ChE132a: Analytical Methods of Chemical Engineering Professor Mike Gordon, Fall 2011 Homework #4: Due Wed. 16 Nov. 2011 (1) PGA p. 503 #1, 5, 6, 9 Explicitly write out an expression for the Fourier coefficients (2) PGA p. 503 #10, 14, 19 Use Mathematica to evaluate the coefficients (3) Use Mathematica to plot the temperature profile for problem #9 at t=0, 1000s, and 1 hr. Use 50 terms in your Fourier series. (4) PGA p. 503 #32 Comment on what the boundary conditions physically mean and describe in words what the temperature profile should look like as time progresses. What is the steady-state temperature of the bar? Use Mathematica for help. (5) [counts a lot] Let’s model the vibration of two different guitar strings. Solve the 1D wave equation with Dirichlet boundary conditions on the domain x=[0,1]. u a t u 2 2 2 2 ∇ = ∂ ∂ Assume that the initial string displacement (i.e., plucking the string) takes the form: 16 )0 , ( x x x u-= (a) What are the boundary and initial conditions? (b) What are the eigenvalues?
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