Problem Set 1. Due Friday December 4, 2009

• Text 1.1
• Text 1.4
• Text 1.5
• Text 1.9
• Text 1.11
• (For graduate students: Text 1.6)

• Text 1.13
• Text 1.21
• Text 1.22
• Text 1.25
• Text 1.29

• Text 2.2. Make an Excel animation for this wave.
• Text 2.6
• Text 2.9.  Make sketches of n = 2 and several m = 1,2,3,4.  Then argue for the value of each integral.
  
• Text 2.12
• Text 2.16
  
• (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 = 0.07 m and 0.11m and between y = 0.17 m and 0.23 m is given a velocity v =0.40 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 Prob 3.3
• Text Prob 3.7
• Convert the strpulse.mws file (emailed) to one where the pulse reflects from the boundaries without flipping over.

• 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 aluminum 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. The waves start at x = 8.0 m and t = 0. Single-stepping through it (at 1 ms/ per step) will let you observe the arrival of various frequencies, each of which travelled about 4 m. You should see the high-frequency, short-wavelength waves arrive first, and 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.)
• Text problem 5.3
• Text problem 5.5
  
• Text problem 5.6

• Text problem 6.2
  
• Text problem 6.3
• Text problem  6.4
• Text problem 6.6
• Use the emailed audacity files for the 0.500 m 1/2" diameter steel drill rod to pull up the lowest four resonant frequencies. This bar is free at both ends, supported by very light sewing thread. We will get an 'exact' bar velocity from class/lab, and it will be quite close to 5000 m/s.  Calculate the four lowest flexural bar frequencies from the theory and make a table with calculated and measured bar freaquencies. Two of the lowest ones will be 'even' modes and two will be 'odd'.  Download Audacity 1.3 beta if you don't already have it.