Standing Waves I
Purpose
In this laboratory exercise you will explore how the frequency of a wave and the stretching force applied to a string affects
the speed of the wave traveling down a vibrating string.
Theory
Standing waves are created when two traveling waves, both having the
speed, amplitude and wavelength interfere with one another while
in the opposite directions. In the diagram to the right one of the waves is
represented by the solid line the other by the dashed line.
same
traveling
One end of the string is vibrated, a wave if sent down the length, L, of
the string
to the other end which is affixed rigidly to a point. This wave is then reflected back in the opposite direction. Standing
waves occurs at a fundament resonant frequency, first harmonic, and also at the harmonics (2,3,4 …) .
Examining the above picture observe the top diagram shows a portion of the wave, solid line, traveling down to the fixed
end then the other portion of the wave, dashed line, traveling back. The two combine to form one complete cycle.
Wavelength, λ, is the distance a wave travels during one complete cycle. In the top diagram it travels down a distance, L,
the back again at a distance, L, to have a wavelength of 2L. If you examine the middle diagram you would see that the
wavelength for the complete cycle is simply L. The third diagram and those that would follow becomes a bit more
difficult to determine the wave length by simple observation.
For a standing wave the wavelength, λ, may be determine from the equation
λ =
2L
[1]
n
Where n is the harmonic of the standing wave [1,2,3...]
With the string vibrating at a frequency, f, to create one of the harmonics the phase velocity, v, of the waves on the string
can be determined from
v = λf.
[2]
where λ is the wavelength for the harmonic and f the frequency at which the string is being vibrated.
When tuning a string instrument you are adjusting the force or tension, T, in the string. The string itself will have a
quantity called the linear density, µ, which is its mass per unit length. It has been determine that the phase velocity, v, of a
wave that travels along the string is a function of these two quantities where that
T
𝑣= �
µ
[3]
In this experiment a tension will be applied to a string and then a frequency will be found to obtain the first harmonic.
Once this tension is set the velocity according with equation [3] should be set and not change.
Since the velocity can now be considered a constant it can be seen in equation [2] that if v is constant and the frequency, f,
is increased the wavelength, λ, should decrease. Experimentally we shall explore this and also explore the effects that
increasing the tension has on the phase velocity.
Procedure:
Turn on the computer.
Run the data Studio Program
Open the file Standing Waves 1.
Finding the first Harmonic
The first harmonic will have a single segment that will have maximum amplitude.
1) .
Pass the loop end of the copper wire through the hole in the metal tab of the string vibrator mechanism.
Pass the wire over the pulley and suspend a mass hanger along with a 200 gram mass from the wire. This
provides the tension, T, to the wire.
Calculate the provide tension T = gravity *mass (in kg) i.e 9.8*0.250kg. The mass hanger is 0.05kg.
2) On the computer examine the panel which was open when you started the Standing Waves module. This panel has both
a amplitude adjustment and a frequency adjustment. We will primarily be concerned with the frequency adjustment.
There are two adjustments, one two decrease or increase the frequency, another to adjust the step by which that the
frequency is adjusted.
Measure the length, L, of the wire from the end of the tab on the string vibrator to the center of the pulley.
Select Start. The frequency should already have been readjusted to 15Hz.
Using a step value of 1Hz slowly change the frequency value. Your goal is to find the fundamental frequency to
achieve the first harmonic for the applied tension.
Once you are in the neighborhood of the harmonic adjust the step value to 0.1 Hz. Fine tune your adjustment so
that the first harmonic is found. The points at each end of the wire should appear to be nearly stationary and the
amplitude of the wire should be as large as possible.
Calculate the wavelength of the first harmonic λ = 2L and the phase velocity, v, v=λf where f is the frequency
that you have just found.
3) Assuming that the mass per unit length of the wire, µ, hasn’t changed then from equation [3] it should be seen that if
the tension, T, remains the same then the phase velocity should also be unchanged. Therefore with the phase velocity
constant then a prediction can be made to determine where other harmonics should be located. Using equation [1],
determine the wavelength for the 2nd and 3rd harmonics
Set the frequency to the one calculated for the 2nd harmonic. Adjust as necessary to obtain the second harmonic.
Calculate the phase velocity using the measured frequency and the wavelength.
Set the frequency to the one determined for the 3rd harmonic. Adjust as necessary to obtain the 3rd harmonic.
Calculate the phase velocity using the measured frequency and the wavelength for the 3rd harmonic.
4)
Determine the average Phase velocity from your measured data.
Calculate the mass per unit length, µ, from your average phase velocity.
Changing the Phase velocity
To change the phase velocity of the wave traveling don the wire we would have to change the Tension in the wire much
like the adjustments on a string instrument. In our case we will increase the mass by known amounts so that quantitative
results can be achieved and measured.
1)
With the 0.250 kg mass suspended change the frequency to obtain the 2nd harmonic.
Calculate the phase velocity for the tension of 2.45 N and use the value of µ determine from the previous
experiment. Record the information into Data Table 2
2)
Double the mass on the mass hanger so that 0.500kg of mass is suspended.
Examine equation [3]. If you were to double the mass one would expected the phase velocity to also increase, but
not double. Think about how much it should increase if we assume that µ remains the same.
Calculate the predicted phase velocity use that prediction to determine the frequency necessary to obtain a second
harmonic standing wave at the new phase velocity.
Set the frequency to the new frequency then make any adjustment to the frequency to obtain a second harmonic
standing wave.
3)
Quadruple the original mass so that 1kg of mass is suspended.
Calculate the predicted phase velocity with the new tension. Determine what frequency would be necessary to
obtain a second harmonic standing wave for the new phase velocity.
Set the frequency to obtain the second harmonic standing wave. Adjust the frequency as necessary to obtain a
second harmonic.
Data Sheet
L = _______ m
Harmonic
Wavelength λ
m
Data Table 1
Predicted Frequency
Hz
Measured Frequency
Hz
Phase velocity
m/s
1
2
3
Average Phase velocity ___________ m/s
µ = _________ kg/m
Compare the predicted and measured values for the 2nd harmonic frequency. If the difference in the values are greater
than 3% postulate what might be affecting the results.
Tension
N
Wavelength λ
m
Data Table 2
Predicted v
Predicted
m/s
Frequency
2.45
4.9
9.8
Does the measure frequency follow predicted results?
If not what might be affecting the results.
Measured
Frequency
Measured
Velocity