Physics 213
Lab 3
The Doppler Shift
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PARTNERS:
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DATE:
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EMAIL ADDRESS:
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Physics 213
Lab 3
Software List
Science Workshop
Microsoft Excel
Equipment List
Science Workshop Interface
Sound Source with power supply
Sound Detector
Long Dynamics Track
Dynamics Cart
2 photogates
Reflective surface
Physics 213
Pre-Lab 3
The Doppler Shift
Name:__________________________
Section:_____
Date:__________
(Read this & answer the questions before coming to lab)
Summary of relevant concepts:
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

When a sound source producing waves at a frequency f and an observer move
relative to one another, the frequency of sound f’ measured by the observer
depends on the details of the relative motion; this change in frequency is known
as the Doppler Shift.
The Doppler shift equations shown below work for any wave moving in a
medium, when the motions of the source and observer are along the line joining
them. In the equations below, v, vo , and vs ,are the speeds of sound, observer and
source relative to the medium; if the medium (e.g. air) is moving, vo and vs
should be measured with respect to the medium.
If the observer is moving and the source is stationary, the measured frequency is:
(v  vO )
Eq. (1)
f   f
v
where the upper sign corresponds to an approaching observer and the lower sign
corresponds to a receding observer.

If the source is moving and the observer is stationary, the measured frequency is:
f  f
v
(v
vS )
,
Eq. (2)
where the upper sign corresponds to the source approaching and the lower sign
corresponds to the source receding from the observer.

More generally, where both the source and observer are moving:
f   f
(v  vO )
(v v S )
Eq. (3)
Note that the signs in the numerator and denominator are independent of each
other. Use the following general rules for the signs: in the numerator, the upper
sign is used if the observer is moving towards the source and the lower sign if
moving away from the source; in the denominator, the upper sign is used if the
source is moving towards the observer and the lower sign if moving away.
Prelab Questions:
Q1. A source of sound generating a tone of frequency f is moving towards a planar
reflecting surface at a constant speed vS. Assume that the motion of the source is
along the perpendicular to the planar surface. If the speed of sound in still air is v,
explain how to calculate the frequency of the sound waves reflected off the planar
surface.
Q2. In Q1, suppose an observer is riding along with source. Determine the frequency of
the reflected sound waves that she would measure. Explain clearly the reasoning
underlying your calculation.
Phys 213
Lab 3: The Doppler Effect
Activity 1: Determining the Frequencies Produced by a Stationary
Source.
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
In this activity, your instructor will produce a stationary tone of known frequency;
your task is to determine the frequency of the tone using a sound sensor and a Fast
Fourier Transform (FFT) analysis of the measured data. An introduction to
Fourier analysis and the basic concepts behind the FFT are given in Appendix C.
This measurement technique will be a key feature of the Doppler shift
measurement in the remaining activities.
Set up the equipment as shown in Figure 1. Install the sound sensor probe on
Science Workshop and create a display of sound amplitude versus time.
Figure 1

Click the record button while the sound source is turned on and remains
stationary. Look at a graph of the sound amplitude vs. time. The amplitude
should remain relatively constant. This is related to the loudness of the sound.
The sound amplitude should remain constant throughout this part of the
experiment as the relative positions of the sound source and sound sensor remain
fixed. A table should be constructed to display the collected data.

Adjust the display to show 5 places to the right of the decimal by using the button
marked 0.0 in a box in Science Workshop.

Open the Excel Fast Fourier template and insert your collected data into the
labeled boxes provided. (The number of data points transferred must be equal to
2n = 2, 4, 8, … 4096 . Choose the largest number of points to transfer.) Use the
Fast Fourier Transform macro button constructed within the Excel spreadsheet
linked to the template to determine the frequency of the sound source for the data
sets taken above. If the Fourier Transform program does not work first, you may
need to install it: On the Excel spreadsheet, click on TOOLS, then ADD-INS and
then ANALYSIS TOOL PACK.

Record the values for the frequencies obtained in the template provided for this
laboratory activity. Insert a properly formatted and sized copy of the Fourier
Decomposition graph that was obtained. Once all of the data have been
transferred to the lab template, you may clear the Excel spreadsheet using the
button provided.
Activity 2: Measuring the Doppler Shift
In our experiment, a sound source and a sensor are mounted on a cart that moves relative
to a reflective surface. As we saw in the pre-lab, the detected Doppler shift is then due to
the combined motion of the source in one case and the observer in the other. The speeds
involved, vo=vs , are the same; namely the speed of the cart. We will assume the speed of
sound in air to be 334 m/s. Then the only unknown speed is that of the cart. We will
measure the cart speed directly using photogate timers set a known distance apart,
measure the shifted frequency using a sound sensor, and apply the Fast Fourier
Transform provided in Excel to determine the shifted frequency.
2A: Measuring the Doppler shift for relative motion away from observer.
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Set the equipment up as shown in the photograph (Figure 2).
Figure 2
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Connect the sound sensor and photogates to the Science Workshop interface box.
Under Sampling Options..., set the sampling rate to 10000 Hz and then set the
start and stop conditions as follows: For a start condition, choose Channel, then
Digital 1, then Low (blocked gate); for a stop condition, choose Channel, then
Digital 2, then Low (blocked gate). Place photogate #2 as near to the sound
sensor as possible. Make certain that the cart trips the photogate before the cart
hits the bumper. Place photogate #1 a convenient distance away from photogate
#2. Move the cart through each of the photogates and take note of the cart
position when each of the gates are tripped. The difference in the two positions is
how far the car travels during the timed period.

Turn on the sound source. Click the record button while the cart remains
stationary. Verify that the frequency is not changing by using the fast Fourier
transform tool built into Data Studio for Science Workshop. You can move the
small window around to select different subsets of the data. Note this base
frequency on your template. It should have a value of approximately 3000 Hz.

You are now ready to record data for the Doppler Shift. Click record, to reset the
master photogate timer. Again, adjust the table display to show 5 places to the
right of the decimal by using the button marked 0.0. Slide the cart away from the
reflective surface to record moving data. Take note of the maximum time
recorded in your data table: it will allow you (along with the distance between
photogates) to calculate the speed of the cart. Label the data files clearly.

Open the Excel Fast Fourier spreadsheet and insert your collected data into the
labeled boxes provided. (The number of data points transferred must be equal to
2n = 2, 4, 8, … 4096 . Choose the largest number of points to transfer.) Use the
Fast Fourier Transform macro button constructed within the Excel template to
determine the frequency of the sound source for the data set taken above.
Transfer the obtained frequencies obtained and graph to your template and clear
the spreadsheet using the clear and reset button provided.
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For the data taken with the cart at rest, record the stationary frequency in the
template provided.
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For the data taken with the cart moving away from the reflective surface, record
the stationary frequency, the moving frequency, the speed and the expected
moving frequency. Enter these values in the activity template provided.
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How do the actual and expected shifted frequencies compare? Is the shift in the
correct direction?
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Comment on possible sources of error and any experimental difficulties you may
have encountered.
2B: Measuring the Doppler Shift for relative motion toward observer.

Repeat the experiment that you just completed -- expect this time, move the cart
towards the reflective surface. Record the time from the master photogate timer: it
will allow you to calculate the speed of the cart. Label the data files clearly.

Open the Excel Fast Fourier spreadsheet and insert your collected data into the
labeled boxes provided. (The number of data points transferred must be equal to
2n = 2, 4, 8, … 4096 . Choose the largest number of points to transfer.) Use the
Fast Fourier Transform macro button constructed within the Excel template to
determine the frequency of the sound source for the data set taken above.
Transfer the obtained frequencies obtained and graph to your template and clear
the spreadsheet using the clear and reset button provided.

For the data taken with the cart at rest, record the stationary frequency in the
template provided.

For the data taken with the cart moving toward the reflective surface, record the
stationary frequency, the moving frequency, the measured speed of the cart and
the expected moving frequency. Enter these values in the template provided.

Is the frequency shift in the correct direction?

Calculate the expected speed of the cart. How do the actual and expected speeds
compare?
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Physics 213
Post-Lab 3
The Doppler Shift
Name:__________________________
Section:_____
Date:__________
Q1. Use the binomial expansion theorem to show that the two equations describing the
Doppler shift when the source is fixed and the observer is moving or vice-versa (eqs.
1 and 2 in the prelab) are identical when the speed of the source or the observer is
small compared with the speed of sound in the medium. Use the assumption that u is
the speed of both the source and the observer (vs = vo = u).
[Note: The binomial expansion theorem is:
nx n( n  1) 2
(1  x ) n  1 

x  ........ (for x 2  1) ]
1!
2!
Q2. One estimate that we could use in our experiment is that as the cart approached the
reflecting surface the Doppler shift detected from the reflecting surface would be
twice the shift expected if we had placed the “observer” on the reflecting surface.
Show that this approximation is valid given your proof in Part A. What would be
the expected frequency using this estimation and the speed of the cart in Activity 2b?
Q3. Determine the difference in the shifted frequency expected using estimate in Part B
and the frequency predicted by Equation 3. Express the difference in hertz and also
as a percentage difference.