Lab 13
Standing Waves
OBJECTIVES
To study the behavior of longitudinal and transverse standing waves in 1- and 2dimensional media.
EQUIPMENT
Mechanical wave driver, bungee cord, pulleys, clamps, supports, weights, meter stick,
springs, water tubes, jugs, tuning forks, Chladni plates, function generator.
THEORY
1. Standing Waves on a String
When a stretched string is plucked it will vibrate in its fundamental mode in a single
segment with nodes on each end. If the string is driven at this fundamental frequency, a
standing wave is formed. Standing waves also form if the string is driven at any integer
multiple of the fundamental frequency. These higher frequencies are called the
harmonics. Each segment is equal to half a wavelength. In general for a given harmonic,
the wavelength n is
n
 2L
n
where L is the length of the stretch string and n is the number of segments in the string.
The linear mass density of the string can be directly measured by weighing a known
length of the string. The linear mass density  is the mass of the string per unit length.
The linear mass density of the string can also be found by studying the relationship
between the tension, frequency, length of the string, and the number of segments in the
standing wave.
 The velocity of any wave is given by v = f where f is the frequency of the wave.
 The velocity of a wave traveling in a string is dependent on the tension, T, in the
string and the linear mass density, µ, of the string: v  T  .
2. Longitudinal Waves on a Spring
The resonant frequencies fn,
fn 
v
n
 nf0 for n  1, 2, 3,... ,
are integer multiples of the lowest resonant frequency or fundamental, f0.
3. Resonant Standing Waves in an Air Column
The resonant frequencies fn for a pipe closed at one end and open at the other are
fn 
v
n
 nf0 for n  1, 3, 5,... ,
where f0 = v/4L. In air at 20OC, the speed of sound is 343 m/s (vsound = 20.1 T m/s).
PROCEDURE
Part 1: Standing waves on a bungee cord
a. Attach one end of a thin bungee cord to a fixed support, hang the other end over a
pulley and suspend a 400 g mass from it. Be sure the mass is supported entirely by
the cord. In DataStudio, double-click the Output icon under Signal Output in the
Experiment Setup window and adjust the output to produce a sine wave with a
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frequency of 40 Hz and amplitude of 1.0 V. Connect the mechanical driver to the
interface output and attach it to the bungee cord near one end.
b. Click the DataStudio Start button and move a "fret" along the cord until you get the
maximum amplitude fundamental standing wave on the portion of the cord between
the driver and fret. Where are the waves reflected in this system, and are the
reflection points nodes or antinodes? Sketch the standing wave in your notes and
record the following data in columns in a table.
 The harmonic number of the wave
 The length of the cord between driver and fret
 The frequency of the standing wave (that you set in DataStudio)
 The wavelength of the standing wave
 The wave speed on the bungee cord (v = f)
c. Create other standing wave patterns as described below. For each, add another row
of data to your table.
 Without changing the frequency, move the fret away from the pulley until a
second harmonic standing wave is produced. Repeat for a third harmonic wave.
 Return the fret to the original (fundamental) position and increase the frequency
until a second harmonic standing wave is produced. Repeat for a third harmonic
wave.
 Change the driving frequency to 20 Hz and find the new fundamental cord length.
Repeat for a driving frequency of 60 Hz.
d. Theoretically, the speed of the waves should depend only on the cord density and
tension (suspended weight), both of which are fixed in this experiment. Is this what
your data show? If so, find the mean and standard deviation of the data in your wave
speed column.
e. Compare your wave speed results with that predicted by the equation v  T  ,
where  = 0.0029 kg/m.
Part 2: Standing longitudinal waves on a spring
a. Use the Mechanical Wave Driver to produce standing waves on a spring. Identify the
nodes and antinodes. Experiment by changing the frequency to obtain various
standing wave patterns.
b. Measure the distance between nodes and determine the wavelength for five different
frequencies. Calculate the wave speed for each case. Does your data support the
prediction that the speed is independent of frequency?
Part 3: Standing sound waves
a. Listen for the fundamental standing sound wave in a tube driven with a tuning fork.
Start with the water level near the top of the tube and lower it until a resonance is
heard, then sweep up and down through the resonance level several times and mark
it with a rubber band. Where are the waves reflected in this system, and are the
reflection points nodes or antinodes? Continue lowering the water until you find the
next resonance (third harmonic). Sketch the third harmonic standing wave in your
notes and record the following data in columns in a table.
 The frequency of the standing wave
 The wavelength of the standing wave (hint: measure the distance between the
rubber bands, not the distances from the top of the tube).
 The speed of sound in the tube.
b. Find additional sound standing waves as described below. For each, add another
row of data, including speed, to your table.
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
With the same tuning fork, find the fifth harmonic resonance level. Measure the
distance between the third and fifth harmonic levels to determine the wavelength.
 With a different frequency tuning fork, determine the wavelength from the
distance between the fundamental and third harmonic resonance levels.
c. Find the mean and standard deviation of the data in your wave speed column and
compare it to the accepted value (at 20°C) of 343 m/s.
Part 4: Two-dimensional standing waves on a Chladni plate
a. Vary the frequency of the function generator until you produce a standing wave on
the Chladni plate. You will be able to tell when you have a standing wave by
sprinkling sand on the vibrating plate as demonstrated by your instructor.
b. Find at least 3 different standing waves and sketch each one in your lab notes. Label
the frequency of the standing wave for each sketch.
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