Formula for phase velocity in dispersive medium

Question

We know that phase velocity of a wave, $v_p=\frac{\omega}{k}$.
For simple sinusoidal waves like - $y=a\sin(kx-\omega t)$.
I can easily understand and prove this by simply setting $ kx-\omega t=c$ ($c$ is some constant) and then differentiating yields that $ \frac{dx}{dt}=\frac{\omega}{k}$. And there are many other simple methods to check or prove this formula. But when we add two sinusoidal waves with different velocities and get a superimposed wave equation like, $y=a\sin(k_1x-\omega_1 t)\cos(k_2x-\omega_2t) $ or something like this, where $\frac{\omega_1}{k_1}\neq\frac{\omega_2}{k_2}$. Then what will be the phase velocity of this wave and why? I am not considering the simplified case like $k_1$~$k_2$ and $\omega_1$~$\omega_2$. What maths do we need to prove the phase velocity formula for this kind of complex waves? In a site I saw the formula for phase velocity for this wave is simply $\frac{\omega_1}{k_1}$, reasoning that phase velocity only depends on sine function terms. [currently unable to provide the link]. So how to prove or visualize this?

[I am very new in this topic and don't know is this question extremely trivial or not, so please pardon me if I have done some mistakes.]

Answer 1

The phase velocity is defined for plane waves (or other propagating eigenmodes, if one can define reasonably well their frequency and wave number). So, for an arbitrary wave shape, we can write $$ f(x,t)=\int \frac{dk}{2\pi}f_ke^{i(kx-\omega_kt)}, $$ and then define $$ v_{ph}=\frac{\omega_k}{k} \text{ (phase velocity)},\\ v_g=\frac{\partial\omega_k}{\partial k} \text{ (group velocity)} $$

See for more details this answer.

Answer 2

From your desmos graphical representation you might visually interpret the result of the product as a single propagating wave with either $\frac{\omega_{1}}{k_{1}}$ or $\frac{\omega_{2}}{k_{2}}$ phase velocity (my guess would be that you track the one with the highest spatial frequency) and interpret it as the "carrier wave"with some noise-like periodical amplitude fluctuations.Different combinations of $ k_{1}, k_{2},\omega_{1},\omega_{2}$ might actually lead to different visual interpretations, not necessarily based on the highest value of $k$. But this is just an interpretation of your initial expression as $A(x,t)\sin(k_{1}x-\omega_{1}t)$ where the "noise amplitude" $A(x,t)$ is $cos(k_{2}x-\omega_{2}t)$, and the interpretation the other way around with the $\cos$ term would obviously be possible as well.

On the other hand the maths tell you that your expression can be casted into the superposition of two harmonic propagating waves each with a different phase velocity. The phase velocity is only defined for a single harmonic wave component and there's no such thing as a single phase velocity for an arbitrary superposition of harmonic waves.

note: I don't see any link with wave propagation in a dispersive medium as your title implies