## Studio Session 12

### 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.

Equipment needed:

• electric string vibrator
• pulley
• base and support rods and clamps
• mass hanger and mass set
• level
• meter stick

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.

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

(b)  Investigate the behavior of a wave pulse.

• 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 to a loose end and the to a fixed end and describe what happens.

(c)  Investigate the behavior of a traveling wave.

• 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.
  amplitude A wavelength λ period T frequency f wave 1 wave 2 wave 3 wave 4 wave 5
• 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.

• Oscillate
• No End
• Amplitude: 0.5 cm
• Frequency: 1.5 Hz
• Damping: one 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, the frequency to zero, and click the restart button.
• 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 zero and 1.5 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 will be generated by a string vibrator.  The waves will all have a frequency of 120 Hz.  Their wavelength is given by λ = v/f.  Since the frequency is fixed, the wavelength of the waves can only be changed by changing the speed of the waves.  You will adjust the tension in the string until 1, 2, or 3 half wavelength of a wave with f = 120 Hz fit into the length of the string.  Then 120 Hz is a natural frequency of the string and the vibrator drives the string into resonance.  The amplitude increases and the standing waves can easily be observed.

Summary:

Given: f = 120 Hz.
Measure: tension F, for λ = L/2, L, 2L/3
Calculate: the mass per unit length μ of the string, using v = λf,  μ = F/v2.

Procedure:

• Mount the vibrator on a rod which is fixed to the table with a clamp.  Mount the pulley onto another rod fixed to the table with a clamp.  Pass a string from the vibrator over the pulley and attach a mass hanger.  Make sure the string is level.  You now have a string with two fixed ends.
• The amplitude of the vibrator arm is so small compared to the amplitude of the string at resonance, that the vibrator is very close to a node.
• Open an Excel spreadsheet and paste the table below into the spreadsheet.

 Fundamental: (n = 1) Second harmonic: (n = 2) Third Harmonic: (n = 3) n measured L (m) λn = 2L/n (m) speed vn = fλn(m/s) hanging mass at resonance (kg) measured F = mg (N) 1 2 3
• Let the length of the string from the vibrator to the top of the pulley be somewhere between 0.8 m and 1.2 m.  Enter the length L into the appropriate cells of the spreadsheet.
• For your chosen length L use the spreadsheet to calculate the wavelength λn = 2L/n and then the speed vn= fλn = 2fL/n of the fundamental and second and third harmonic for f = 120 Hz.
• Turn on the vibrator.  Try to produce the fundamental standing wave on the string.  Adjust the amount of mass hanging from the string until the string is driven into resonance.  Enter the measured value for the hanging mass.  Calculate the measured force F = mg.  (g = 9.8 m/s2)
• Repeat for the second and third harmonic and fill in the table.

Data Analysis:

Calculate the mass per unit length μ of the string using μ = F/v2.  Average the values obtained from your three 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 three 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 session report, certify with you signature that you have actively participated, and hand it to your instructor.