While I was studying beats, I tried to find a displacement function of any particle in the most generalized form. I ended up with $$y=2A\sin(\pi(t-x/v)(f_1+f_2))\cos(\pi(t-x/v)(f_1-f_2)).$$
Now, given such a function wherein you cannot compare to a general equation, how do you find the wave speed and the phase speed? And, I think, some clarification on how "group velocity" applies to such a function or phenomenon would be appreciated .
PS: Can we first break down this function using Fourier analysis and then maybe simplify things a bit?