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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 zaxis ' (1) Find the first 10 zeros ofJo and 11. (2) Show that a FourierBessel series approximation to f(r)= r2 can be constructed using Bessel functions
of any order. Make two ZOterm FourierBessel 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 ZOterm 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 FourierBessel series. (c) Make a table of the first 10 coefficients. (d) Construct a FourierBessel 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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 Fall '08
 Gordon,M

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