Standing Waves on a String
(Power Amplifier)
Waves and Sound: Standing wave properties
DataStudio file: 50 Standing Waves.ds
Equipment List
Qty
1
1
1
2m
2m
2m
2
1
1
1
1
1
1
1
1
Items
PASCO 750 Interface
Power Amplifier
String Vibrator
String
Braided Cord, Yellow
Elastic Cord
Banana Plug Patch Cord
Super Pulley
Pulley Mounting Rod
Universal Table Clamp
Mass and Hanger Set
“C” Clamp
Balance
Tape Measure
Strobe (optional)
Part Numbers
CI-7500
CI-6552A
WA-9857
SE-8050
ME-9876
SE-9409
SE-9750 or SE-9751
ME-9450
SA-9242
ME-9376B
ME-9348
SE-7286
SE-8723
SE-8712
SF-9211
Introduction
The purpose of this activity is to explore standing waves on a string and to determine how the
speed of a wave in a vibrating string relates to various characteristics of the string such as length,
linear density (mass per unit of length), tension, and frequency.
Background
Most vibrations of extended bodies such as the prongs of a tuning fork or the strings of a piano
are standing waves. Standing waves (sometimes called stationary waves) are produced by the
constructive and destructive interference of two traveling waves that have the same amplitude,
wavelength, and speed but opposite directions. Standing waves can be created on a stretched
string by sending waves from one end that travel the length of the string, reflect from the other
end, and then interfere with the oncoming waves.
The stretched string can have many frequencies that produce a standing wave pattern. Regions of
maximum amplitude called antinodes and points that appear to ‘stand still’ called nodes
characterize the standing wave pattern.
For a stretched string, the fundamental frequency is the frequency at which the entire string
vibrates up and down “as one piece”. At this frequency, the wavelength of the wave on the string
is twice the length of the string, or 2 L (where L is
the length of the string).
If the frequency is adjusted so the same length of
string vibrates with two antinodes separated by a
node, the wavelength of the wave on the string is the
same as the length of the string. If the standing wave pattern has three antinodes, the wavelength
is two-thirds of the length of the string.
For any wave, wave speed can be calculated from wavelength and frequency by v  f where v
is wave speed,  is wavelength, and f is frequency.
Predictions
For a standing wave on a string, what is the relationship between the frequency and the number
of segments in the pattern? What is the relationship between the fundamental frequency and the
frequencies that produce standing wave patterns?
Setup
In the first part of this activity, explore the relationship between wavelength and frequency. In
the second part, determine the relationship between wave speed and string density.
The diagram shows the arrangement of the apparatus.
Pulley
String
Vibrator
String
Hanging
Mass
Set up the equipment as shown in the picture.
1.
Set up the PASCO 750 Interface and computer and
start DataStudio.
2.
Connect the Power Amplifier to Analog Channel A.
Connect the power cord into the back of the Power
Amplifier and plug it in.
3.
Open the DataStudio file: 50 Standing Waves.ds.

The DataStudio file has a Signal
Generator window that controls the
output of the Power Amplifier.
4.
Clamp the String Vibrator and the pulley
about 1.2 m apart. Tie one end of a 1.5 m
piece of string to the blade on the String
Vibrator. Put the string over the pulley and hang about 0.150 kg (150 g) from it.
5.
Measure the distance from the knot where the string attaches to the vibrator to the top of
the pulley. Record this distance as “L” (in the Lab Report section). Note that “L” is not the
total length of the string, only the part of the string that vibrates.
6.
Use two banana plug patch cords to connect the SIGNAL OUTPUT of the Power
Amplifier to the String Vibrator.
Part 1: Wavelength and Frequency
Procedure
1.
Click ‘Start’ in DataStudio. The output from the Power Amplifier starts automatically.
about midway. Change the frequency increment in the Signal Generator to ‘1’ and adjust
the frequency so that the string vibrates in one segment. Adjust the amplitude and
frequency to obtain a large-amplitude wave, but also check the end of the vibrating blade.
The point where the string attaches should be a node.
Frequency
increment.
Adjust the
frequency.
2.
Record the fundamental frequency in the Lab Report section.
3.
Adjust the frequency so the standing wave pattern has two segments (two antinodes with a
node in the center). Record the new frequency.

NOTE: As you adjust the frequency, you may also need to adjust the Amplitude so that the
String Vibrator blade does not hit the housing.
4.
Calculate the ratio of the two-segment frequency to the fundamental frequency and record
the ratio.

If a strobe light is available, use it to illuminate the vibrating string. Adjust the strobe’s
frequency so it matches the String Vibrator’s frequency.
5.
Use the equation for wave speed to calculate the wave speed of the one-segment standing
wave. Record the wave speed in the Lab Report section.
6.
Calculate the wave speed of the two-segment standing wave and record the calculation.
7.
Adjust the frequency so the string vibrates in three segments (three antinodes). Record the
frequency. Calculate the ratio of the three-segment frequency to the fundamental frequency
and record the ratio.
8.
Calculate the wave speed of the three-segment wave and record the calculation. Click
‘Stop’ to end data recording.
Part 2: Wave Speed and String Density
Background
For any wave, speed is related to the wavelength and the frequency. For a wave on a string, the
speed is also related to the tension, F, in the string, and the linear density, , where  is the mass
of the string divided by its length. In theory, the wave speed, v, is given by v 
F

.
For the standing wave on a string, the tension, F, is the weight of the hanging mass (mg). The
formula for wave speed becomes v 
mg

where m is the hanging mass.
In this part of the activity you will adjust the frequency so that the wave pattern always has four
segments. The length, L, of the string will equal two wavelengths (L = 2).
Combining L = 2 and F = mg with v 
mg

f2 
and v = f gives the following:
4g
m
L2
where f is the driving frequency, g is the acceleration due to gravity, m is the hanging mass,  is
the linear density of the string, and L is the length of the vibrating part of the string.
Procedure
1.
Use the same setup as in Part 1, but hang about 0.050 kg (50 g) from the string over the
pulley. Measure the distance from the knot on the vibrator blade to the top of the pulley
and record this as L.
2.
Measure and record the total hanging mass, including the mass hanger.
3.
Click ‘Start’. Use the Signal Generator window to adjust the frequency of the Power
Amplifier so the string vibrates with four segments. Record the frequency.
4.
Add 50 g to the hanging mass and repeat steps 2 and 3 (measure and record total hanging
mass; adjust and record the frequency).
5.
Continue to add increments of 50 g to the hanging mass up to 250 g. Click ‘Stop’ to end
data recording. Record your data in the Lab Report section.
Analysis
1.
Make a graph of frequency squared (f2) versus hanging mass, m. Use kilograms as the unit
for mass.
2.
Find the slope of the best-fit line for your data. As shown in the Background section, the
slope of the f2 versus m graph is:
slope 
4g
L2
3.
Using the slope of the graph, calculate the linear density, , of the string.
4.
Determine the actual linear density of the string by measuring the mass of a known length
and dividing the mass by the length. (If you do not have a balance that can measure to 0.01
g, use several meters of the string.)
5.
Compare the actual linear density to the measured linear density determined from the
graph. Compute the percent difference:
%diff 
measured  actual
X100%
actual