Vibrating Drum. A vibrating circular membrane of unit

Chapter 10, Problem 21E

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Vibrating Drum. A vibrating circular membrane of unit radius whose edges are held fixed in a plane and whose displacement \(u(r, t)\) depends only on the radial distance \(r\) from the center and on the time  \(t\) is governed by the initial-boundary value problem.

                    \(\frac{\partial^{2} u}{\partial t^{2}}=\alpha^{2}\left(\frac{\partial^{2} u}{\partial r^{2}}+\frac{1}{r} \frac{\partial u}{\partial r}\right)\),

                      \(0<r<1, \quad t>0\),

                    \(u(1, t)=0, \quad t>0\),

                    \(u(r, t) \text { remains finite as } r \rightarrow 0^{+}\) ,

                    \(u(r, 0)=f(r), \quad 0<r<1\)

                   \(\frac{\partial u}{\partial t}(r, 0)=g(r), \quad 0<r<1\)           

 

where \(f\) and \(g\) are the initial displacements and velocities, respectively. Use the method of separation of variables to derive a formal solution to the vibrating drum problem. [Hint: Show that there is a family of solutions of the form].

                    \(u_{n}(r, t)=\left[a_{n} \cos \left(k_{n} \alpha t\right)+b_{n} \sin \left(k_{n} \alpha t\right)\right] J_{0}\left(k_{n} r\right)\),

where \(J_{0}\) is the Bessel function of the first kind of order zero and \(0<k_{1}<k_{2}<\cdots<k_{n}<\cdots\) are the positive zeros of \(J_{0}\). Now use superposition.] See Figure 10.24.

         

                                     Figure 10.24 Mode shapes for vibrating drum.

                          (a) \(J_{0}(2.405 r)\),       (b) \(J_{0}(5.520 r)\),    (c) \(J_{0}(8.654 r)\)

Equation Transcription:

Text Transcription:

u(r,t)

r

t

partial^2u/partial t^2=alpha^2(partial^2u/partialr^2+1/r+partial u/partial r

0<r<1,  t>0

u(1,t)=0, t>0

u(r,t) remains finite as r rightarrow 0^+

u(r,0)=f(r), 0<r<1

Partial u/partial t(r,0)=g(r), 0<r<1

f

un(r,t)=[a_n cos (k_n alpha t)+b_n sin (k_n alpha t)] J_0(k_nr

J_0

0 <k_1<k_2<cdot<k_n<cdot

J_0

J_0(2.405r)

J_0(5.520r)

J_0(8.654r)

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