N ame ________________________________
Partner(s): ________________________________
Experiment 11: Addition of Waves
Objectives
Understand the addition of w aves using the superposition principle, through
manifestations of tw o source interference, standing w aves, and spectral analysis.
Equipment
Computer w ith LoggerPro and Labview , speakers, microphone, tubes, meter
stick
Pre-Lab
A list of A ctivities is to be completed before this Lab. Completing these Pre-Lab
assignments w ill expedite your progress during the lab period. A s part of your
preparation you are also expected to read through the lab manual before the lab.
You w ill be pressed for time during the lab. Since successful completion of all lab
activities counts tow ards your final lab grade it w ill be important to be w ell
prepared by doing Pre-Lab assignments and reading the entire lab before
attending the lab.
Points earned today
Lab
____
Challenge
____
Total
____
Physics 1200
Instructor Initials ____
XI-1
Pre-Lab for LAB#11
Complete the following before you attend class:
Under most conditions, the addition of w aves follow s the principle of linear superposition,
w hich states that the sum of tw o or more w aves at a particular point in space and at a particular
time is equal to the sum of the individual w ave amplitudes. This principle is general to all types
of w aves; in this lab, you w ill study several aspects related to the addition of sound w aves.
For example, standing w aves result from the addition of tw o w aves traveling in opposite
directions. This situation often arises w hen w aves reflect from boundaries, such as the closed or
open ends of a tube. You can visit this w ebsite for an applet that w ill help you visualize this
process:
http:/ / w w w .w alter-fendt.de/ ph14e/ stw averefl.htm
Consider a pipe that is closed at one end. Sketch the standing w ave pattern in each of the
follow ing situations; show ing the regions of high and low air pressure variations (pressure
antinodes and pressure nodes). Then formulate equations that relate the w avelength λ and
frequency f to the length L of the pipe.
a) Tube w ith one open end: fundamental.
λ1 =
f1 =
b) Tube w ith one open end: first overtone (3rd harmonic).
λ3 =
f3 =
c) Find the ratio of the first overtone and fundamental frequencies: f3/ f1 =
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d) Tube w ith both ends open: fundamental.
λ1 =
f1 =
e) Tube w ith both ends open: first overtone (2nd harmonic).
λ2 =
f2 =
f) Find the ratio betw een fundamental and first overtone frequencies: f2/ f1 =
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List of Today’s Activities
Standing waves
Problem Solving
Lab Activity
Group Work Problem
M easuring standing w ave resonances in a tube
Spectral analysis (aka Fourier analysis)
Problem Solving
D emonstration
Lab Activity
Problem Solving
Physics 1200
Discuss concept questions
M easuring the Fast Fourier Transform (FFT) of a single tone
Comparison of piano and flute tones – musical ‘timbre’
Beats - A dding w aves w ith different frequency
FFT of a snap
FFT of tube resonances
Group Work Problem
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Laboratory
Activity 1
Group Work Problem – Schumann resonances (Earth’s standing waves)
Radio w aves can reflect off a layer of the atmosphere called the ionosphere, w here a large
portion of the atoms and molecules have been ionized by solar UV. Radio w aves can also reflect
off the Earth’s surface. In effect, radio w aves bounce back and forth off the surface and
ionosphere, just as visible light can bounce betw een tw o metal mirrors. Lightning and other
natural phenomena generate radio w aves w ith a range of frequencies. Those frequencies that
are just right w ill travel around the earth, meet themselves in phase, and form standing w aves.
The set of frequencies that w ill do this are know n as Schumann resonances, in honor of
Winfried Otto Schumann (1888-1974, Germany), w ho predicted their existence in 1952.
In this exercise you w ill estimate the first 5 Schumann resonances using w hat you know
about standing w aves. The picture below illustrates the standing w ave pattern for one of the
resonances. Keep in mind that the atmosphere is really just a thin skin surrounding the Earth, so
that the w aves really circle the Earth at close to the Earth’s radius, w hich is r e ~ 6378 km.
1) H ow many w avelengths fit around the loop in
this picture?
2) What is the w avelength?
λ=
(
) units?
3) Write a general equation for the standing
w aves:
nλ =
Using this equation, fill in the follow ing table. Compare your calculated frequencies (fcalc) w ith
the observations (fobs) at:
http:/ / w w w .glcoherence.org/ monitoring-system/ earth-rhythms.html
resonance
w avelength
(km)
fcalc (H z)
fobs (H z)
1
2
3
4
5
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Activity 2
Standing waves in a tube
In this activity you w ill use a microphone to directly measure the standing w ave resonances in
an open ended tube and compare this to calculated values.
Set the microphone near the speaker, and open the LoggerPro template ‘microphone.cmbl’. This
program allow s you to directly monitor a microphone’s output as a function of time. First, run
the program by clicking on the green ‘Collect’ button. You should see the microphone’s output
on the screen as it picks up background noise from the room.
N ow open the Labview program ‘output sound.vi’, w hich w ill allow you to generate a single
tone from the laptop’s speakers. To run the program, hit Ctrl-R, or the right arrow button at the
top of the w indow . To stop, hit Ctrl-. or one of the stop buttons at the top of the w indow . You
can adjust the sound frequency by clicking on the slider control, or by using the digital
indicator. Verify that the microphone output is a clear sinusoidal signal.
Place your speaker at one end of the plastic tube, and the microphone at the other end. Adjust
the volume to be as low as possible, while still producing a clear signal on the computer.
This will minimize interference with neighboring groups (and prevent headaches). You can
adjust the horizontal and vertical axes as needed.
Vary the frequency betw een 0-1000 H z as you monitor the microphone’s output on the
computer. You can precisely adjust the frequency by using the digital indicator on the tone
generator program. A t one or more ‘resonant’ frequencies w ithin this range, standing w aves are
formed, and the microphone output should be particularly large. The process is similar to
blow ing gently over a soda bottle, w here for certain conditions, a pronounced sound is heard. If
you don’t notice any values w here the microphone output is larger, try varying the frequency
over this range more slow ly. The low est frequency resonance you can pick out is likely to be the
tube’s fundamental frequency.
Estimate how precisely you can locate the fundamental frequency, by having your partner(s)
repeat the process.
Fundamental frequency = _____________
Error estimate = ± _______
When you have precisely identified the fundamental frequency, record this value in Table 1 and
repeat this process for the next 3-4 resonances. The higher resonances may be more difficult to
identify.
To accurately predict the resonance frequencies for comparison, it is necessary to make an end
correction, w hich compensates for the fact that the vibrating air column extends slightly beyond
the tw o ends of the tube. A s a result, the tube has an effective length, Leff , given by:
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Leff = L + 2 × 0.3 × d , w here L is the measured length of the tube, and d is the diameter. Record
these values in Table 1, and calculate Leff.
N ow calculate the standing w ave frequencies for your tube (fn calc in Table 1). Use the equations
you derived in the PreLab, and take Leff as the length of the tube. Use the speed of sound that
you measured in class last w eek. Enter these values in Table 1.
TA BLE 1
Resonance #
f n (H z)
Ratio: fn / f1
n=1
f1/f1 = 1
2
f2/f1 =
3
f3/f1 =
4
f4/f1 =
5
f5/f1 =
L=
d=
fn calc (H z)
Percent error =
| fn- fn calc | / fn calc
Leff =
Compute the ratio of measured resonance frequency, fn , to the measured fundamental
frequency, f1. Does the ratio fn/f1 follow the pattern you expect from the PreLab? Why or w hy
not?
Were your measured frequencies close to your calculated values? Why or w hy not?
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H ave your instructor check your w ork before you proceed.
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Activity 3
Spectral Analysis
The human brain/ ear system is a remarkable instrument. Sounds w ith intensity spanning 12
orders of magnitude can be detected, allow ing us to hear soft w hispers one minute, and jet
engines the next. The brain/ ear system also performs spectral analysis of sounds in real-time.
This ability allow s us to pick out individual instruments in a symphony, and eavesdrop on a
conversation in a crow ded room. This activity w ill introduce you to some concepts related to
spectral analysis.
The Fast Fourier Transform (FFT): The FFT of a signal produces a spectrum, which is a plot of
amplitude versus frequency.
A ny time-domain signal f (t) can be represented as the sum of sine w aves w ith different
frequency (cosines are simply shifted sine w aves):
The more complicated the signal, the more sine w aves are needed. It is often convenient to view
the frequency spectrum of complicated signals, w hich essentially is a plot of the coefficients an,
bn as a function of frequency. The process of calculating these coefficients from f (t) is know n as
a Fourier transform. The “ fast Fourier transform” is a convenient algorithm for performing this
calculation using your computer. You w ill use Logger Pro to calculate the FFT frequency
spectrum of familiar sound w aves.
FFT of a single tone: A pure sine w ave tone has the simplest FFT, because only one sine w ave
in the Fourier series is required. Load the LoggerPro template “ FFT.cmbl” . You w ill see tw o
graphs: one is the time domain signal picked up by your microphone, the second is the
calculated FFT frequency spectrum.
Position your microphone near the speaker (the tube is not needed for this part), and set your
computer to output a tone w ith frequency betw een 100-1000 H z. Run the LoggerPro program
by clicking on the Collect button. Sketch w hat you see in the tw o panels:
Time domain
Frequency domain
Check that the FFT signal corresponds to the frequency output:
Output frequency = _________ (
Physics 1200
)
FFT peak = ______________ (
)
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Physics 1200
XI-10
Activity: Beats (adding waves with different frequency)
A ctivate the second output in the Labview program ‘output sound.vi’. You can now output tw o
sine w aves at the same time. Set one of the frequencies and adjust the other to match. A s the
tw o frequencies begin to match, you w ill hear a slow modulation in amplitude. This is called a
beat phenomenon. M easure the spectrum using the LoggerPro FFT program, and sketch the
time and frequency domain signals you observe. For clarity, the tw o frequencies should differ
by more than 4 H z. Be sure to adjust the axes in the graphs to clearly observe the beats in
both domains.
Time domain
Frequency domain
M easure the beat period in the time domain signal: T beat= __________ (
Record the frequencies:
f1 = __________ (
N ow compute the expected beat period:
),
Tcalc =
f2 __________ (
1
= ___________ (
f 2 − f1
)
)
)
Do the numbers match?
Beat phenomena provide a useful w ay to measure the frequency of unknow n signals. With the
Labview program running, have a partner set one of the output frequencies to an unknow n
value, and toggle the sw itch to hide the slider. You can now indirectly measure this frequency
by tuning the frequency of the first output until you hear very slow beats. H ow close did you
get?
M atching frequency = __________
(
)
‘Unknow n’ frequency = _________ (
)
Sw itch roles w ith your partners until everyone has had a chance. This is an example of
“ heterodyne” detection, w hich is w idely used to detect high frequency signals in electronics.
M usical ‘timbre’ (comparing a piano and a flute): H ow do w e tell the difference betw een a
piano and a flute? M usical notes played on real instruments are not pure sine w ave tones like
you’ve been using today: they also contain additional harmonics. You w ill find tw o mp3 files on
your desktop that contain examples of the same note played on a piano and on a flute.
Physics 1200
XI-11
To compare the spectrum of each instrument using the LoggerPro FFT program, you first need
to activate the trigger function by clicking on the “ Data Collection” button just to the left of the
green Collect button. On the Collection tab, set the length to 0.06 s, and on the Triggering tab,
enable triggering on a sensor value that is ~ 0.03 au larger than the background noise level
picked up by the microphone (this number appears in the bottom left corner of the w indow ).
A fter setting these values, exit the dialogue box. N ow w hen you click the green Collect button,
the program w ill w ait for a trigger.
Play an mp3 file to trigger the data collection, and sketch the FFT spectra from 100-2kH z:
Piano
Flute
What is different betw een the tw o sets of spectra?
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Which note is being played? (you may w ant to refer to Lab 10)
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N ote = __________
XI-13
FFT of a complicated signal: N ow trigger data collection w ith a sharp impulse of sound, such
as that produced w hen you snap your fingers or clap your hands. M any more sine w ave
components are present in such a complicated signal. A s a result, the FFT show s a spectrum
that extends over the w hole range of human hearing (and beyond!).
Sketch w hat you see in the tw o panels from 100-20kH z:
Time domain
Frequency domain
Activity: FFT of tube resonances
N ow that you have successfully measured the standing w ave resonances of a tube the hard
w ay, you can use the FFT to do the same measurement much more quickly. A s you’ve seen, a
continuum of sine w aves is generated w hen you snap your fingers. Frequencies w ithin this
continuum that correspond to standing w ave resonances w ill be preferentially transmitted
through the tube.
Position your microphone at one end of your tube. Trigger data collection by snapping your
fingers (or clapping your hands) at the other end of the tube. The FFT should show a series of
peaks at the tube’s resonant frequencies. You may need to repeat this a few times to get clear
peaks in the FFT. Record the frequencies fn FFT in Table 2, and compare these values w ith those
you measured (fn) and calculated (fn calc) earlier (Table 1).
Table 2
Resonance #
f n FFT (H z)
f n (H z)
fn calc (H z)
n=1
2
3
4
5
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Do your values agree? If not, speculate on w hat could cause any discrepancies.
This is an example of Fourier Transform spectroscopy, w hich is w idely used in modern infrared
spectrometers (FT-IR) and nuclear magnetic resonance (N M R and M RI). These techniques have
applications throughout the physical and life sciences.
End of Lab 11
When you are finished, close both LoggerPro and Labview. D o not save any changes.
H andy links:
Fourier synthesis applet: http:/ / w w w .falstad.com/ fourier/
End corrections in w ind instruments: http:/ / w w w .phys.unsw .edu.au/ jw / musFA Q.html#end
Flute acoustics: http:/ / w w w .phys.unsw .edu.au/ jw / fluteacoustics.html
Basics of digital filtering: http:/ / w w w .falstad.com/ dfilter/
Physics 1200
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