Name: ___________________
PHYSICS 119 – SPRING 2008
EXPERIMENT 3: WAVES, PHASE DIFFERENCE, SPEED OF SOUND
S.N.: ____________________
SECTION:
Experiment 3
PARTNER:
DATE:
WAVES, PHASE DIFFERENCE, SPEED OF SOUND
Objectives
•
•
•
•
•
Study traveling waves using sound: wavelength, period, frequency, and phase velocity.
Set up oscilloscope to observe signals from a microphone and from a signal generator.
Determine frequency of sinusoidal signals on an oscilloscope screen.
Determine the phase difference between two sinusoidal signals on an oscilloscope screen.
Measure the speed of sound by observing signals from microphones at different distances from an audio speaker.
Traveling waves
Traveling waves (some examples: sound waves, water waves, waves on a string, light) are periodic variations of
some physical quantity (pressure for sound, displacement for water and string, electric field for light) for which the
quantity moves some distance x in time t. Standing waves on a string were studied in Lab 2.
Traveling waves are characterized by a period T, the time that it takes a particular pattern of oscillation to repeat
itself at a fixed value of x, a frequency f, the rate at which the oscillation takes place (f = 1/T), the wavelength λ,
the distance in the direction x that it takes the oscillation to repeat itself for fixed time t, the phase velocity υ,
which is the speed at which any portion of the oscillation pattern propagates and the amplitude A, which is the
magnitude of the physical quantity that is traveling. Sinusoidal waves are waves in which the physical quantity
that varies in space and time can be described by a sine function of position x and time t. For example, the
displacement y of a sinusoidal wave on a string that is traveling in the +x direction with velocity υ, period T, and
amplitude A can be represented by
2( #
& 2(
y = A sin $
x'
t ! . For these sinusoidal waves, the phase difference
T "
% )
between two waves of the same frequency is the difference in radians between two points of the same amplitude
and slope. One wavelength is 2π radians. Two waves that are “in phase” (phase difference of 0 radians) have their
crests at the same position at the same time. The “phase velocity” υ for these sinusoidal waves is the speed at
which a crest moves.
As a preliminary exercise, use the transparencies in the envelope by your lab station to construct a) two waves “in
phase”, b) two waves 90 degrees (π/2) out of phase, and c) two waves 180 degrees (π) out of phase. Draw the
three examples in the space below. The horizontal axis represents either distance x or time t.
in phase a)
→(x or t)
π/2 out of phase b)
→(x or t)
π out of phase c)
→(x or t)
119LAB3-PHASE.DOC 2008-03-21
3-1
PHYSICS 119 – SPRING 2008
EXPERIMENT 3: WAVES, PHASE DIFFERENCE, SPEED OF SOUND
Sound Waves
A speaker converts an electrical signal into an audible sound wave and a microphone converts the audible sound
wave into an electrical signal. A pure audio tone (a tone with one constant frequency) can be heard by connecting
a function generator (set to produce sine waves) to a speaker. When a microphone picks up this sound, it
converts it into an electrical signal that can be observed as a sine curve on an oscilloscope. More complex
sounds, such as your voice, will appear on the oscilloscope as a more complex wave.
For most of this lab, we will be dealing with an especially simple form of sound wave: traveling, periodic waves
that are almost perfectly sinusoidal. The pattern you see on the oscilloscope keeps repeating itself. It shows, as a
function of time, the variation of the air pressure at a microphone. The parameters for such waves, wavelength λ
frequency f, and speed υ. are related by the simple equation υ. = λf.
Experimental Procedure
To familiarize yourself with the oscilloscope, work through the three page tutorial provided in the lab.
When you have finished, you will be prepared to observe the sound waves created by your voice.
The microphones and amplifiers used in this experiment are sensitive and you will probably experience
interference from other speakers, other students and moving chairs. To minimize this interference, please talk as
quietly as possible and do not scoot your chairs across the floor.
Connect one microphone to the Ch 1 input of the scope. Turn the Ch 1 VOLTS/DIV knob clockwise to the 5mV
setting (shown at the bottom left of the scope display screen) and turn the SEC/DIV knob to the 5 ms setting
(shown next to “M” below the scope display screen). With these settings, you should be able to observe your
“voice pattern” as you speak into the microphone.
1.
Describe the differences in the pattern when you speak into the microphone softly and not so softly?
2.
Explain what you understand by the term “pitch” of a sound? Describe the differences in the pattern do you notice when you
whistle or hum at a high pitch compared to a low pitch?
Use a tuning fork to obtain a sine curve on the oscilloscope. Hit the tuning fork gently with the rubber hammer, in the plane
of the tuning fork near the top. If you hit it too hard or use the table instead of the rubber hammer, you may hear a very
high-pitched sound. This is the result of higher harmonics of the fundamental frequency, not the note you are looking for.
These forks, when properly “bumped”, produce accurate frequencies in the “middle” of the piano range. The frequencies in
CPS (cycles per second) are stamped on the side of the tuning forks. The name CPS has been replaced by Hz, an
-1
abbreviation for Hertz. The units of Hz are s .
After striking the tuning fork, hold it close the microphone and you should see a decaying oscillation on the scope. When
the oscillation on the scope “looks good”, press the RUN/STOP button to “freeze the data”. You may want to try this
several times until you get a good trace on the scope. (Press the RUN/STOP button again to take new data.) You should
try to get a trace on the scope that looks like a sine wave with 5-10 cycles of oscillations. You may need to adjust the
SEC/DIV knob to change the time base to get 5-10 oscillations.
We want to use the scope trace to measure the frequency of the sound wave produced by the tuning fork and to compare
our measurement to the “stamp” on the tuning fork. To do this, we need to keep track of our measurement errors. We will
assume that the time base of the scope is accurate to 3% (as is true for other Tektronix scopes).
119LAB3-PHASE.DOC 2008-03-21
3-2
PHYSICS 119 – SPRING 2008
EXPERIMENT 3: WAVES, PHASE DIFFERENCE, SPEED OF SOUND
3.
Record the setting of the oscilloscope time base and the time base units. In the ( ), enter 3% of the time base as its
uncertainty.
time base setting =
(
)
4.
Determine the time interval between N full oscillations of the sine curve. N should be an integer between 5 and 10. You can
use the scope to make this measurement: Press the CURSOR Main Menu button and press the first button to the right of the
on-screen Menu Option TYPE until TIME is displayed. Two vertical dotted lines (cursor lines) will appear on the scope. You
can move the left vertical cursor line with the Ch 1 POSITION knob. Move it to a point where the rising slope of the sine
wave passes through 0. Now use the Ch 2 POSITION knob to move the right cursor line to a point where the rising slope of
the sine wave crosses through 0 five to ten full oscillations later. The times corresponding to the cursor lines are displayed in
the bottom two entries of the Menu Option field. (Time “0” is shown as a small arrow at the top of the display screen.) Delta
is magnitude of the difference between these cursor times. How does this interval compare to the time between N adjacent
maxima? Explain.
Record your results below.
number N of full oscillations
time for N full oscillations
s
We now need to assign an uncertainty to our measurement. There are two sources of uncertainty: the 3% uncertainty of our
time base and the uncertainty in placing the cursors. Multiply the “time for N oscillations” you recorded above by 3% and enter the
result as the uncertainty from the time base below. We will ignore the uncertainty from placing the cursors to save time – in any
case we could reduce the cursor uncertainty by taking several measurements and averaging the results.
Uncertainty from time base
s
5.
What is the name commonly used to describe the time interval between maxima? Calculate this time interval and its
uncertainty from your data in 4. Show your work.
6.
Determine the frequency of the sine wave and the uncertainty in the frequency. Show your work. Compare this with the
value stamped on the tuning fork, ffork, below. To determine δf, the uncertainty in the frequency, we need to know how much
the frequency changes if the period changes by its uncertainty. A simple way to estimate δf is to take
δf = 1/period – 1/(period + δ period) where δ period is the uncertainty you found for the period.
f = 1/period =
Hz
ffork =
119LAB3-PHASE.DOC 2008-03-21
±
δf =
Hz
Hz
3-3
PHYSICS 119 – SPRING 2008
EXPERIMENT 3: WAVES, PHASE DIFFERENCE, SPEED OF SOUND
Velocity of Sound
In the following steps, you will use a signal generator connected to a speaker. The steady tone coming from the
speaker can be annoying. Please unplug your speaker or turn the function generator output down except when
making observations.
Figure 2-1. Arrangement for determining the speed of sound.
Connect the two microphones, the speaker, the oscilloscope and the function generator as shown in Fig. 2.1. Place the
speaker near one end of the railing. Note the position of the speaker by reading the railing scale through the hole in the
mounting. Place mike #1 15.0 cm away as read on the scale. Before turning on the function generator, check to see that
the AMPLITUDE control is turned fully counterclockwise to the MIN position. [Note: The function generator is capable of
putting out sufficient power to destroy the speaker. Therefore, you should always turn the AMPLITUDE control to MIN
before turning on the function generator and return it to MIN when you turn it off]. Set the frequency to 2000.0 Hz. Increase
the function generator output until an audible (but not too loud) signal is heard from the speaker.
Set the Ch 1 and Ch 2 VOLTS/DIV settings on the scope to 5 mV, and the time base (SEC/DIV) to 250 µs. Press the
RUN/STOP button to enter data taking mode and you should see two traces on the scope. For proper triggering, press the
TRIG MENU button and press the button to the right of the “SOURCE” Menu Option until “EXT” is displayed in the
SOURCE box. This tells the scope to trigger from its external input. For the rest of this lab session, keep the
oscilloscope triggered on the external input from the signal generator. Also check that the TYPE Menu Option
displays EDGE.
The signals may look fuzzy. This is not a problem because the scope can average many sweeps to reduce the noise. To
do this, press the ACQUIRE Main Menu button, and press the button to the right of the AVERAGE Menu Option to select
averaging. The number of sweeps to average is the bottom box of the Menu Option field, and a choice of 16 works well.
7.
Sketch the pattern seen on the oscilloscope for two different signal generator frequencies, f1 = 2000 Hz and f2 = 3000 Hz.
Record the time base setting (SEC/DIV) and the vertical sensitivity (VOLTS/DIV) for CH1.
SEC/DIV =
119LAB3-PHASE.DOC 2008-03-21
CH1 VOLTS/DIV =
3-4
PHYSICS 119 – SPRING 2008
EXPERIMENT 3: WAVES, PHASE DIFFERENCE, SPEED OF SOUND
8.
Which of the following can you not determine from the oscilloscope pattern and knob settings alone: wavelength, period,
frequency, and speed? Explain.
Set the frequency to 2000.0 Hz. Move mike #1 30cm from the speaker, then hold mike #2 alongside mike #1, the same
distance from the speaker. Press the MEASURE Main Menu button and use the Ch 2 POSITION control knob to move
the Ch 2 trace above or below the Ch 1 trace.
9.
Sketch the pattern you see below. Label your traces Ch 1 and Ch 2.
SEC/DIV =
CH1 VOLTS/DIV =
CH2 VOLTS/DIV =
10. What is the phase difference (radians) between the two signals? Explain.
Move mike #1 to a position 15cm from the speaker. Place mike #2 10cm behind mike #1 and observe the pattern of the
two traces. [Note that, although the mikes are not exactly at the position read from the scale, the separation between them
is the difference in their scale readings.]
11. Sketch the pattern you see on the next page. Label the horizontal axis with the sweep rate (sec/div).
119LAB3-PHASE.DOC 2008-03-21
3-5
PHYSICS 119 – SPRING 2008
EXPERIMENT 3: WAVES, PHASE DIFFERENCE, SPEED OF SOUND
12. What is the phase difference (radians) between the two signals? Explain your reasoning..
13. For the following experiment, we will attempt to measure the wavelength of the sound wave. We will measure and record the
positions of Mike #2 where the Mike #2 scope trace is 1800 out of phase with the Mike #1 trace. We will repeat the
measurement several times to get an estimate of the measurement error.
Start by positioning Mike #1 about 15 cm from the speaker. (Read the position from the scale on the rails – there is a hole in
the mike mount that makes the scale visible.) Record the position of Mike #1 under Trial #1, Mike 1 Pos. in the Table below.
With Mike #2 close to Mike #1, slowly move Mike #2 away from Mike #1 until the phase of the Mike #2 scope trace is 1800
out of phase with the Mike #1 scope trace. Record the position of Mike #2 under Trial #1, Mike 2 Pos. a in the Table below.
Now move Mike #2 even further from Mike #1 until again the scope traces are 1800 out of phase. Record this Mike #2
position underTrial #1 Mike 2 Pos. b in the Table. Move Mike #2 back even further until again the scope traces are 1800 out
of phase and record this as Mike 2 Pos. c.
Now move Mike #1 about 2 cm closer to the speaker, record its position under Trial #2. Find and record the first 3 positions
of Mike #2 where its trace is 1800 out of phase with the Mike #1 trace, as you did above, to fill in the Trial #2 line of the Table.
Finally, move Mike #1 about 4 cm further away from the speaker (about 17 cm total from the speaker), and record its position
under Trial #3. Find the 3 positions of Mike #2 where the signals are out of phase to fill in the Trial #3 line of the Table.
Complete the Table by recording the difference between Mike #2 Pos. a and Pos. b, and Pos. b minus Pos. c for each trial.
These last 2 columns provide you with 6 measurements of the wavelength of the sound wave.
Trial #
Mike 1 Pos.
Mike 2 Pos. a
Mike 2 Pos. b
1
2
3
119LAB3-PHASE.DOC 2008-03-21
3-6
Mike 2 Pos. c
Pos. a – Pos. b
Pos. b – Pos. c
PHYSICS 119 – SPRING 2008
EXPERIMENT 3: WAVES, PHASE DIFFERENCE, SPEED OF SOUND
14. From the Table above, determine the wavelength of the sound wave. Because your 6 measurements don’t all agree, your best
estimate for the wavelength will be the average of the 6 measurements. What is the uncertainty? There is a formal procedure
for determining the uncertainty, δ, from a collection of measurements (that involves calculating the “standard deviation). Here
N
1
( xi ! x ) 2
it is: # =
where N=6 is the number of measurements, xi is an individual measurement
"
N ( N ! 1) i =1
and x is the average value of the measurements. Compute δ from the expression above (remember to take the square root of
2
δ2). Does your uncertainty seem reasonable?
λ=
±
m
15. Calculate the speed of sound from your value of the wavelength in 16 and from another known quantity.
Show your work. (For the uncertainty, you may ignore the uncertainty in the other known quantity.)
υ=
±
m/s
16. Record the temperature and barometric pressure in the lab. Calculate the absolute temperature corresponding to this
temperature. Roughly estimate the temperature and pressure uncertainties from the number of significant figures.
T=
±
˚C
T=
17. Calculate the speed of sound using the formula
±
K
! = 332.13
P0 =
±
Pa
T 101325
m /s . T is the temperature in degrees
273.15 P0
Kelvin and P0 is the atmospheric pressure in Pascals. Calculate the uncertainty in the speed. Do this by separately
recalculating the speed for the temperature increased by its uncertainty, then for the pressure increased by its uncertainty, in
each case subtracting the original speed to get two uncertainty contributions to the speed (“propagate the errors”!) Now take
the square root of the sum of the squares of these two uncertainties to get the total speed uncertainty.
Show your work.
υ=
±
m/s
18. Compare the measured speed from 17 with the calculated speed from 19. Comment.
119LAB3-PHASE.DOC 2008-03-21
3-7