Electronic – Bessel Differential Equation and Angle Modulation

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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

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: -

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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: -

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Or this: -

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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: -

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