Now using the hook, the rubber bands (the medium-weight rubber bands are appropriate for this experiment), the paper clip and the ruler, as demonstrated on the video clip, tension the string with a single rubber band and pluck it to see how long it takes the pulse to travel down the string and back.

• Set the pendulum timer at a length such that a complete cycle of the pendulum corresponds approximately to the motion of the pulse down the string and back.

• Be sure you are using a complete cycle of the pendulum and not a half-cycle.

• Watching the timer and your pulses, adjust the length of the pendulum until its motion  is as synchronized as possible with the pulse, being certain to maintain the 1-rubber-band tension in the string.

• It is best to synchronize the pendulum with as many return pulses as you can detect.  For example, if you can see or feel four return pulses, you adjust the timer until it stays with all four pulses.

• Take the readings necessary to determine the length of the pendulum (and, hence later the time required for the motion of the pulse).
• Repeat the process, this time using 2 rubber bands.
• Repeat again for 3 rubber bands, then for 4.

You should have 1-2 meter lengths of each string or cord; if not you may cut a 1 meter sample from the end of each string or cord:

• Determine the mass and length of each of these string or cord samples.

For each string sketch a graph of L = pendulum length vs. T = string tension, and infer the proportionality between pulse velocity and string tension.

• Recall that the frequency of the pendulum, in complete cycles/second, is f = 2 `pi / `omega, where `omega = `sqrt(g / L) with g = acceleration of gravity and L = pendulum length.
• It follows that frequency is inversely proportional to the square root of pendulum length.
• This greater pendulum frequency imply greater pulse velocity or lesser pulse velocity?

• What then is the proportionality between pendulum frequency and pulse velocity?
• What must then be the proportionality between pulse velocity and string tension?

Using pendulum length and string length data, determine the velocity of a pulse in each string under each of the four tensions.

• From each pendulum length you can determine the period of the pendulum, and hence the time for a round trip of the pulse.

• You may, if you wish, use the formula T = 2 `pi `sqrt ( L / g ) for the period T of a pendulum.

• From this time and the length of the string you can easily determine the velocity of the pulse.

Sketch graphs of pulse velocity vs. tension and pulse velocity vs. mass density:

• Sketch on one set of coordinate axes a graph of v = pulse velocity vs. T = tension for each of the strings.
• Sketch a graph of v = pulse velocity vs. `mu = mass per unit length of string for a tension corresponding to 1 rubber band. On the same set of coordinate axes sketch the graphs for 2 rubber bands, for 3 rubber bands and for 4 rubber bands.

Linearize the graphs:

• For each string linearize the v vs. T graph using a power-function transformation on v.

• Determine the power p of v such that v^p vs. T forms the most nearly straight line thru the origin.   (Hint:  good starting values for p are + or - 1/2, 1 and 2.
• If v^p vs. T forms a straight line thru the origin, within experimental error, then we conclude that within experimental error v is proportional to T^(1/p).

• For each tension linearize the v vs. `mu graph using a power-function transformation on v.

• Follow the same guidelines as for v vs. T.

For each situation, pulse velocity should be v = `sqrt( T / `mu ), where T is the tension and `mu the mass per unit length.

• By hanging weights from your weight set from the rubber bands you used, determine the tension in Newtons for each number of rubber bands (the tension in the rubber band is equal to the weight of the suspended mass; figure out how to get the rubber bands to stretch to the same distance they did during the experiment).
• `mu will be the mass of the string sample divided by its length.

Determine the predicted velocity for each string and each tension.

• Be sure to use appropriate units for all quantities.

You will set up a fairly heavy cord between two supports, under a 2-rubber-band tension.  You will then introduce a periodic disturbance at one end (i.e., you will wiggle that end back and forth) at a slowly increasing rate and note the rates when the cord oscillates in a coherent and predictable fashion.  You will time the oscillations using the program HARMTIME.

• The name HARMTIME stands for 'harmonic timer', not for something sinister. If you haven't downloaded the program, click now and download it.
• When you run the program, you will be asked for the initial time between beeps and percent decrease in time between beeps, per minute of running time.
• It is suggested that you use an initial time of 1 second between beeps and a 50% decrease per minute.
• With experience you might decide to use different values in subsequent runs of the experiment.

First, fold the thickest cord so that its two ends are touching; then fold it once more so that this cord is quadrupled.

Now Enter to start the beeping, and begin smoothly wiggling the string back and forth in a direction perpendicular to the string.

• Displace the string only a centimeter or two in each direction as you move it back and forth.
• You might think 'right, left, right, left, ... ' as you move the string to the right and to the left; you should complete one right-left movement for each beep (e.g., you should be at 'right' every time the computer beeps).
• As the beeps get faster, you will complete your right-left cycles more and more quickly, and the string will respond to the frequency of your disturbance.
• Sometimes the string will feel like it's working against you, and sometimes it will resonate with your efforts.
• When the string resonates, it will form a smooth and relatively consistent pattern, with nodes and antinodes at relatively consistent positions.
• When the relatively consistent pattern emerges, note the number of antinodes (antinodes are the regions where the string swings back and forth with maximum amplitude), and note the time of a complete cycle as indicated on the computer screen.
• You should be able to observe the following patterns:

• The fundamental mode with a single antinode in the middle of the string,
• the first 'overtone' with two antinodes moving back in forth in opposite directions, and
• the second overtone with three antinodes, one in the middle and one between the middle and each end.

Use the procedure of Experiment 10 to determine the speed of a pulse in the string; carefully note the length of the string.

The resonant vibrations you have observed are the result of periodic waves traveling down the string and meeting the reflected wave in such a way that the two waves reinforce one another to form a coherent and predictable standing wave.

• The wavelength of the disturbance is four times the distance between a node and an antinode.
• The node-antinode configuration of the fundamental mode of vibration for the string is Node-Antinode-Node, consisting of two node-antinode distances over the length of the string.

• The first overtone consists of Node-Antinode-Node-Antinode-Node, or four node-antinode distances, corresponding to the length of the string.
• The second overtone pattern is Node-Antinode-Node-Antinode-Node-Antinode-Node, so that the string length is equal to 6 node-antinode distances.

How much uncertainty do you think there is in each the periods you observed for each resonant frequency (i.e., how close to the 'most resonant' frequency, or the middle of the resonant frequency range, were your results in each case)?

• What is the corresponding range of possible actual wave velocities corresponding to each resonant frequency (i.e., calculate the wave velocity for the lowest and for the highest possible value of each actual resonant frequency, based on the error of your measurement)?
• What do you think is the uncertainty in the pulse velocity you measured?
• Do the ranges of the pulse velocity and of the calculated wave velocities for the 2-rubber-band tension overlap?
• Do the ranges of the pulse velocity and of the calculated wave velocities for the 4-rubber-band tension overlap?

For the 2-rubber-band tension, what frequency would be required to obtain 4 antinodes?

• What frequency would be required to obtain 5 andinodes, and what frequency would be required to obtain 6 antinodes?
• What frequency would be required to obtain n antinodes?

Give a synopsis of how the frequency of a standing wave in a string under a constant tension is related to the number of antinodes.

The maximum and minimum frequencies of a buzzer being spun in a circle of known radius are observed using a computer program to generate changing frequencies.  The results are analyzed to validate the predicated Doppler shift for velocities which are low compared to wave velocity.

In this experiment you will use the sound of the buzzer on the video clip to verify the magnitude of the Doppler shift as the buzzer spins toward and away from you. The accuracy of your results will depend, among other things, on how well your ear can match the frequencies generated by your computer with those of the buzzer.

• Begin by playing the video clip for the experiment. Listen to the directions and make notes. Then find the part(s) of the clip where the buzzer is being spun toward and away from you.

Using the program DOPPLER, you will first match the frequency of the spin and that of the sound generated by the program, then you will match the frequencies of the highest and lowest sounds made by the buzzer. You can run the program in one window while playing the video clip in another so as to match the high and low sounds, as instructed below. You can manipulate the volume on your speakers to match the volume of the sound generated by the program.

• First match the frequency of the stationary buzzer with that of the computer program FREQ.

• The program asks you for a frequency and a duration.   Frequency can be any number between 40 and 10,000.  Duration is in seconds.   Enter a frequency and a duration then enter at the third prompt to hear the sound.
• Keep trying different frequencies until one sounds like it matches that of the buzzer.

• Next match the high frequency of the buzzer, as it approaches you, with that of the computer program FREQ.

• Repeat for the low frequency heard as the buzzer moves away from you.
• Using the program TIMER, determine the angular frequency with which the buzzer is spun, in cycles per second.

• Run the program DOPPLER.

• At the first prompt, you give the angular frequency you just determined, in cycles / second.
• At the third prompt, enter the high and low frequencies you observed (as instructed, type in the frequencies separated by commas then Enter)
• The simulation won't sound much like the buzzer, because the computer will give pure tones and the buzzer has a much richer tone.  So you should probably select to hear just the high and low tones, which are easier to match. .
• Listen to the program and the video clip simultaneously to see if they seem to match.   You might have to try a couple of times to synchronize the programs with the clip.
• Reset the frequencies to match the high and low frequencies as your hear them.

Analysis of Data

For each circle radius and period, you will determine the speed of the buzzer as it moves toward you and away from you. You will then compare the predicted Doppler shift with the observed shift.

• From the radius of the circle made by the buzzer and the period of the motion around the circle, determine the speed of the buzzer (i.e., divide the circumference of the circle by the time required to complete a circle).
• Assuming that the speed of sound is approximately 340 m/s, determine the ratio of the speed of the buzzer to that of sound.
•  Determine the high and low frequencies expected if the frequency increases and decreases by the ratio of buzzer speed to sound speed.
• Why is it reasonable to expect that at the speed of the buzzer, which is very low compared to that of sound, the observed frequency increase or decrease should be in the same ratio to the actual buzzer frequency as the speed of the buzzer to that of sound?

Determine the high and low frequencies expected from the actual Doppler shift f' = f / (1 - vBuzzer / vSound), where vBuzzer is positive if the buzzer is moving toward and negative if the buzzer is moving away from the observer.

• How close were your observations to the frequencies predicted by the Doppler shift equations?
• Why did we use the speed of the buzzer around the circle for the approaching speed and for the receding speed of the buzzer?

Both transverse and longitudinal standing waves are created when an aluminum rod is held at its center and struck in the longitudinal direction on one of its ends.  The frequencies of these waves can be matched to frequencies generated by a simple computer program.  The frequency of the longitudinal wave is compared to that predicted from the mass density and bulk modulus of aluminum.

This experiment depends on your being able to match the frequencies you hear from the video clip with those you hear from the program FREQ. 

• Some people have better ears than others, so your results will depend on the quality of your ear (or the ear of the person you coerce into listening and helping you match frequencies).

From the video clip, matching the pitch of the sound of the vibrating rod with the pitch generated by your computer using the program FREQ, determine the frequency of the transverse disturbance in the aluminum rod (the transverse disturbance is the one set up by striking the end of the rod in a direction perpendicular to the rod; the rod 'flaps' back and forth about the middle, like the wings of an airplane in turbulent air).

From the video clip, using a similar procedure, determine the frequency of the longitudinal disturbance in the aluminum rod (the longitudinal disturbance is the one set up by striking the end of the rod on the floor, and has a much higher pitch than the transverse disturbance).

Assuming that the frequency you determined in each case is the fundamental frequency, corresponding to a node at the middle where I was holding the rod and to antinodes at the ends, which are free, determine the velocity of the transverse pulse and the velocity of the longitudinal pulse in the aluminum rod.

How much uncertainty do you think there was in your determination of frequencies?

Three of four tubular chimes cut from standard steel conduit pipe are tuned to a familiar tune; the fourth emits a frequency which makes an interval of a tritone with the lowest-frequency chime.  These frequencies are identified and observed using a computer program that emits user-specified frequencies.   Frequency ratios are analyzed and compared with the length ratios of the chimes.

As with the last experiment, the accuracy of your results on this one depends to an extent on the quality of your ear.

You have received four tubular chimes.

On three of the chimes you can play the first three notes of The Star Spangled Banner ('Oh-O-Say'). If you use the fourth the sound after the lowest it will probably sound out of tune. Determine which three give you the tune.

• Measure the lengths of these three chimes, as well as that of the fourth.
• Using the program FREQ, determine the frequency of each chime.
• Determine the ratio of the frequency of each chime to that of the longest.
• Sketch a graph of chime frequency vs. length.
• Determine the ratio of the length of each chime to that of the longest.

2^(1/12) is approximately 1.059, accurate four significant figures. Using your calculator determine the value of 2 ^ (1/12) to five-significant-figure accuracy, and use this result in all subsequent calculations (even if the instructions use the four-figure version).

• Each of your frequency ratios should be close to an integer power of 2 ^ (1/12) -- e.g., for four significant figures, to 1.059^2, 1.059^3, 1.059^4, etc..

• For each frequency ratio you measured, determine this integer power.

• If you have a keyboard, you could match the pitch of two keys with the FREQ program and match the ratio of the pitches with a power of 2 ^ (1/12).

• The power would be equal to the number of 1-key intervals (having both black and white keys) between the two keys you selected.

Determine whether the frequency vs. length data is best linearized by squaring the lengths, by taking the square root of the lengths, by taking the inverse square of the lengths, or by taking the inverse square root of the lengths.

• Your conclusion will be that frequency is proportional to some power of the length: f = k * L^p, where the power p is the linearizing power.
• k will be the slope of the graph of f vs. L^p.

• Sketch a good graph of f vs. L^p, and determine the value of k.

If you wish to make your own set of chimes, which you might well do (though it is not required, if you have time and inclination you are encouraged to do so--all you need is a few pieces of electrical conduit at a couple of dollars for a 10 foot length and a hacksaw), you could use the f = k L^p function to determine how long to cut the chimes.

You don't have to make the chimes, but you do have to answer the following:

• If you were indeed going to make the chimes, and you wished to match the frequency ratios 1.059^2, 1.059^3, 1.059^4, ..., 1.059^12, then what lengths would you need if your shortest piece was 20 cm long?

• (Hint: If you wanted to match 1.0594^4, you would need f2 / f1 = 1.0594^4. In that case what would the length ratio L2 / L1 be?
• The algebra is based on the fact that f2 / f1 = [ k L2 ^ p ] / [ k L1 ^ p ] = ( L2 / L1 ) ^ p.)