I am reading Matter and Light by de Broglie and a little bit confused about how he deduced the equation of
$v_g := \frac{\mathrm d\omega}{\mathrm dk} = v$
I understand that a wave that is static in one inertial frame could be written as
$ \phi=e^{2\pi i \nu_0 (t-\tau_0)}$
and if you see this wave from moving frame of reference (that moves $v$ to $-x$)
$ \phi=e^{2\pi i \nu_0 \left(\frac{{t^\prime-vx^\prime/c^2}}{\sqrt{1-v^2/c^2}} -\tau_0\right)}$
this way you get
$\nu = \frac{\nu_0}{\sqrt{1-v^2/c^2}}$
and
$ \lambda = \left ( \frac{\nu_0 v}{c^2 \sqrt{1-v^2/c^2}} \right )^{-1}$
and you could calculate $\frac{\mathrm d\omega}{\mathrm dk}$ using the facts that $\omega = 2\pi \nu$ and $ k = \frac{2\pi}{\lambda}$ and the chain rule
$ \frac{\mathrm d\omega}{\mathrm dk} = \frac{\frac{\mathrm d\omega}{\mathrm dv}}{\frac{\mathrm dk}{\mathrm dv}}$
and get the answer
$ \frac{\mathrm d\omega}{\mathrm dk} = v$
but I don't understand the last part. what is the physical meaning of applying chain rule below.
$\large \frac{\mathrm d\omega}{\mathrm dk} = \frac{\frac{\mathrm d\omega}{\mathrm dv}}{\frac{\mathrm dk}{\mathrm dv}}$
I am confused because we are looking at the electron from one frame of reference and $v$ is constant. Constant means not changing. For example if we use chain rule by $ \nu_0$ we get something like this.
$\large \frac{\mathrm d\omega}{\mathrm dk} = \frac{\frac{\mathrm d\omega}{\mathrm d\nu_0}}{\frac{\mathrm dk}{\mathrm d\nu_0}}$
and we get different result
$\large \frac{\mathrm d\omega}{\mathrm dk} = \frac{c^2}{v}$
I wonder the difference between them.
