This laboratory experiment is designed to study the parameters that affect standing (stationary) waves in various strings. The effects of string tension and density on wavelength and frequency will be studied. Three experiments will be conducted to find (1) the frequency of the electric vibrator, (2) the density of Kevlar string and (3) an unknown tension on the Kevlar string.

A wave is the propagation of a disturbance through a medium. The physical properties of that medium (e.g., density and elasticity) will dictate how the wave travels within it. A wave may be described by its basic properties of amplitude, wavelength, frequency and period T. Figure 1 displays all of these properties. The amplitude, , is the heigh of a crest or the depth of a trough of that wave. The wavelength, , is the distance between successive crests or successive troughs. The time required for a wave to travel one wavelength is called the period, . The frequency, , is , and is defined as the number cycles (or crests) that pass a given point per unit time.

Figure 1. Properties of a standing wave.

Since the wave travels one wavelength in one period, the wave velocity is defined as . The wave velocity can then be written as

(1)

Relationships between these basic properties give the angular properties of the wave: and , where is the wave number and is the angular frequency. We can then write a one dimensional sinusoidal wave function for a wave traveling to the right as .

In this experiment, we will introduce an oscillating disturbance to a length of string with the use of an electric vibrator. The vibrator shakes the string back and forth, creating a disturbance perpendicular to the string's length. This disturbance, then, propagates along the string until it hits the stationary pulley about one meter away. This wave is known as a transverse wave since its disturbance is perpendicular its motion.

When the wave reaches the pulley-end of the string it is reflected back toward the vibrator-end of the string. In doing so, the disturbance is not only reflected back along the string, but it is also reflected over the axis of propagation. This is shown in the figure.

Figure 2. When a wave encounters a physical barrier such as the wall in this cartoon, it is reflected backwards and is inverted over the axis of propagation. (Reprinted with permission from Laney Mills.)

When a vibrating body produces waves along a tightly stretched string (click on the image to the right), the waves are reflected at the end of the string which cause two oppositely traveling waves to exist on the string at the same time. These two waves interfere with each other, creating both constructive and destructive interference in the vibrating string. If the two waves have identical amplitudes, wavelengths and velocities, a standing wave, or stationary wave, is created. The constructive and destructive interference patterns caused by the superposition of the two waves create points of minimum displacement called nodes, or nodal positions, and points of maximum displacement called antinodes. If we define the distance between two nodes (or between two antinodes) to be , then the wavelength of the standing wave is . Figure 3 illustrates the case where the length of string vibrates with 5 nodes and 4 antinodes.

Figure 3. A standing wave is created when an incident and reflected wave have identical amplitudes, wavelengths and velocities.

It is possible to obtain many discrete vibrational modes in a stretched string. That is, for a string to vibrate with a specific wavelength, the tension applied to the string must have a certain value. It is possible for the string to vibrate with another specific wavelength, but the tension must be adjusted until that particular mode is reached. If the tension is such that it is between vibrational modes, the string will not exhibit the standing wave phenomenon and we won't see a standing wave. When the frequency of the vibrating body is the same as that of the particular vibrational mode of the string, resonance is established.

The wave velocity of a standing wave is dependent on the medium through which the wave travels. The velocity of standing waves propagating through a taunt string, for instance, is dependent on the tension in the string, , and the linear density of the string, . For waves of small amplitude this velocity is given by

(2)

where the linear density, , is the mass of the string, , divided by its length, ,

(3)

Setting Equations 1 and 2 equal and solving for , we find the relationship between wavelength and tension, namely

(4)

1. Use the white household string and the standing wave apparatus to find the vibration frequency, , of the electric vibrator.

2. Use the standing wave apparatus to independently determine the linear density, , of the yellow Kevlar string. Use the accepted frequency of the driver in your calculations for this objective (f = 120 Hz). (N.B.: You will need to use the light-weight plastic cup as your shot container when using the Kevlar string.)

3. Replace the lead shot container with the unknown mass. Then use the standing wave apparatus along with the Kevlar string and the movable vibration stop to determine this unknown mass.

(Figure 1.) The pulley-end of the standing wave apparatus. (Figure 2.) The electric vibrator of the standing wave apparatus. (Figure 3.) The triple-beam balance with counter weights. The counter weights are used to measure masses over 610g. (Figure 4.) These containers are attached to strings shown in Figure 6. Filling the containers with lead shot (see Figure 5) creates the necessary tension in the string. One container is made of metal and the other is a lighter one made of plastic. The C-bolt (pictured) is used to attach the containers to the strings. (Figure 5.) The supply of lead shot. (Figure 6.) Two types of string will be used in this experiment: household string (white) and Kevlar string (yellow). (Figure 7.) The lead shot container is suspended from the string. Note the bucket is positioned so that if the container were to fall, lead shot will not spill all over the floor. (Figure 8.) The vibration stop. The stop's position may be adjusted along the length of the meter stick. (Figure 9.) A weight of unknown mass. (Figure 10.) An image of a standing wave on the household string. C-clamp. Additional meter stick.

[Click on images to enlarge.] 1 2 3 4 5 6 7 8 9 10

1. Caution!!! Wash hands after handling the lead shot.

2. Keep the white lead shot bucket under the hanging container to prevent lead shot from rolling all over the lab room floor in the event the container should unexpectedly fall.

3. It may be useful to use the unit of dynes rather than Newtons when measuring the tension force. Note that .

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Each lab group should download the Lab Report Template and fill in the relevant information as you perform the experiment. Each person in the group should print-out the Questions section and answer them individually. Since each lab group will turn in an electronic copy of the lab report, be sure to rename the lab report template file. The naming convention is as follows:

For example the group at lab table #5 working on the Ideal Gas Law experiment would rename their template file as "5 Gas Law.doc".

Each student should download the questions. Each person in the group should print out the questions and answer them individually. Discussing the questions as a group is acceptable, but each student is responsible for turning in answers that represent their own work, not the work of others.

These Nudge Questions are to be answered by your group and checked by your TA as you do the lab. They should be answered in your lab notebook.

General Nudges

1. Are standing waves produced for any given string tension?
2. How will you measure the wavelength of the vibrating string?
3. What is the uncertainty in your measurement of the wavelength of the vibrating string?
4. What steps did you take to decrease the error or uncertainty in the wavelength measurements?
5. How will you determine the tension in the string?
6. What is the uncertainty in the tension measurement?
7. What units will you use in this lab, m-k-s or c-g-s?

1. How will you determine the linear density, , of the household string?
2. Based on measurement uncertainties, what are the maximum and minimum values of ?
3. How many measurements will you record for this Objective? Is there a finite or an infinite number of measurements that can be made?
4. What parameters will you graph in order to measure the frequency of the electric vibrator?
5. Did your best-fit line fall within the data points' error bars?
6. From your graph, what is the nominal measured value for the vibrator frequency, ?
7. Using your findings in Nudge #2 above, what are the maximum and minimum values for ?
8. Knowing that electrical current in the United States oscillates at 60Hz, does your value for make sense?

1. What is the maximum number of antinodes that you were able to create with this setup?
2. Were you able to obtain one antinode?
3. Is there a relationship between the string tension and the number of antinodes created?
4. What parameters will you graph in order to measure the density of the Kevlar string?
5. Did your best-fit line fall within the data points' error bars?
6. What is your value for the density of the Kevlar string. (Use = 120Hz.)
7. How does your value of compare to the accepted value?

1. What is the distance between the vibrator and the vibrator stop necessary to produce a standing wave? What is the distance, , between nodes?
2. How can you reduce the uncertainty in ?
3. Are there other distances at which standing waves are formed? If so, is constant or does it depend on the position of the vibrator stop? Is this what you expect?
4. How did you determine the wavelength, , for this setup?
5. Using your calculated values for , and , explain how you can determine the value of the unknown mass without using a scale.
6. What is your calculated value of the unknown mass?
7. How does your calculated value of the mass compare to a measured value?

• Watch the students' time; this could be a long lab (especially since it is the first one) and the number of Objectives may need to be decreased.
• While it is possible for one antinode to be formed with the Kevlar string, it is beyond the strength of the household string to support the weight necessary to produce one node. Caution students to stop at 2 antinodes when using the household string, otherwise you will be picking up lead shot for quite a while!

As of now, there are no CUPOL experiments associated with this experiment.