X-Nico

Vibrations of a circular membrane

The vibrations of an idealized circular drum head—essentially an elastic membrane of uniform thickness attached to a rigid circular frame—are solutions of the wave equation with zero boundary conditions.

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Going back to the equation for $R(r),$ its solution is a linear combination of Bessel functions $J\; m$ and $Y\; m.$

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Turning to the equation for $R(r),$ with the observation that $K=-\backslash lambda^2,$ all solutions of this second-order differential equation are a linear combination of Bessel functions of order 0,

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