Problem Set 1. Due December 5, 2003
1. Survey chapter 1.
a) Write down one major advance in acoustics in
each of the centuries from the 17th through the 20th.
b) Come up (maybe from the web) with some more detail
on one of the early advances in acoustics (Mersenne, Chladni, etc.)
2. If P1= A1 exp(i omega t), P2 = A2
exp (i omega t), and P3 = A3 exp (i omega t) and
P1 + P2 = P3, and P1-P2
= 2 P3, find the phase difference between P1 and P2 and the ratio
of amplitudes |P1/P2|.
3. Meeks 1.7
4. Meeks 1.8.
Problem Set 2. Due December 12, 2003
1. Raichel, prob 4.4, p. 86.
2. A 350-kg machine operates at a frequency of 628 rad/sec (100 Hz). Find the required spring constant k so that only 0.5% of the machine’s vibration will be transmitted to the floor. First, use the transmissibility spreadsheet from the acoustics web page to figure out the frequency of the mass-spring system. Then determine the damping constant so that the maximum transmissibility will be 5.0. [ Fig 18.8 should tell you that the mass-spring frequency should be less than 10% of 628 rad/sec.] You can use the transmissibility spreadsheet (from the acoustics web page) to work this out.
3. Download the maple worksheet on the travelling pulse from the acoustics web page. Then modify it so that the pulse is not travelling initially, but standing still. Then re-work the calculations and animation. You should see the pulse break into two which travel in opposite directions, like Raichel shows on p. 72. Email me the spreadsheet. Call it nnn staticpulse, where nnn is your name.
4. Download the fit program for the driven harmonic oscillator. Adjust the parameters to give the best fit to the data, using the sliders. When you are done, calculate the Q of this driven damped oscillator.
5. Create a spreadsheet which shows a travelling wave. The instructions are under the web page for physics 3, spring 2003. Then add in a wave travelling in the opposite direction, so that you have a graph of a standing wave (the sum of the wave travelling to the right and the one travelling to the left).
Problem Set 3. Due December 19, 2003
1. Raichel, prob 6.4, p. 128.
2. A rectangular membrane fixed on all four sides has a length of 0.55 m and a width of 0.45 m. Its lowest resonant frequency is 175 Hz, and the membrane surface density is 0.0042 kg/m^2. Find the 'tension' (in N/m) of the membrane.
3. For the membrane of the previous problem vibrating in its lowest mode, and having a displacement in the center of 0.005 m, calculate the average displacement of the membrane (requires integrating over the membrane).
4. An aluminum rod of length 0.70 m has a compressive load of 50,000
N applied to each end.
a) By how much is the rod compressed? [Needs data on aluminum.]
b) If the rod radius is 4,0 mm with no compression, what will its radius
be under the 5E4 N compressive load?
Problem Set 4. Due January 9, 2004
1. (7) From your data on the 1/2" aluminum bar (L = 0.523 m),
a) measure the lowest four resonant frequencies, and calculate the lowest
four resonant frequencies.
b) indicate what you used for the longitudinal velocity of sound in
aluminum, for your calculations.
c) put your results in a table.
d) examine the table and see if you might have used a longitudinal
velocity a little too high or a little too low.
2. (7) Ditto for the 1/2 brass bar (L = 0.543 m).
3. (5) A marimba is a musical instrument using plastic or wooden bars whose width is constant, but whose thickness varies, being thinnest in the middle and thickest at the ends.
a) Would you expect the velocity of transverse waves in the bar to be
greater in the middle than at the ends, or smaller in the middle than at
the ends. Give a specific semi-quantitative reason for your answer.
b) Would you expect the lowest resonant frequency of a marimba bar
to be higher or lower than for a bar whose cross-sectional area is constant,
having the same area as at the end of the actual marimba bar? Give a reason
based on one or more of the equations we have developed.
4. (5) A solid bar of oak (r = 720 kg/m3,
c=4000 m/s) is free at both ends. It is 0.040 m wide, 0.300 m long, and
0.0080 m thick.
a) Calculate the lowest two longitudinal frequencies of this bar, free
at both ends.
b) If you wanted to tune the lowest frequency of this free-free bar
to the musical note F4 (349.23 Hz, p. 478), how might you do
it?
Problem Set 5 for January 23, 2004
1. Show that by multiplying matrices you get Raichel's equation (8.27),
p. 158. The first matrix is at x=0, so f=1.
The second matrix is at x=L, so f = exp(ikL), and f^2 = exp(2ikL).
Remember that T = 1/|M_11|^2.
You also need to realize that something like 1- exp(-2ikL)
simplifies to
exp(-ikL) [exp(ikL) - exp(-ikL)] =
2i exp(-ikL) sin(kL)
2. Run the filter spreadsheet acoustic filter f.exe for Raichel's Fig 8.7, p. 160. When you have a curve close to what he has, then let the side branch length be 10 cm long. Report your results for this filter from 0 to 2400 Hz. [ Include your graph on the hw sheet.]
3. Use a helmholtz resonator as a sidebranch in the filter spreadsheet,
using Raichel's data for his graph on p. 162. His graph shows a min at
250 Hz,
but I think you will find the resonant frequency is more like 570 Hz.
[The filter program thinks it is dividing by zero at zero frequency, so
don't worry
about that point.]. You should get a graph like Raichel's, except the
zero is at around 570 Hz. Include it on your homework.
4 a) Raichel 9.1 b) Raichel 9.2 { Read the text to see how this goes. I think you'll find it quite straightforward.}