I'm trying to undestand the role of the Bessel differential equation in calculating the side bands of a phase modulated signal. I started out with the
textbook example $$u_{PM} = cos(w_C \cdot t + \Delta \phi \cdot sin(w_m \cdot t)).$$ With trigonometric identities and the equation \$e^{j \Delta \phi \cdot sin(x)} = \sum_{n=-\infty}^{\infty} J_n(\Delta \phi) \cdot e^{jnx}\$, with J being the Bessel function, i came to the identical expression $$u_{PM} = \sum_{n=-\infty}^{\infty} J_n(\Delta \phi) \cdot cos(w_C \cdot t + n \cdot w_n \cdot t).$$ I read, that
the Bessel differential equation describes for example the oscillations on a circular membrane.
My question is, why does the Bessel function appear here? What is the meaning of the Bessel differential equation in this context?
Best Answer
The bessel functions expressed graphically: -
Now think of the Y axis (x = 0) as a line passing up and down through the centre of the drum membrane. Then. If you hit a circular drum membrane in the centre it might do this: -
Or this: -
The 2nd scenario is what would be expected from a bass drum - it's resonating in the lowest frequency mode possible. The 1st scenario is when it's resonating at the next highest mode.
Look familiar? What about circular waveguides and the internal E fields: -