then the wave speed is the same for all waves of that type.

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Why is it $v_{wave}$ instead of $c$?  Is it because the wave has mass or because some attribute of the wave is not accounted for in the equation (such as its density)? – Jonathan Hickman Oct 28 '12 at 12:34
@JonathanHickman The speed of the wave depends on the nature of the medium carrying the disturbance. For electromagnetic waves the disturbance is carried along in the electric and magnetic fields obeying Maxwell's equations, and the wave speed is determined by the vacuum permittivity and vacuum permeability that appear as physical constants in those equations. But waves show up in other media, too. For a wave on a string, the speed will be determined by the tension of the string and its mass/length, and this speed will be very different from the speed of light $c$. – kleingordon Oct 28 '12 at 23:48
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