(Solved): Please Help Consider a cylindrical drum with a vibrating circular membrane. When the drum is struck ...

Consider a cylindrical drum with a vibrating circular membrane. When the drum is struck exactly at the center of the membrane, the oscillations will be described by a vertical displacement \( u(r, \theta, t) \) given by: \[ u(r, \theta, t)=\sum_{n=1}^{\infty} a_{m} J_{0}\left(k_{m} r\right) e^{i \omega_{m} t} . \] In this equation \( J_{0}(x) \) is the Bessel function of order \( 0, k_{m}=\alpha_{m} / R, R \) is the radius of the drum, and \( \alpha_{m} \) denote the zeros of \( J_{0} \) (that is, \( J_{0}\left(\alpha_{m}\right)=0 \) ). The first three zeros are: \[ \begin{array}{l} \alpha_{1} \approx 2.4048 \\ \alpha_{2} \approx 5.5201 \\ \alpha_{3} \approx 8.6537 \end{array} \] Calculate the frequency \( \nu_{m} \) (in \( 1 / \mathrm{sec} \) or Hertz) of the sound emitted by the drum for \( m=1,2 \), and 3 , assuming \( R=20 \mathrm{~cm} \) and a sound velocity \( c=125 \mathrm{~m} / \mathrm{s} \).