# We can also nd the corresponding eigenvectors for the

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Unformatted text preview: r) cos 2θ. u(1, θ, t) = 0, 5.8.3. (For students with access to Mathematica) a. Run the following Mathematica program to sketch the Bessel function J0 (x): n=0; Plot[ BesselJ[n,x], {x,0,15}] b. From the graph it is clear that the ﬁrst root of the equation J0 (x) = 0 is near 2. Run the following Mathematica program to ﬁnd the ﬁrst root α0,1 of the Bessel function J0 (x): n=0; FindRoot[ BesselJ[n,x] == 0 ,{x,2}] Find the next two nonzero roots of the Bessel function J0 (x). c. Modify the programs to sketch the Bessel functions J1 (x), . . . , J5 (x), and determine the ﬁrst three nonzero roots of each of these Bessel functions. 5.8.4. Which has a lower fundamental frequency of vibration, a square drum or a circular drum of the same area? 5.9 Fourier analysis for the circular vibrating membrane* To ﬁnish up the solution to the initial value problem for arbitrary initial displacements of the vibrating membrane, we need to develop a theory of generalized 150 Fourier series which...
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## This document was uploaded on 01/12/2014.

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