Resonance and standing waves
Investigation 15D: Wavelength and standing waves
A standing wave is a wave that has been trapped between two boundaries, such as the fixed ends of an
elastic string. Because standing waves persist for long periods of time they make an excellent
example for investigating wavelength and frequency.
Part 1: Setting up a standing wave
1. Launch the interactive simulation of a vibrating string driven by an oscillator.
2. Change the oscillation frequency until you create a standing wave.
3. Change the frequency to see different standing wave patterns with different numbers of nodes
and antinodes.
Questions
a. Do the standing waves appear at all frequencies or just specific ones? Explain this using the
idea of resonance.
answer: Standing waves appear at only certain resonant frequencies for which the length of the
string is a multiple of the half-wavelength.
b. Describe the role of reflection and interference in creating standing waves.
answer: Incoming waves reflect off the boundaries of the system, which are the fixed ends of
the string. At resonant frequencies, the reflected waves interfere constructively with the
incoming waves. If the system dimensions are some multiple of a wavelength, then this
constructive interference creates standing waves.
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c. Sketch two or three different standing wave patterns and indicate the wavelength of each.
Answers will vary. Examples:
Part 2: Relationship between wavelength and frequency
1. Adjust the frequency until you get the largest amplitude standing wave you can that has a
single node between its fixed ends. This is the second harmonic because there are two
antinodes.
2. Using the measuring scale, estimate the wavelength of the standing wave and record its
wavelength and frequency.
3. Adjust the frequency to find the standing waves from the first through the eighth harmonics
and record the wavelength and frequency of each.
Table 1: Wavelength and frequency
harmonic
frequency
f (Hz)
16 32 f×λ
1st 2nd wavelength
λ (m)
2.0 1.0 3rd 4th 0.67 0.50 47 63 31.6 31.6 5th 0.40 79 31.6 6th 7th 0.33 0.29 96 111 31.6 31.6 8th 0.25 126 31.6 31.6 31.6 The frequency values in this table are based on
the default values for tension and mass/length.
Questions
a. What is the relationship between frequency and wavelength? State it with a mathematical
formula.
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answer: The higher the frequency, the shorter the wavelength.
f!
Wavelength and frequency are inversely proportional.
!
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b. What units does the product of frequency × wavelength have? What is the interpretation of
this quantity? Fill in the third column of your table with the value of f × λ.
cycles meters
meters
f ! !=
!
=
second cycle
second
answer: Frequency multiplied by the wavelength is the speed of the wave in the string.
It has units of m/s.
c. How could you control the frequency of a vibrating object through its wavelength? Describe at
least two applications of this principle.
answer: You can control the natural frequency by controlling the distance between the
boundaries. Musical instruments and MRI imaging machines employ this strategy.
For example, when a trombone player extends the slide, the result can be a longer
wavelength and lower note.
d. Going further: What happens to the fundamental frequency of the standing wave if you
quadruple the tension in the string? What happens to the fundamental frequency if you
quadruple the mass per unit length of the string?
If you quadruple the tension, the frequency doubles.
If you quadruple the mass of the string, the frequency is halved.
What do these results imply about the speed of waves in the string?
The wavelength of the fundamental stays constant at 2.0 m. Since
v = λf then quadrupling the tension must double the wave speed.
T
Quadrupling the mass must halve the wave speed. Wave speed
v!
m
must be proportional to the square root of T/m.
e. Listen to the sound produced by the vibrating string. When you increase the tension in the
string, does the pitch rise or fall? When you increase the mass of the string, does the pitch rise
or fall? Can you relate this to playing an instrument?
Increasing tension raises the pitch. Increasing the mass lowers the pitch.
If you tighten a guitar string you increase the tension and the note gets higher.
If you pluck a thicker, more massive guitar string, it makes a lower note.
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