A standing wave is a pattern which results from the interference of two or more waves traveling in the same medium.  All standing waves are characterized by positions along the medium which are standing still.  Such positions are referred to as nodes.  Standing waves are also characterized by antinodes.  These are positions along the medium where the particles oscillate about their equilibrium position with maximum amplitude.  Standing wave patterns are always characterized by an alternating pattern of nodes and antinodes.

Transverse waves on a string

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 fits 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.

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.  Students 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.

#### Equipment needed:

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

Experiment:  Standing waves on a string

• 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
 n measured L (m) λn = 2L/n (m) speed vn = fλn (m/s) hanging mass at resonance (kg) measured F = mg (N) Fundamental: (n = 1) 1 Second harmonic: (n = 2) 2 Third Harmonic: (n = 3) 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 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?

Open Microsoft Word and prepare a short report.