Bessel Differential Equation and Angle Modulation
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?


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: -