Standing Waves and Resonance on an Elastic Cord


Standing Waves

Standing waves, sometimes called stationary waves, are waves that remain in a constant position in space. They are commonly formed when waves traveling in opposite directions meet and interfere with one another. Points in the medium that appear to be standing still are called nodes, while points that undergo maximum displacement are called antinodes.

Standing waves can be formed in a variety of media:

…as sound waves in a column of air, as in an organ pipe

…on the surface of a drum, as complex two-dimensional vibrations

…as seismic waves, in which portions of the earth are oscillating

…as light waves produced by a laser

…as mechanical waves in a vibrating string, in which the traveling wave is reflected at the far end of the string

In this experiment you will use a littleBits servo to cause a length of elastic cord to exhibit standing waves. When the servo is vibrating at the natural frequencies of the stretched elastic cord, you will observe standing wave oscillations with a relatively large amplitude. This phenomenon is known as resonance.

There are two m4v movies attached to this lesson showing standing waves on an elastic cord.

Objectives:

Students will be able to:

– Investigate standing waves using littleBits

– Use a basic law of physics relating wave speed, frequency, and wavelength to determine the speed of the wave in the elastic cord.

Assessment:

Use the Standing Wave Data Table (a PDF attached to this lesson) to assess students’ ability to accurately collect data and complete the experiment. Consider adding your own questions to the sheet to further assess student understanding of targeted concepts.

Standards:

NGSS Disciplinary Core Ideas

– PS3: Energy

– PS4: Waves and their applications in technologies for information transfer

NGSS Science and Engineering Practices

– SEP3. Planning and carrying out investigations

SEP4. Analyzing and interpreting data

Duration: 1 hour class period

GRADE LEVEL
High School (ages 14-17)
College/University (age 18+)

SUBJECT
Technology
Science

MODULES & ACCESSORIES USED (7)
slide dimmer (1)
led (1)
power (1)
servo (1)
Arduino (1)
number (1)
battery + cable (1)

LESSON GUIDE

STEP 1 : Introduce the activity

If students are not familiar with standing waves, you may need to provide some background information before students can successfully complete the activity.

Use a jump rope or other cord to make a standing wave. Introduce or review important vocabulary including, nodes, antinodes, wavelength, wave velocity, frequency, and resonance.

Explain that students will investigate standing waves using littleBits. As they create standing waves with an increasing number of antinodes, they will look for patterns and relationships between frequency, wavelength, and wave velocity.

Consider having students learn more about standing waves from online sources like the Physics Classroom.

You may also choose to show the videos included with the activity to help illustrate the concepts.

STEP 2 : Set-up the circuit

Standing Waves1

The picture above shows the circuit student groups will build to investigate standing waves. The power module is connected to a slide dimmer that is attached to the Arduino a0 input. The servo is connected to the d5 output of the Arduino, and the servo switch is set to turn (NOT swing). Optionally, students can also connect an LED to the d5 output to see flashing in sync with the servo’s vibration. A number module is connected to the d9 output of the Arduino. The number module will display the rate of vibration of the servo in cycles per second (also known as a Hertz). The slide dimmer will allow you to vary the cycles per second from 1 to 25 Hertz.

The white elastic cord, about 1 mm in diameter, will not fit into the holes of the servo’s arm. Students will have to insert a wire that does fit into the end hole of the servo arm and twist tightly. The elastic cord, about 2 to 3 meters long, can then be looped through the wire and knotted to hold it tightly to the servo. You can purchase the round elastic cord from most fabric and sewing supply stores.

One person in each lab group can hold the servo motor, while a second person holds the free end of the 4 meter elastic cord.

STEP 3 : Ready the Arduino module

Note: The Arduino module’s micro-USB cable must be connected to a computer with the Arduino IDE software installed. THE ARDUINO SWITCHES SHOULD BOTH BE SET TO ANALOG.

Next, students power up the Arduino module and start the Arduino IDE software on the connected computer. Students should select Tools>Board>Arduino Leonardo. Then students select Tools>Serial Port and select the serial port that the Arduino will use for communication. The name of the port depends on the computer’s operating system. On a Mac, it will start with /dev/tty.usbmodem… and a PC will start with COMM… In Windows, you can look for the USB serial device in the ports section of the Windows Device Manager. If you are on a Linux machine, the port will look like /dev/ttyUSB…

Students open the sketch file called StandingWaves.ino and upload the sketch to the Arduino module. They will see the yellow rx/tx LEDs blink on the Arduino module while the sketch is uploaded. After the blinking stops, the micro-USB cable can be removed from the Arduino. Have students check that the sketch is working correctly by first attaching the LED only to the d5 pin and the numbers module to the d9 pin. Using the slide dimmer, they should be able to see the LED blinking from a low of 1 Hertz (cycle per second) to a maximum of 25 Hertz, and the numbers module should display the Hertz value.

Once they have verified that the sketch is running properly, they can connect the servo module to the d5 pin on the Arduino.

STEP 4 : Collect data

Standing Waves2

As mentioned earlier, one member of each lab group can hold the servo motor (with the elastic cord attached) while another lab partner holds the other end of the elastic cord. The cord should be stretched only a very small amount, just so that there is no slack in the elastic cord. Students first set the “cycles per second” rate to 3 Hertz, and then adjust the stretch of the elastic cord until it continually displays the fundamental mode of vibration, with nodes at each end of the cord, and an antinode at the center of the cord, as in the top figure (purple waves) in the main image for this lesson. Have students record the frequency in Hertz in the appropriate number of antinodes row of the data table shown above. (A PDF is included for class duplication of the table.) Also, have students record the length of the stretched elastic cord in the space provided at the top of the data table.

Without changing the amount that the cord is stretched, students now increase the cycles per second until they see the cord vibrating with the first overtone, as shown in the middle (the blue waves) of the main image for this lesson and then add new data to their table.

Students repeat this process for as many overtones as they can, recording data in the table. Make sure that students keep the length of the stretched elastic cord constant. The servo is a bit quirky in its behavior at speeds much greater than about 15 hertz. Also, note that the servo motor does get somewhat warm when running, so remind student to give it a “cooling off” time periodically.

STEP 5 : Analyze the data

Standing Waves3

The wavelength is the length of one complete up-and-down cycle, as shown in the diagram below.

If we let the length of the cord be L, we see that for the fundamental frequency, L is half the wavelength. For the first overtone, L is equal to the wavelength. For the second overtone, L is 3λ/2. In general, L = nλ/2, or λ = 2L/n. With students, calculate and record the values for the wavelengths in meters of each of the standing waves they created in the third column of the table.

Now students are ready to determine the velocity of the waves in the elastic cord. Students will make use of a well-known equation in physics that relates velocity, frequency, and wavelength: v = fλ. The Greek letter lambda (λ) is commonly used to represent the wavelength of the wave. Finally, have students compute the velocity of the waves in meters/second and record their calculations in the fourth column of the table.

The velocity should be the same regardless of the value of n, within the limits of error in this experiment!

With students, discuss, the the sources of error in this experiment.