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

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?

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Where did the $n_2$ come from in your original definition of the wave vector $k_2 = n_2w/c_0$? – John Rennie Apr 18 '12 at 14:50
I don't know where it comes from but particular formula is related to nonlinear optics and according to the lecture notes $k_2=n_2 \omega/c_0$ is the wave vector of the Second harmonic generation wave. – alvoutila Apr 18 '12 at 15:00
Mentioning "nonlinear optics" was very helpful here. Now we know where to begin our search. – Pygmalion Apr 18 '12 at 15:04
If $\omega$ is the frequency of the fundamental then it's wave vector would be $\omega / c_0$, the wave vector of the first harmonic would be $2\omega / c_0$ because it's twice the frequency. The second harmonic would be $3\omega / c_0$ and so on. – John Rennie Apr 18 '12 at 15:06
Please include reference and link for the lecture notes if possible. – Qmechanic Apr 22 '12 at 15:30

It looks like your mistake lies in the assumption that the various $\lambda$ are the same. It might help to write:
$$k_2 = n_2\frac{2\pi}{\lambda_0}$$
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.