Materials Included In Kit

Additional Materials Required

Prelab Preparation

Cut the nylon string into 10-cm pieces, one for each group.

Safety Precautions

Take care, and do not suddenly release a stretched Slinky. The spring may snap back rapidly, which may cause personal injury or damage to the Slinky. Wear safety glasses. Do not extend the Slinky more than 4 meters. Please follow all laboratory safety guidelines.

Lab Hints

• This laboratory activity can be completed in two 50-minute class periods. It is important to allow time between the Introductory Activity and the Guided-Inquiry Activity for students to discuss and design the guided-inquiry procedures. Also, all student-designed procedures must be approved for safety before students are allowed to implement them in the lab. Prelab Questions may be completed before lab begins the first day.
• This activity requires a great deal of space. Each group must be able to stretch the Slinky to a maximum of 4 meters and move it side to side for a total of at least 40 cm. Some space may be conserved by placing the spring on the floor and tying one end of the spring securely to the leg of a sturdy table. Tie more than one coil together to prevent the end coil from bending. A ball of string is included for this purpose.
• If the activity is conducted on a tile floor, students can use the tile boundaries to ensure a consistent amplitude for each wave pulse.

Teacher Tips

• The Guided-Inquiry Design and Procedure discussion questions were developed to help students reach the conclusion that amplitude does not affect wave speed and therefore does not need to be one of the variables tested. Some students may desire to test amplitude to verify this conclusion.

• Students may discover that changing the tension in the spring affects the wave speed. It is important that the tension in the spring remains constant for each trial in both the Introductory Activity and the Guided-Inquiry Activity.

Further Extensions

Alignment to the Curriculum Framework for AP^® Physics 1

Enduring Understandings and Essential Knowledge
A wave is a traveling disturbance that transfers energy and momentum. (6A)
6A1: Waves can propagate via different oscillation modes such as transverse and longitudinal.
6A3: The amplitude is the maximum displacement of a wave from its equilibrium value.

A periodic wave is one that repeats as a function of both time and position and can be described by its amplitude, frequency, wavelength, speed, and energy. (6B)
6B1: For a periodic wave, the period is the repeat time of the wave. The frequency is the number of repetitions of the wave per unit time.
6B2: For a periodic wave, the wavelength is the repeat distance of the wave.
6B4: For a periodic wave, wavelength is the ratio of speed over frequency.

Learning Objectives
6A1.1 The student is able to use a visual representation to construct an explanation of the distinction between transverse and longitudinal waves by focusing on the vibration that generates the wave.
6A1.2 The student is able to describe representations of transverse and longitudinal waves.
6A3.1 The student is able to use graphical representation of a periodic mechanical wave to determine the amplitude of the wave.
6B1.1 The student is able to use a graphical representation of a periodic mechanical wave (position versus time) to determine the period and frequency of the wave and describe how a change in the frequency would modify features of the representation.
6B2.1 The student is able to use a visual representation of a periodic mechanical wave to determine wavelength of the wave.
6B4.1 The student is able to design an experiment to determine the relationship between periodic wave speed, wavelength and frequency and relate these concepts to everyday examples.

Science Practices
1.2 The student can describe representations and models of natural or man-made phenomena and systems in the domain.
1.4 The student can use representations and models to analyze situations or solve problems qualitatively and quantitatively.
2.2 The student can apply mathematical routines to quantities that describe natural phenomena.
4.2 The student can design a plan for collecting data to answer a particular scientific question.
5.1 The student can analyze data to identify patterns or relationships.
6.2 The student can construct explanations of phenomena based on evidence produced through scientific practices.

Answers to Prelab Questions

1. Compare and contrast transverse and longitudinal waves.

   Particles in a transverse wave are displaced perpendicular to the direction of the wave propagation, whereas particles in a longitudinal wave are displaced parallel to the direction of the wave propagation. Transverse waves exhibit crests and troughs, areas of maximum displacement from a point of equilibrium, and longitudinal waves exhibit compression and rarefaction, which are also areas of maximum displacement from a point of equilibrium.

2. A fishing bobber floating on the water bobs up and down and back to a point of equilibrium once every 1.2 seconds.
   1. Explain what type of wave is causing the bobber to move up and down.

      The fishing bobber is being displaced by a transverse wave, as the motion of the bobber is perpendicular to the propagation of the waves passing by.

   2. What is the frequency of the water wave?

      The bobber moving up and down corresponds to one complete wave cycle, which occurs every 1.2 seconds. Therefore, the frequency is 1/1.2 s or 0.83 Hz.

   3. What is the period of the wave?

      The period, T, is the time is takes for one complete cycle, or 1.2 s.

3. Refer to the following diagram in which the vertical distance from the highest point on the wave to the lowest point represents 0.1 m, and the horizontal distance from one letter to the next is 0.05 m.
   {13793_PreLab_Figure_6}
   1. What is the amplitude of the wave?

      The amplitude is 0.05 m.

   2. Describe the wavelength in terms of letter intervals. What distance does this represent?

      The wavelength is from A to E (or B to F, etc.) and is 0.2 m.

   3. If the wave travels from point A to point G in 4.5 seconds, what is the speed of the wave?

      Since v = d/t, v = 0.3 m/4.5 s or 0.07 m/s

   4. Determine the period of the wave described previously.

      The interval from A to G represents 1.5 wave cycles. Since it takes 4.5 seconds to complete 1.5 wave cycles, it would take 3 seconds to complete one wave cycle.

   5. Calculate the frequency of the wave.

      f = 1/T = 1/3 s = 0.33 Hz

Sample Data

Introductory Activity

Part A. Transverse Waves

As the transverse wave pulse travelled along the spring, the nylon string moved to the side and back, perpendicular to the length of the spring. When the pulse reached the end, it reflected back in the opposite direction and on the opposite side of the spring. The amplitude of the reflected pulse was smaller than the incident pulse.

Part B. Longitudinal Waves

As the longitudinal wave pulse travelled along the spring, the nylon string moved forward and back in the same direction as the pulse. When the pulse reached the end, it reflected back the same way.

Part C. Standing Wave

In order to increase the number of antinodes in the standing wave from one to two, the end of the spring needed to be shaken back and forth faster. This increases the frequency of the waves.

A standing wave with one antinode (and two nodes) is ½ wavelength. Since the spring was stretched to 3 m, the wavelength would be 3 m x 2 = 6 m. A standing wave with two antinodes represents one wavelength or 3 m.

Guided-Inquiry Activity

Effect of Amplitude on Wave Speed

The amplitude has no effect on the wave speed, within experimental error.

Standing Waves

A pattern appears between the number of antinodes and the wavelength. As the number of antinodes increases, the wavelength decreases.

Graph A

Graph B

Analysis and Conclusions

As the frequency increases, the wavelength decreases. Graph B shows a linear relationship between wavelength and period, which is 1/f. (The data point of 0,0 was included as a period with a value of 0 would result in 0 wavelength.) The slope of the graph represents the wave speed. The value of 6.6 m /s is very close to the measured wave speed of the transverse wave pulse created in the Introductory Activity, 6.5 m /s.

Answers to Questions

Guided-Inquiry Discussion Questions

1. Consider the wave pulses that were propagated in the Introductory Activity.
   1. Did the properties of the wave pulse change in any way as the pulse traveled through the spring?

      The wave speed seemed to remain constant. The size of the pulse, its amplitude, diminished as the wave continued to travel along the spring, especially after it reached the end and reflected back.

   2. What eventually happened to the wave pulse? Explain.

      The wave pulse eventually disappeared and the spring stopped vibrating. As a wave is propagated through a medium, it loses energy to the medium. In addition, some energy is transferred to the floor (or benchtop) due to friction. When the wave reached the end, some energy was transmitted to the person holding the spring.

2. Given the equation for velocity, as well as your observations from the Introductory Activity, predict how amplitude, frequency and wavelength are related to wave speed.

   Amplitude has no effect on the speed of the wave. The amplitude continually decreased as the wave moved throughout the spring, but the speed seemed to remain constant. Velocity is distance/time. Since the wavelength is a measure of distance and the period is a measure of time, it would reason that speed is directly proportional to λ/T. As T = 1/f, then wave speed is directly proportional to λ • f.

3. Plan, discuss, execute and evaluate an experiment to test your prediction stated above.
   1. Discuss and design a controlled experiment to test the prediction. Which variable will be manipulated? Which variables will remain constant?

      The frequency of the wave is easily manipulated. The same spring should always be used and the length of the stretched spring should remain constant.

   2. How reproducible were the measurements in Parts A and B from the Introductory Activity? How many trials would you recommend for future experiments?

      Timing a single pulse for Parts A and B was difficult to do with consistency and is a source of random error. At least 10 trials are recommended to determine the wave speed.

   3. Keep in mind that a standing wave is a wave pattern caused by the interference of incident and reflected waves of the same frequency and wavelength. What properties of a standing wave can be measured directly?

      The wavelength of the standing wave can be measured, knowing the length of the stretched spring and observing the number of antinodes. The period of the wave can be measured by timing a number of cycles and determining the time for one cycle. The frequency of the standing wave can then be calculated.

   4. Which type of wave will you propagate? Explain the reasoning for your choice.

      Since the wavelength is most easily measured in a standing wave, that would be the logical choice.

   5. How will the data be presented?

      Graphing wavelength versus frequency and wavelength versus period (1/f)would be the best way to represent the data.

Review Questions for AP^® Physics 1

1. Which of the following properties, amplitude, frequency and wavelength, affect the wave speed of the stretched spring?

   The properties of amplitude, frequency or wavelength have no effect on the wave speed of the stretched spring. As long as the spring was stretched to the same length, the wave speed was constant. Frequency and wavelength are inversely proportional. Therefore, v = λ • f.

2. A longitudinal wave is propagated through a medium. The distance from one maximum compression to the next is x meters and its speed is y m/s. Express the frequency of the waves in terms of x and y.

   f = y/x

3. A wave is propagated through a spring. A point on the spring moves perpendicular to the motion of the wave a total of 32 cm during one complete wave cycle. What is the amplitude of the wave?

   Starting at equilibrium, a point on the spring would move vertically to a maximum displacement, back to equilibrium, to another point of maximum displacement and back to equilibrium in one complete cycle. Therefore, the amplitude is 32 cm/4 or 8 cm.

4. A standing wave is created in a vibrating string in which the length, L, is 1.5 m. The string is fixed at each end and displays three antinodes.
   1. Determine the wavelength of the standing wave.

      A standing wave with three antinodes represents 3/2 wavelengths or λ = 2/3 L. The wavelength is 1.0 m.

   2. The period of the vibrating string is 0.004 seconds. What is the frequency of the standing wave?

      f = 1/0.004 s = 250 Hz

   3. What is the speed of the propagated transverse waves?

      v = 1.0 m x 250 Hz = 250 m /s

   4. At what frequency would the same string need to vibrate in order to establish one antinode?

      One antinode represents ½ λ, or λ = 2L.

      {13793_Answers_Equation_1}
   5. The motor creating the vibrations in the string can reach a maximum frequency of 300 Hz. Can a standing wave with four antinodes be created with the string as it is using this motor? Explain.

      A standing wave with four antinodes cannot be created with this motor and string setup. A standing wave can be created only at certain frequencies. In order for four antinodes to form, the wavelength must be 0.75 m.

      {13793_Answers_Equation_2}

      This is greater than the maximum frequency the motor can achieve.

5. Explain how a guitar player can produce so many different notes with only six strings.
   
   Each string vibrates at a particular frequency. When the player presses on a fret of the guitar, the length of the vibrating guitar string is shortened, effectively shortening the wavelength. This results in a higher frequency and a different pitch.

References

AP Physics 1: Algebra-Based and Physics 2: Algebra-Based Curriculum Framework; The College Board: New York, NY, 2014

Introduction

Ocean waves, seismic waves, sound waves, electromagnetic waves—waves are all around us. The majority of information we receive on a daily basis reaches us in the form of waves. All waves transmit energy; however, mechanical waves can only be propagated through a medium. Use a Slinky^® to investigate properties of mechanical waves and determine the relationship among properties of mechanical waves.

Concepts

• Longitudinal wave
• Transverse wave
• Standing wave
• Wavelength
• Frequency
• Amplitude

Background

A wave is a displacement or disturbance (vibration) that moves through a medium or space. A wave pulse is a single vibratory disturbance that travels from one point to another and a periodic wave is a series of evenly timed uniform vibrations. As a wave propagates, energy is transferred through the medium or space. Mechanical waves move through materials such as springs, air, and water. As the mechanical wave moves through the medium, it loses energy to that medium.

Waves are often categorized based on how they travel. Particles in transverse waves vibrate perpendicular to the direction of the wave propagation. Longitudinal waves vibrate particles parallel to the direction of the propagation (see Figure 1).

Transverse waves display the common characteristic properties of crests and troughs (see Figure 2), like ripples on a pond. Longitudinal waves display the characteristic properties of compressions, or areas of high molecular density and pressure and rarefactions, or areas of low molecular density and pressure (see Figure 2). Sound waves are longitudinal. A wave’s amplitude is directly related to the amount of energy transmitted by the wave. For a transverse wave, the amplitude is the maximum displacement above or below its position of equilibrium. The amplitude of a longitudinal wave is the maximum increase or decrease in pressure in the medium as the wave travels through. The wavelength (λ) is the distance from one point on a wave to the same point on the next wave, e.g., crest to crest or compression to compression (see Figure 2).

The number of waves that pass a fixed point in a given amount of time is known as the frequency, f. In other words, frequency is a measure of how often particles vibrate as the wave travels through the medium. Frequency is measured in cycles per second or waves/second. One cycle per second is 1 hertz (Hz). The period of a wave is the time it takes for one part of a wave to make one complete cycle and is measured in seconds per cycle. The period, T, is equivalent to 1/f. Conversely, f = 1/T. Frequency is different than wave speed (v), which is the distance traveled by a point on a wave in a given amount of time (v = d/t, measured in m/s).

All traveling waves follow the principle of superposition. When two or more waves meet at the same location, the waves overlap with each other and the amplitude of the new wave form is the sum of the amplitudes of the individual waves. However, the original wave patterns are not lost. Instead, they travel through each other, interact with superposition, and then emerge with the same original shape (see Figure 3). The superposition of two or more waves creates two types of interference—constructive interference and destructive interference. Constructive interference occurs when the superposition of two or more waves produces a wave form with a larger amplitude than any of the original waves. Destructive interference occurs when the superposition of two or more waves produces a wave with a lower amplitude than any of the original waves (see Figure 4). When two continuous waves that have the same frequency interact while travelling toward each other from opposite directions, an interesting wave form can be created. If the waves are the correct frequency, a standing wave is produced (see Figure 5). A standing wave is characterized by the presense of nodes and antinodes. A node is a point in a standing wave that appears to be stationary. This is due to complete destructive interference. An antinode is a point in a standing wave, halfway between two nodes, at which the largest amplitude occurs (see Figure 5).

Experiment Overview

The purpose of this advanced inquiry lab is to investigate the properties of transverse and longitudinal waves using a Slinky spring. The investigation begins with an introductory activity to observe the motion of transverse, longitudinal and standing waves propagated through the spring. The procedure provides a model for guided-inquiry design of experiments to determine which of the following wave properties affect the wave speed: frequency, amplitude and wavelength.

Materials

Prelab Questions

1. Compare and contrast transverse and longitudinal waves.
2. A fishing bobber floating on the water bobs up and down and back to a point of equilibrium once every 1.2 seconds.
   1. Explain what type of wave is causing the bobber to move up and down.
   2. What is the frequency of the water wave?
   3. What is the period of the wave?
3. Refer to the diagram below in which the vertical distance from the highest point on the wave to the lowest point represents 0.1 m, and the horizontal distance from one letter to the next is 0.05 m.
   {13793_PreLab_Figure_6}
   1. What is the amplitude of the wave?
   2. Describe the wavelength in terms of letter intervals. What distance does this represent?
   3. If the wave travels from point A to point G in 4.5 seconds, what is the speed of the wave?
   4. Determine the period of the wave described above.
   5. Calculate the frequency of the wave.

Safety Precautions

Take care and do not suddenly release a stretched Slinky. The spring may snap back rapidly, which may cause personal injury or damage to the Slinky. Wear safety glasses. Do not extend the Slinky more than 4 meters. Please follow all laboratory safety guidelines.

Procedure

Introductory Activity

Read the entire procedure before beginning. Construct an appropriate data table to record observations and measurements.

Part A. Transverse Waves

1. Choose a coil of the Slinky about half way from each end, and tie the nylon string tightly in a single knot.
2. Place the Slinky on the floor or on a smooth lab benchtop.
3. While a lab partner holds one end of the Slinky securely, grasp the other end and stretch the spring 3 m across the floor. Note: Take care to hold the ends of the Slinky securely. Holding more than one coil will prevent the end coil from bending out of shape. Do not allow the stretched Slinky to release suddenly.
4. With a rapid motion, shake one end of the Slinky sideways and back to its original position to create a single pulse with an amplitude of about 20 cm.
5. Observe the transverse wave as it travels the length of the Slinky, noting the motion of the string. What happens when the pulse reaches the end of the spring? Record all observations.
6. Repeat step 5, timing how long it takes for the wave to travel a specific distance. This may require timing the wave from one end of the spring to the other and back again.
7. Repeat step 6 nine more times, keeping the speed of the sideways shake motion constant.

Part B. Longitudinal Waves

8. Position the Slinky on the floor and stretch it to 3 meters as in steps 1–3.
9. With a free hand, carefully gather up a set of coils approximately 20 cm from the end and compress them tightly.
10. Making sure you and your partner are holding the opposite ends of the Slinky securely, release the compressed coils without letting go of the end coil. Observe the compression wave as it travels the length of the Slinky, and note especially the motion of the string. What happens when the wave pulse reaches the end of the spring? Record all observations.
11. Repeat step 10, timing how long it takes for the wave to travel a specific distance. This may require timing the wave from one end of the spring to the other and back again.
12. Repeat step 11 nine more times.

Part C. Standing Waves

13. Place the spring on the floor or on a smooth lab benchtop and stretch the spring to 3 meters, making sure each end of the spring is held securely.
14. Create a periodic standing wave by making periodic transverse waves, consistently shaking one end of the spring back and forth until one antinode is formed.
15. Repeat step 14, creating a standing wave with two antinodes.

Analyze the Results

• Calculate the average wave speed for the transverse and longitudinal waves.
• Describe how the motion of your hands changed in order to increase the number of nodes in the standing wave. What wave property does this represent?
• Determine the wavelength of each standing wave that was created.

Guided-Inquiry Design and Procedure

Form a working group with other students and discuss the following questions.

1. Consider the wave pulses that were propagated in the Introductory Activity.
   1. Did the properties of the wave pulse change in any way as the pulse traveled through the spring?
   2. What eventually happened to the wave pulse? Explain.
2. Given the equation for velocity, as well as your observations from the Introductory Activity, predict how amplitude, frequency and wavelength are related to wave speed.
3. Plan, discuss, execute and evaluate an experiment to test your prediction stated above.
   1. Discuss and design a controlled experiment to test the prediction. Which variable will be manipulated? Which variables will remain constant?
   2. How reproducible were the measurements in Parts A and B from the Introductory Activity? How many trials would you recommend for future experiments?
   3. Keep in mind that a standing wave is a wave pattern caused by the interference of incident and reflected waves of the same frequency and wavelength. What properties of a standing wave can be measured directly?
   4. Which type of wave will you propagate? Explain the reasoning for your choice.
   5. How will the data be presented?
   6. List any safety concerns and the precautions that will be implemented to keep yourself, your classmates and your instructor safe during the experimental phase of the laboratory.

Opportunities for Inquiry

The same spring was used for each experiment in this lab. If the spring was always stretched to the same length, the tension in the spring would have remained constant. Design an experiment to test the effect of the tension in the spring on the mechanical wave speed.

Student Worksheet PDF

13793_Student1.pdf