Mechanical Waves

A wave pulse is a disturbance that moves through a medium.  A periodic wave is a periodic disturbance that moves through a medium.  The medium itself goes nowhere.  The individual atoms and molecules in the medium oscillate about their equilibrium position, but their average position does not change.  If the displacement of the individual atoms or molecules is perpendicular to the direction the wave is traveling, the wave is called a transverse wave.  If the displacement is parallel to the direction of travel the wave is called a longitudinal wave or a compression wave.

Waves can transport energy and information.  Examples of mechanical waves are water waves, sound waves, and seismic waves. All waves are described mathematically in terms of a wave function, and reflection, refraction and diffraction and interference a characteristic behaviors of all types of waves.

In this session you will study wave motion in one dimension only, to learn about several of these characteristic behaviors.  You will study the motion of waves on a string.

Open a Microsoft Word document to keep a log of your experimental procedures, results and discussions.  Address the points highlighted in blue.  Answer all questions.

Exploration

Use an on-line simulation from the University of Colorado PhET group to explore the behavior of waves on a string.
Link to the simulation: http://phet.colorado.edu/en/simulations/wave-on-a-string

(a)  Explore the interface.  Try the different controls.

(b)  Investigate the behavior of a wave pulse.

• Start with the following settings:
  • Pulse
  • No End
  • Amplitude: 0.5 cm
  • Pulse Width: 0.5 s
  • Damping: none
  • Tension: high
  • Ruler and timer: checked
• Observe this pulse and measure its speed in cm/s.  (You can start and stop the pulse and use slow motion.)
• Vary (one at a time) amplitude, pulse width, damping, and tension, and describe what happens.
• Return to the original pulse but change the end first to a loose end and then to a fixed end and describe what happens.

(c)  Investigate the behavior of a traveling wave.

• Start with the following settings:
  • Oscillate
  • No End
  • Amplitude: 0.5 cm
  • Frequency: 1.5 Hz
  • Damping: none
  • Tension: high
  • Ruler and timer: checked
• Observe this wave (wave 1) and make measurements.  Enter your measurements into the table.
  • Measure the amplitude of the wave in cm.
  • Measure the wavelength in cm.
  • Measure the period in s. 
  • Find the speed of the wave in cm/s.

• Move the amplitude slider to 0.25 cm (wave 2) and then to 0.75 cm (wave 3).  Make new measurements.  Describe what changes.
• With the amplitude slider at 0.5 cm, move the frequency slider to 1 Hz (wave 4) and then to 2 Hz (wave 5).  Make new measurements.  Describe what changes.
• Discuss the relationships between wavelength and frequency, period and frequency, amplitude and frequency and speed and frequency.
• Can you change the speed of the wave?  What can you do to produce a wave that moves with approximately 1/4 the speed of wave 1.
• Describe what happens when you include damping.

(d)  Investigate the behavior of a standing wave.

• Start with the following settings:
  • Oscillate
  • No End
  • Amplitude: 0.5 cm
  • Frequency: 0.75Hz
  • Damping: 1 notch
  • Tension: high
  • Ruler and timer: checked
• Describe the wave.  What is its maximum amplitude in cm?  Does this maximum amplitude change when you change the frequency slider?
• Change the end to a fixed end.
  • Slowly increase the frequency.  Wait several seconds to see what happens to the maximum displacement from equilibrium of the wave for each frequency setting.
  • For which frequencies between 0.75 Hz and 1.7 Hz is the maximum displacement from equilibrium largest (larger than that of the traveling wave)?  In other words, for which frequencies do you hit resonance?  What happens if you set the damping to zero at those frequencies?
  • What can you say about the wavelengths associated with those frequencies?

Experiment

Standing waves of many different wavelengths can be produced on a string with two fixed ends, as long as an integral number of half wavelengths fit into the length of the string.  For a standing wave on a string of length L with two fixed ends

L = n(λ/2),  n = 1,2,3,...  .

• Fundamental: L = λ/2,  n = 1, 1/2 wavelength fits into the length of the string.
• Second harmonic: L = λ  n = 2, one wavelength fits into the length of the string.
• Third harmonic: L = 3λ/2,  n = 3, 3/2 wavelengths fit into the length of the string.

For a string the speed of the waves is a function of the mass per unit length μ = m/L of the string and the tension F in the string.

v = √(F/μ).

In this lab, waves on a string with two fixed ends are be generated by a the Pasco Mechanical Wave Driver connected to a Sine Wave Generator.  The generator frequency displayed in units of Hz in the display of the generator box.  Students will watch two video clips.  In these clips the frequency of the waves is increased in steps of 1 Hz.  The wavelength of the waves is given by λ = v/f.  The speed of the waves can be changed by changing the string tension.  For two different tensions students will determine the frequency f of the sine wave generator when 1, 2, 3 or 4 half wavelength of a wave fit into the length of the string.  Then f is a natural frequency of the string and the generator drives the string into resonance.  The amplitude increases and the standing waves can easily be observed.

Summary:

Choose the the video clip for the specific tension F.
Measure the frequency f, for λ = 2L, L, 2L/3, L/2.
Calculate the mass per unit length μ of the string, using v = λf,  μ = F/v².

Procedure:

Open an Excel spreadsheet and paste the table below into the spreadsheet.

• Chose video clip 1.
• The mass hanger's mass is 50 g and a 100 g mass is added to the hanger.  Calculate the tension F = mg and enter this value into the spreadsheet.  (g = 9.8 m/s²)
• Enter the length L of the string into the appropriate cells of the spreadsheet.
• Use the spreadsheet to calculate the wavelength λ_n = 2L/n and then the speed v_n= fλ_n = 2fL/n of the fundamental and second and third harmonic for f = 120 Hz.
• Play the video clip.  Find the frequency f of the fundamental standing wave on the string.  Use the slider and start and stop the video clip near resonance until you find the frequency f for which you observe observe a stable resonance with a large amplitude.  Enter the value of f into the spreadsheets.
• Repeat for the second and third and fourth harmonic and fill in the table.
• Repeat the whole procedure for video clip 2.  In clip 2 a 150 g mass added to the mass hanger.

Data Analysis:

Calculate the mass per unit length μ of the string using μ = F/v².  Average the values obtained from your eight measurements and estimate the uncertainty in this average value.

Discuss:

• Were you able to clearly identify the resonances?
• How do your values of μ obtained from the eight measurements compare?  In your opinion, are they equal within experimental uncertainties.  If not, what do you think can explain the differences?

Convert your log into a lab report.

Name:
E-mail address:

Laboratory 11 Report

• In one or two sentences, state the goal of this lab.
• Make sure you completed the entire lab and answered all parts.  Make sure you show your work and inserted and properly labeled relevant tables and plots.
• Add a reflection at the end of your report in a short essay format.

Save your Word document (your name_lab11.docx), go to Canvas, Assignments, Lab 11, and submit your document.