Vibrating Drum. A vibrating circular membrane of unit
Chapter 10, Problem 21E(choose chapter or problem)
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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