Problem Set 1. Due Friday December 6, 2013

• Text 1.1
• Text 1.3
• Text 1.4
• Text 1.5
• Text 1.7
• (For graduate students: Text 1.6)

• Text 1.9
• Text 1.11
  
• Text 1.12
• Text 1.13
• Text 1.23
• Text 1.25 a), b) d)
  

• Text 1.9 (if not turned in with set 2)
• Text  1.29
• Text  2.6

• Text Prob 3.1
• Text Prob 3.3
• Convert the strpulse.mws file (emailed) to one where the pulse reflects from the boundaries by flipping over at the LH end (x = 0) but not at the RH end.(x = L). It would also be OK to have the pulse remain upright at each end, not flipping over at all.
• Text prob 3.5
• Text Prob 4.1

• 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
• Show for a bar free at both ends for the lowest transverse vibration frequency, that the nodes occur at 0.225 L from either end of the bar of length L. (Start from Eq (12), p. 5  in the handout on flexural vibrations)

• Text Prob 7.1
• Text Prob 7.7
• Text Prob 7.9
• Text Prob 7.12

• A helmholtz resonator has volume of 0.0230 m³, and a neck area of 0.0025 m². Determine its neck length L if its resonant frequency is 140.0 Hz. 
• Determine the speed of sound in helium at 100^o C. (Helium is monatomic, so gamma = 5/3. Its molecular weight is 4.0 g/mol). 
• A pvc pipe of 3” diameter is 0.800 m long and open at both ends. a) Find the frequency of the lowest resonance in this pipe, using 0.800 for its length and not including ‘extra’ length at the ends. b) Find the lowest resonant frequency assuming an extra length at each end is 0.60 times the pipe radius.
• The pipe in part b) of  the previous problem is excited by an external loudspeaker at 20^o C. Take the acoustic velocity at each end of the pipe to be 0.350 m/s, and assume it is constant over the cross-sectional area. a) Calculate the ‘source strength’ Q at each end, in m³/s. b) Calculate the total radiated power from the ends of the pipe. (Ans:  P_{each end} = 0.58 mW)