I am trying to get a better grasp of the meaning behind Bessel Functions of the First Kind ($J_n(\beta)$) for applications to frequency modulation.
My understanding of it is that $J_n(\beta)$ is used to determine the amplitude of each frequency component resulting from frequency modulation, so we can easily see that the value of $J_n(\beta)$ contributes to a change in magnitude. In other words, given some function, the $J_n(\beta)$ will act as a scalar multiplier and change the amplitude of said function.
My question is then how does $J_n(\beta)$ affect the phase of the function? I can see that the value of $J_n(\beta)$ can be positive or negative, so does that mean the phase of the function jumps 180 degrees every time there is a sign change? This seems strange to me, as it would seem that means there could be many many jumps between 0 and 180 degrees phase, depending on the value of $n$ and $\beta$.
If anybody has some insight on this, it is greatly appreciated.