Problem Set 1. Due Friday November 30, 2007

• Text 1.2
• Text 1.3
• Text 1.5
  
• Text 1.9
• Text 1.10
• (For graduate students: Text 1.6)

• Text 1.11
• Text 1.13
  
• Text 1.21
• Text 1.23
• Text 1.25
• Text 1.29

• Make an animation for a wave like in text 2.2, with y = R^2 - (x-ct)^2, but not negative.
• Text 2.5
  
• Text 2.7
• Text 2.16
• Text 2.17
• For example 2.3, find coefficients A[1] through A[9]. Don't turn in more than 1 page of output.
  
• (For graduate students: Text 2.15. At very low frequencies, this mass will simply swing back and forth like a pendulum, so your equation for omega ought (at low frequencies) to give the frequency of a pendulum.)

• A rectangular membrane has length Ly = 0.40 m and width Lx = 0.20 m. Its lowest resonant frequency is 126 Hz. This membrane is given a sharp blow so that a small region between x = 0.08 m and 0.10m and between y = 0.18 m and 0.22 m is given a velocity v =0.38 m/s while the rest of the membrane is at rest. Find the coefficients of the 6 lowest-frequency modes of this membrane. This requires an analysis like was done in Section 2.5, pp. 47-49. You will have to carry out integrals in Maple over both x and y.
• Text problem 3.1
• Convert the strpulse.mws file (emailed) to one where the pulse reflects from the boundaries without flipping over.
• The fitting of data  from the speaker (you did this mostly Tuesday Dec 11). Email the spreadsheet.
  

• Work out the odd-symmetry solutions for the flexural vibrations of a rod or bar free at both ends
• Write down the odd-symmetry wave functions
• Express the y'' = 0 boundary condition at =/- L/2, and that for y''' = 0.
  
• From these two (y'' = 0, y''' = 0) write down the equation in kL/2 implied by satisfying free-end boundary conditions
• Find the lowest two odd-symmetry frequencies for a 12.7-mm diameter aluninum bar 0.8 m long.
  
• Use the maple animation of a pulse at x = 8 m at t=0 on a 16-m long wire to observe the waves reaching  x = 12  m.  The pulse is made up of 500 fourier components, each frequency of which depends on k^2. Single-step through the animation and observe the high-frequency components arriving first, lower frequencies later. Record the measurements you make, and verify that the velocity of the arriving waves is the 'group velocity' Explain what you measured, and show a sample calculation. (Remember k = 2 Pi / wavelength.).
• Give your work and answers to the three question on the Stress and strain, bar vibrations handout. (The answers are hiding under the boxes, but don't peek till you work them out.)
• Write out the wave function for the lowest-frequency even solution of the clamped-clamped pipe. Plot this in Maple and you should get the figure in the handout. Then do the lowest-frequency odd wave function and plot it in Maple.  Email in the Maple.
  
• Text problem 5.9
• Show the radius of gyration for a pipe of inner radius r1 and outer radius r2 is equal to sqrt(r1^2 +r2^2)/2.Hint: see p. 2 of the flexural waves handout. [If we let r1->0, we have a solid pipe, and kappa must reduce to r2/2, which it does.]
  

• a) Show that Eq. 15.6 follows from the previous equation, as indicated in the text. (work out the details)
  
• b) A highway noise barrier of 6x6 (0.14 m x 0.14 m) wood members is erected. Take the density of the wood (use oak if you haven't already done the problem), and treat this barrier as a partition as was done in the beginning of sect 15.4. Calculate the transmission coefficient at 50 Hz and again at 500 Hz.
• Text problem 7.1. Also calculate the maximum particle displacement.
• Text problem 7.7
• Text problem 7.25 [See Sections 7.5 and 7.7 in the text. I arrive at his answer for part a) by getting the magnitude of the signal then finding a minimum in the magnitude. But for part b) I get a ratio of 0.88 ]

• Write a report for each of the four experiments you did in week 7. Something like 2 pages each.