Lab Exercise II
- Timo Budarz
Standing Waves on a String
The goal of this lab is to analyze standing waves on a string under tension. The string will be driven by a Pasco wave driver, which is just a speaker with a post attached to the speaker cone. You drive the speaker cone with a sinusoidal voltage with a chosen frequency. The post bobs up and down at whatever frequency you choose. The post is to be in light contact with the string such that the string is shaken at that same frequency. With a correctly chosen frequency you will see standing wave resonance modes on the string. Go back to the section on these modes to remind yourself of the physics.
Part 1 - Harmonic Frequencies of a String under Tension
By driving the string with the wave driver with a 1.0V input sinusoidal wave of a frequency you choose, change the frequency at first by one hertz at a time until an obvious standing wave is seen. If the frequency goes too high or too low, but is close to the resonant frequency the amplitude will be diminished. Tune the frequency to the nearest tenth of a hertz to excite a maximum amplitude. Find the frequencies of the first three harmonics in this way. Be sure to record the length of your string from support to support as well as the mass of your hanging weight.
Part 2 - Frequency versus Tension
By using three other hanging masses, find the fundamental frequency for each mass. Record the mass and frequency. Make sure you have the length of the string recorded.
Part 3 - Frequency Response Curve
Select one of the hanging masses you already used in the previous parts. You already know where the resonances occur, but I want you to plot a frequency response curve for the string. This is a plot of oscillation amplitude on the y-axis vs. driving frequency on the x-axis. Start at 10Hz and make a plot of amplitude of oscillation (half of peak to peak height) versus driving frequency. Please plot a single Hz at a time anywhere near the harmonic frequencies. Feel free to skip 5Hz at a time in the gaps between them where no appreciable amplitude arises. To connect data points so that this looks like a function you can highlight the data in the spreadsheet view, right-click and choose create polyline. It will "connect the dots" in the sequence they're entered in the spreadsheet.
For the data from part 1 above, plot the harmonic frequencies versus harmonic number to deduce the fundamental from the slope. Use an appropriate curve fit for these data points and display the equation on the plot with at least three significant digits. Using the length of the string, the hanging weight and the fundamental frequency, find the linear density of the string. For the data from part 2 above, for the fundamental frequencies versus hanging weight data, plot frequency versus tension for your four data points. Fit this data with the correct functional form according to the equations in the chapter. Display your fit equation on the chart with three significant digits.
1. What happens to the amplitude of the disturbance on the string as the frequency approaches the one of the harmonics? 2. What form of damping do you suppose exists in this system? 3. Based on your answer to number two, what should the answer to number one be for a string oscillating in vacuum? 4. Make a hand-drawn plot of amplitude versus frequency of driving force for your string while labeling important frequencies. Try to roughly represent the amplitudes you saw in lab, including the bandwidth of excitation. 5. How does the tension affect the resonant frequency on a string? 6. How does your answer explain why stringed instruments are tuned the way they are? 7. From the tension in the string and your measured frequencies of resonance, find the linear density of the string. It may help to look ahead to section 20.1 for details of the physics of wave propagation along a string. 8. How many Hz away from the first harmonic can you drive the system and still get half the peak amplitude of oscillation that occurs when you drive it exactly at its first harmonic frequency?