So where is mistake in the formula of wave number (magnitude of wave vector)?

Question

I have following form for wave vector $k_2=n_2 \omega/c_0$. Now because $\omega=2 \pi c/ \lambda$, then $k_2=n_2 \omega/c_0=\frac{n_2 2 \pi c_0}{c_0 \lambda}=\frac{n_2 2 \pi}{\lambda}$. But problem is that $k$ (is magnitude of wave vector~ wave number) and according to wikipedia $k=\frac{2 \pi}{\lambda}$. So where is mistake?

Answer 1

It looks like your mistake lies in the assumption that the various $\lambda$ are the same. It might help to write:

$$k_2 = n_2\frac{2\pi}{\lambda_0}$$

Answer 2

This is just a different convention. Some people use k to mean radians per meter, and $\omega $ to mean radians per second (angular wavenumber and frequency) while other people use it to mean cycles per meter or per second.

If you are reading a text that defined it in some way, then that is what it means in that text. Each author is free to define any symbols any way he likes. Wikipedia is not some sort of absolute authority on notation.