## 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.

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 first to a loose end and then 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: 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 experiment, 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.  You will analyze video clips in which the string tension F is fixed and the length of the string is being varied.  You will measure the length of the string when the string supports a standing wave such that 1, 2, or 3 half wavelength of a wave 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.

You will analyze three video clips, string_x.mp4, x = 1 - 3.  To play a video clip or to step through it frame-by-frame click the "Begin" button.  The "Video Analysis" web page will open.

Begin

In the video clips the vibrator is mounted onto a rod which is fixed to the table with a clamp.  A pulley is mounted onto another rod on a movable stand.  One end of a string is attached to the vibrator.  The string is passed from the vibrator over the pulley and a mass is attached to its other end.  The string is level.

The string is 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.  The other node is the top of the pulley.

• Step through the video clip string_1.mp4 frame by frame.  You will see the vibrator drive the string into resonance when the length of the string becomes equal to exactly 1/2 wavelength of a standing wave.  Locate the frame in which this happens and hold the clip at that frame.
• Calibrate x.  Use the 0.2794 m long marker for the calibration.
• Click "Start Taking Data".
• Click on the node at the vibrator for your first data point.  Then click "Step Down" once, to return to the same frame and click on the node on top of the pulley for your second data point.   Stop taking data.  Find the distance between the two nodes L (the difference between the two x values).
• Open Excel and construct a spreadsheet as shown below.   Enter L, λ and the string tension F given in the clip.  For the fundamental λ = 2L.
 Clip String 1 String 2 String 3 f (Hz) L (m) λ (m) v = fλ (m/s) v2 (m/s)2 F (N) 120 120 120

• Repeat the experiment using the video clips string_2.mp4 and string_3.mp4.  These clips show the second and third harmonic, respectively.  For the second harmonic λ = L (two half wavelengths fit into the length of the string) and for the third harmonic λ = 2L/3 (three half wavelengths fit into the length of the string).

Data Analysis:

Calculate the measured wave speed v = fλ for each string.  For waves on a string we have F = μv2.  This tension is provided by a hanging mass, F = mg.  Calculate v2 and plot F versus v2.  Use Exce's trendline to find the best fitting straight line to this plot.  This slope of this line is equal to the mass per unit length of the string μ in kg/m?

Discuss:

• Were you able to clearly identify the resonances?
• What value of μ did you obtain?  Convert to units of g/m.  Does your value of μ seem reasonable?

Convert your log into a session report, certify with you signature that you have actively participated, and hand it to your instructor at the beginning of the next studio session.