Lab 13: Standing Waves
Name: __________________________
Goals:
- Measure resonant wavelengths and resonant frequencies in a column of air in order to compute the speed of sound.
- Find resonant frequencies on a string in order to measure the wave velocity.
Frequency, wavelength and velocity:
The frequency f , wavelength l and velocity v of a wave are related by the simple formula:
v = fl
In this lab, we will measure frequencies and wavelengths and use them to compute wave velocities.
Standing waves in a half-open pipe:
In a half-open pipe, the displacement amplitude is maximum at the open end (because the air does not encounter any
boundaries), and the displacement amplitude is zero at the closed end (air at the closed end can’t move). Thus, the standing
waves in a half-open column of air correspond to wavelengths with an anti-node at one end and a node at the other. Several
examples are shown below for a pipe of fixed length:
We can determine the wavelength of each standing wave by simply counting how many waves are contained on the length of
the pipe in each case.
In the first part of the lab, we will change the length of the pipe instead of changing the wavelength of the sound waves. Using a
tuning fork to agitate the air in the column, we can find the lengths corresponding to resonances in the pipe. A water reservoir
is used to change the length, and the amplitude of the sound will be substantially louder at the location of each resonance.
As you can see, the distance between adjacent resonances corresponds to one-half the wavelength of the tone, so we can
quickly deduce the wavelength by simply doubling that distance.
Temperature dependence of the speed of sound:
The speed of sound depends on both the medium and the temperature. For air at 1 atm of pressure, the speed of sound varies
with temperature approximately like the formula:
v = (331+ 0.6 ×T ) m/s
where T is the temperature in degrees Celsius.
Standing waves on a string:
When a string is fixed at both ends, the resonant wavelengths correspond to having a node at both ends of the
string. Several examples are shown below:
The wavelength can be determined quickly if you know the length of the string.
Wave speed, tension and mass per unit length:
The wave speed on a string increases with tension and decreases with the “heaviness” of the string. The tension is
measured in Newtons (as usual), and the “heaviness” of the string is measured in kg/m. This quantity is usually
called “linear density” and is indicated by m or m / L . The wave speed is determined by the simple formula:
v=
T
m
In this lab, the string is very stretchy, so the linear density changes when you unhook the apparatus. You will have
to find a way to accurately measure m for the stretched string!
Half-open pipe with changing length:
1. Set up your half-open pipe with the water reservoir as shown below. The reservoir can be raised or lowered to
change the water level, and you may want to use an extra can of water if you find the capacity of the reservoir is not
large enough. The tuning fork should be oriented so it pushes air in the direction of the pipe – remember, sound is
caused by pressure waves!
2. Choose a tuning fork with a frequency of 300 Hz or higher, then note the following:
Frequency: ______ Hz
Distance between two adjacent nodes: _______ m
Wavelength: _______ m
3. Compute the speed of sound based on the distance between resonances for your tuning fork.
Speed of sound (based on standing wave): ________ m/s
4. Compute the speed of sound based on the air temperature and find a percent difference.
Speed of sound (based on temperature): ________ m/s
% difference: _________
Half-open pipe with changing frequency:
1. Using the same apparatus, lower the water level until the air column is exactly 1 m in length. Clamp the speaker so it
is pushing air down the pipe, and connect the speaker to a frequency generator.
2. Find the two lowest resonances in the pipe. You can convince yourself they are the first two resonances by making
sure one of them is three times the frequency of the other.
Frequency of the first harmonic:
_______ Hz
Frequency of the second harmonic: _______ Hz
3. Now use the first harmonic to compute the speed of sound. Show your work carefully.
Wavelength of the first harmonic:
Speed of sound: ________ m/s
________ m
String fixed at both ends:
1. Set up the vibrating string as shown below:
mechanical vibrator
frequency generator
500 g
2. Use the frequency generator to find the first four harmonics on the string, and note the quantities below:
First harmonic:
Frequency _________ Hz
Wavelength: ________ m
Wave speed: ___________ m/s
Wavelength: ________ m
Wave speed: ___________ m/s
Wavelength: ________ m
Wave speed: ___________ m/s
Wavelength: ________ m
Wave speed: ___________ m/s
Second harmonic:
Frequency _________ Hz
Third harmonic:
Frequency _________ Hz
Fourth harmonic:
Frequency _________ Hz
Average wave speed: __________ m/s
3. Now compute the wave speed by using the physical properties of the system:
Linear density: ________ kg/m
Tension:
________ N
Wave speed:
________ m/s
% difference:
________