# Bessel Differential Equation and Angle Modulation

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?

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: -

• +1 for pretty pictures. Could stand to be more descriptive though :) – Daniel Jan 13 '16 at 21:24
• Thanks, the examples were already helpful. So the sum over the Bessel function represents the y-movement of an oscillating snare drum at point x for example. Can then anyone make clearer how the snare drum/phase modulation-analogy works? What exactly modulates the phase here? Why isnt it a pure sine? – Peet Into Jan 14 '16 at 22:12
• I followed the first part of the comment then you say the "the sum over" bit and lost me. The bessel graph represents a snap shot of the ripples if that helps? Looking for the analogy might be trying to find an analogy between right angle triangles and the way incoherent noise signals add - they both use pythagorous for working out the hypotenuse of course but there isn't a clear direct analogy. – Andy aka Jan 14 '16 at 22:26
• Ok, that was my fault. Then its pure coincidence. – Peet Into Jan 14 '16 at 23:57
• I'm not sure it's philosophically correct to say that either LOL. One universe, one big bang etc... – Andy aka Jan 15 '16 at 10:01