# Question about the dispersion relation

I'm a Non-physics major student. Recently, I am puzzling by the dispersion relation. For a wave, we have $v=f\lambda$, where $v$, $f$ and $\lambda$ is the velocity, frequency and wave length of the wave, respectively. According to this relation, velocity should depends on the frequency. Then why should we have the definition of dispersive medium, as in my opinion everything is dispersive. Because from the formula above, the velocity should automatically depends on the frequency. Can anybody help me with my question? Thanks a lot!

This question has an interesting connection to a post a few days ago. I can't find that question at the moment. The questioner there asked if it made any difference if one writes $F=ma$ or $a=F/m$. Of course, the math is the same, but the way the expression is written suggests "cause" and "effect".

Here's an example where the format of the formula might have helped. For simple or ideal cases, the speed of a wave in medium is a constant independent of frequency. So the left hand side of $v=\lambda f$ should never change. It looks like it changes with frequency, but when you change the frequency, the wavelength changes at the same time in such a way that $v$ does not change.

Perhaps it would be wiser to be acknowledge the constant speed and write $$\lambda = \frac{v}{f}$$.
Here the typography makes clear the inverse relationship between frequency and wavelength.

Dispersion occurs in media where the speed is not constant. Strictly speaking, all media are dispersive. Taking the speed as constant is an approximation that works very well for some purposes. (Of course, vacuum is not a medium, so that statement does not apply to electromagnetic radiation in the vacuum.) We would have $$\lambda(f) = \frac{v(f)}{f}$$ The format of the formula doesn't matter from the point of view of math, but it does suggest something about interpretation. I suppose the linear versions $F=ma$ and $v=\lambda f$ are more common because it's easier and takes less space to typeset the formula on one line. (That's just a guess.) Hope that helps!

• Thanks garyp! You analogy makes me clear. I have another analogy, where an object is tranlating in one direction. We have $v(t)t=s(t)$, here $f$ plays the same role as $t$. At any $t$we have one $v(t)$. Similarly, for a specific medium, for a wave with fixed frequency $f$, we will get a $v$ according to $v(f)$. Actually, $v=f\lambda$ just comes from the relation between displacement, time and velocity. Commented Aug 23, 2016 at 12:29

For media where the speed does not depend on frequency, the equation $v = f \lambda$ does not mean $v$ depends on the frequency. It is a relationship between $f$ and $\lambda$ such that when you multiply them together they give the speed $v$.

For a non-dispersive medium if you know the frequency of the wave and its wavelength, then you can work out the speed and this number is the same for all frequencies and wavelengths.

This means that if you change the frequency of the wave then (since you also know the speed $v$ as a fixed number) you can work out the wavelength (similarly if you have a wave with a different wavelength you can determine the frequency).

For media described as dispersive then you write this as $v(f)$. This means if you know the wavelength for one particular frequency you can calculate the speed of the wave for that combination.

But, if the frequency changes and you do not know how the speed depends on frequency, then you cannot work out the wavelength for a different frequency (since you do not know what the speed is, as it changes with frequency and you have only worked it out for one frequency).

• Thanks Jim! So in non-dispersive medium $f$ and $\lambda$ always holds the relation that $constant=f\lambda$ . That is $f$ is always proportion of $1/\lambda$ Commented Aug 23, 2016 at 11:57
• Yes, that's right.
– jim
Commented Aug 23, 2016 at 17:52