SC-1  Transverse Standing Waves on a String

OBJECTIVE: To study propagation of transverse waves in a stretched string.

INTRODUCTION:

A standing wave in a string stretched between two points is equivalent to superposing two traveling waves on the string of equal frequency and amplitude, but opposite directions. The distance between nodes (points of minimum motion) is one half wavelength, ($\lambda$/2).

The wave velocity, $v$, for a stretched string is $v = \sqrt{ F/ \mu}$ where F = tension in the string and $\mu$ = mass per unit length. But v = f$\lambda$ and hence

$$f = \frac{\sqrt{F/ \mu }}{\lambda }  .$$ (1)

Figure 1: The Modes of a String

Figure 2: A close-up

PART A: Waves from a mechanical driver (i.e. a speaker)

APPARATUS:

Basic equipment: Electrically driven speaker; pulley & table clamp assembly; weight holder & selection of slotted masses; black Dacron string; electronic balance; stroboscope.

Computer equipment: Personal computer; PASCO $^{\mbox \copyright}$ interface module; power amplifier module; various electrical connectors.

The set-up consists of an electrically driven speaker which sets up a standing wave in a string stretched between the speaker driver stem and a pulley. Hanging weights on the end of the string past the pulley provides the tension.

The computer is configured to generate a digitally synthesized sine wave (in volts versus time) with adjustable frequency and amplitude (max: $\sim$10 V).
PASCO interface: This transforms the digital signal into a smooth analog signal for input into the power amplifier.

Power amplifier: The amplifier transforms the voltage sine wave single into a current suitable to drive the loudspeaker. (A few exotic speakers, often referred to as electrostatic speakers, actually utilize high voltages directly to produce sound.)

Precautions: Decrease the amplitude of the signal if the speaker makes a rattling sound, or if the red pilot light on the amplifier is lit. The generator is set to produce sine waves; do not change the waveform.

Note: Although the speaker is intended to excite string vibrations only in a plane, the resultant motion often includes a rotation of this plane. This arises from non-linear effects since the string tension cannot remain constant under the finite amplitude of displacement. [See Elliot, Am. J Phys. 50, 1148, (1982)]. Other oscillatory effects arise from coupling to resonant vibrations of the string between pulley and the weight holder; hence keep this length short.

Figure 3: The apparatus

SUGGESTED EXPERIMENTS:
PROCEDURE I: Checking Equation (1)

1. Place the sheet of paper provided on the table; this will make it easier to see the vibration of the string. Measure accurately the distance, $L$, between the bridge and the pin of the speaker using the two meter ruler; record this in your lab notebook. Click on the LAUNCH EXPERIMENT icon (i.e., the telescope), from the on-line lab manual. The computer monitor will appear as shown in Fig. 4.

2. You will see that the computer is set to produce a 60 $Hz$ sine wave with an amplitude of 3 $V$. To start the string vibrating CLICK the ``ON'' button.

3. CLICK on the up/down arrow in order to change the amplitude or the frequency of the signal although this produces rather large steps.
   In order to produce other step increments see the table below or simply double click right on the display number itself to use numeric input.

   Figure 4: The PASCO display

   

   
   

   Table 10: Controlling the Signal Generator    

   $$f$$ Keyboard Amp. Change

   Shift key 100 Hz about 2.00 V

   No key 10 Hz about 1.00 V

   Ctrl key 1 Hz about 0.50 V

   Alt key 0.10 Hz about 0.10 V

   Alt + Ctrl 0.01 Hz about 0.02 V     keys

   
   

4. At 60 $Hz$ check eqn. 1 by first calculating the necessary string tension to produce a standing wave in the third or forth mode. Weigh the string to get $\mu$. Your instructor will provide you with a one meter length of string. (Dacron 30# has $\sim$ 0.283 g/m.) Note that the hanger itself has a 50 $g$ mass so it may not be easy to access the forth mode (depending on $L$).

   Check your results by adjusting the string tension by increasing/decreasing the weight to find the tension which results in the largest amplitude vibrations. How do the two values (calculated and measured) compare?

5. Now put a 200 $g$ mass on the mass hanger and restart the signal generator. Record the total mass and tension in your lab book.

6. Adjust the frequency so that the amplitude of the oscillation is at its maximum by changing the frequency in 1 $Hz$ steps. This is best done as follows: First decrease the frequency until the amplitude of the string is very small.
   Then increase the frequency in 1 $Hz$ steps, observe that the amplitude first increases and then decreases. Record the best frequency $f_2$ in your table.

7. Change the frequency to observe the third mode. Find and record the best frequency (using 10 $Hz$ steps at first may be faster).

8. Find and record the frequency of the higher modes.

9. OPTIONAL: Check the frequency $f$ of the string in its 2nd mode with the stroboscope. Note that the stroboscope is calibrated in RPM or cycles per minute, NOT $Hz$ (cycles per second). You should find a value close to 70 $Hz$.

ANALYSIS:

1. Divide the various frequencies $f_n$ by $n$ and enter the values in a table. Calculate the average value of $f_n/n$; this is the expected value of the frequency of the first mode.

2. Calculate the velocity of propagation on the string using the appropriate equation.

3. Calculate the mass per unit length of the string. How do the two values for the string mass per unit length compare?

PROCEDURE II: $f_n$ vs string tension

In this section you will investigate the dependence of the resonant frequency of a string as a function of the applied tension.

1.

Choose six masses between 100 gm and 1 kg and enter the values in the data table.

2.

Determine the resonant frequency of the second mode of the string under these different tensions and record your results. (Hint: increasing the mass by a factor of two increases $f_n$ by nominally a factor of $\sqrt{2}$.)

3.

Plot a graph of frequency versus mass, $m$, and include the zero value.

4.

Plot a graph of frequency versus $\sqrt{m}$ and again include the zero value.

QUESTIONS:

1. Which of the two graphs can be fitted with a straight line? A parabola? Why?

2. From the slope of the graph having the linear relationship obtain the mass per unit length of the string and compare to your previous result.

PART B: ``Virtual'' waves on a drum head

PROCEDURE III: (If time permits)

Vibrations of a circular drum head In this section you will examine, via a virtual demonstration, the vibrational modes of a two dimensional drum head.

The (0,1) Mode	1. Click on the icon at left to down-load and initiate the MPEG movie plug-in to observe the ``first'' mode. 2. Use the replay and step frame functions to view the motion. 3. Where is the displacement at a maximum? Always at a minimum?
The (0,2) Mode	1. Click on the icon at left to download and initiate the MPEG movie plug-in to observe the first of the two ``second'' modes. 2. Use the replay and step frame functions to view the motion. 3. Where is the displacement at a maximum? Always at a minimum?
The (1,1) Mode	1. Click on the icon at left to download and initiate the MPEG movie viewer to observe the second of the two ``second'' modes. 2. Use the replay and step frame functions to view the motion. 3. Where is the displacement at a maximum? Always at a minimum? All mpegs created by Prof. Dan Russel, Kettering University