che132a-rec-8

# che132a-rec-8 - Chemical Engineering 132A 1 Analytical...

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Unformatted text preview: Chemical Engineering 132A 1 Analytical Methods of Chemical Engineering Professor Mike Gordon, Fall 2011 Mathematica Assignment #8: Fun with Bessel Functions! Commands for today Bessell[n,r] Jn(r), Bessel function of first kind, order n BessellZero[n,i]//N find the ith zero of l,,(r) numerically Animate[Plot[u[r,t],{r,0,1}],{t,0,50,0.1}] Animate a plot of u[r,t] wrt time over the domain r=[0,1] RevolutionPlot3D[f[r],{r,0,1}] Make a surface of revolution in 30 by rotatingf[r] around the z-axis ' (1) Find the first 10 zeros ofJo and 11. (2) Show that a Fourier-Bessel series approximation to f(r)= r2 can be constructed using Bessel functions of any order. Make two ZO-term Fourier-Bessel series for f(r) on the domain [0,1] using Jo or l; in the expansion. Remember that u" = i, where j; is the ith zero of J..(r). Make a table for the u” eigenvalues "0 which you will use to calculate the series coefficients. Use Nlntegrate with the AccuracyGoal->20 and MaxRecursion->50 options. Now make a table of these coefficients to use in the sum. Plot ﬁr) and both of the ZO-term series on the same graph. Comment on how well your series represent f(r). (3) Let’s analyze the vibrations of a circular drum. Solve the wave equation in cylindrical coordinates with wave veIOCity a :1 and drum radius of 1;, =1: Wu 2 2 — = a V u 8:2 2 Assume that the initial hit to the drum gives the following displacement: u(r,0) = l —(L) "o (a) What are the boundary conditions, initial conditions? (b) Create a function that calculatesthe rnth coefficient of the Fourier-Bessel series. (c) Make a table of the first 10 coefficients. (d) Construct a Fourier-Bessel series solution for u(r,t) with 10 terms using your table of Jo zeroes and‘ coefficients. - (e) Plot u(r,0) from r=[-1,1] to make sure your series solution gives the right initial condition. (f) Animate the u(r,t) plot for r=[-1,1V] from time=0 to 50, step 0.05. Slow down the solution so you can really see how the drum cross section moves in tirne. (g) Make a RevolutionPlot3D for u(r,0) with r=[0,1]. (h) Animate the RevolutionPlotSD for time=0 to 50, step 0.05. This takes a while, so be patient. ...
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