chapter3 experiment1: sound - northwestern...

Chapter 3

Experiment 1:Sound

3.1 Introduction

Sound is classified under the topic of mechanical waves. A mechanical wave is a term whichrefers to a displacement of elements in a medium from their equilibrium state; but to bea wave this displacement must then propagate through the medium. The speed at whichthe wave propagates is inversely related to the mass density of the propagating medium anddirectly related to the forces attempting to restore the equilibrium condition.

A mechanical wave can propagate through any state of matter: solid, liquid, and gas.Mechanical waves can be of two types: transverse or longitudinal. A transverse wave ischaracterized by a displacement from equilibrium which takes place at right angles to thedirection the wave propagates; longitudinal waves have the displacement from equilibriumalong the axis of propagation.

Since two directions are perpendicular to the direction of propagation, transverse waveshave two independent polarization directions. The form of the equations describing thesetwo types of waves is very similar. However, transverse waves can only exist in solidmedia, where intermolecular bonds prevent molecules from sliding past one another easily.Such sliding motion is called shear. Solids support shear forces and will spring back ratherthan continue to slide; this intermolecular connection will transmit the transverse wave frommolecule to molecule.

Longitudinal waves rely only on pressure and can exist in both solids and fluids. Theydepend on the compressibility of the media. Solids and fluids all show a resistance to com-pression. Sound waves are longitudinal waves that are transmitted as a result of compressiondisplacement of molecules of the medium. We usually discuss sound in air, but sound travelsin everything except empty space. A sound wave can be generated in solids, liquids, or gassesand can continue to propagate in a different medium.

The equation used to describe a simple sinusoidal function that propagates in space isgiven by

Y(x, t) = A0 sin[k(x− vt)

]p (3.1)

25

CHAPTER 3: EXPERIMENT 1

where Y is the time and position dependent displacement of the media from equilibrium,A0 is the maximum displacement or amplitude of the medium’s motion, v is the velocity ofthe wave which depends on the characteristics of the media. This particular wave travelsalong the x-axis. . .x must increase at speed v to keep up with vt. p is the polarization ofthe wave. The case a longitudinal wave has p = x and a transverse wave has p = pyy + pzzsome combination of y and/or z polarization. k is a constant that is determined by both thespeed of the wave and the frequency of the wave. The constant k is usually expressed as

k = 2πλ, (3.2)

where λ is the wavelength. The wavelength is related to the wave velocity v and the wavefrequency, f , by the expression

v = λf. (3.3)

A periodic mechanical wave is characterized by a frequency of oscillation, f , which isdetermined by the source of vibration motion that creates the disturbance. Thus, thefrequency and the speed of the wave in the media determines the wavelength. The sourcecan choose to oscillate at any frequency it chooses, but the medium decides the velocity ofpropagation.

CheckpointWhat is the difference between a displacement wave and a pressure wave?

CheckpointIs sound a displacement wave, a pressure wave, or may it be considered as both?

Equation (3.1) describes the oscillations of particles with equilibrium position x. Theseequations describe either longitudinal or transverse waves. The difference lies in the inter-pretation of the displacement which is described in the equation. For a transverse wave,Equation (3.1) describes oscillations of the y and/or z coordinates of the particles at x.For transverse waves the actual wave looks very similar to the plot of the displacement andis easily visualized. For a longitudinal wave, Equation (3.1) describes oscillations of the xcoordinate of the particles at x in equilibrium. This results in a sinusoidal variation in thedensity of media along the axis of propagation. This generally is much harder to visualize,and there are few natural examples that can be easily observed. One such example wouldbe the pulse of compression which can be generated in a slinky spring.

A sound wave is a longitudinal wave and since the displacement of the wave causes avariation in the density of air molecules along the direction of the wave, it can be viewed aseither a displacement wave or a pressure wave. The above equation may be used to describeeither picture. The displacement maximum is usually 90 degrees out of phase with the

26


(a)

(b)

(c)

Pressure Wave

Displacement WaveFigure 3.1: An illustration of the two representations of a longitudinal wave. Thedisplacement representation is the position of particles with respect to their average positions,but the pressure increases as the particles move toward each other (compression) anddecreases as the particles move away from each other (rarefaction). The displacement waveleads the pressure wave by 90◦.

pressure maximum as shown in Figure 3.1. A sound wave is shown with both displacementand pressure representations. The picture represents the density of the medium as the wavepasses through it.

CheckpointWhat determines the pitch of a sound wave? The source, the medium? Whichdetermines the speed of sound, the sound generator, the medium, or both? Whichdetermines the wavelength of sound, the sound generator, the medium, or both?

3.1.1 Superposition

When two sound waves happen to propagate into the same region of a medium, the instan-taneous displacement of the molecules of the medium is normally the algebraic sum of thedisplacements of the two waves as they overlap. If at one time and place each individualwave would happen to be at a maximum amplitude, say Y1max and Y2max the net result wouldbe a displacement of the medium at a value equal to the sum of Y1max and Y2max . This is

27


(a)

(b)

(c)

Y1

Y2

Y1 +

Y2

Figure 3.2: An illustration of constructive interference. Y1 and Y2 are in phase at all timesso that their sum has amplitude equal to the sum of Y1’s and Y2’s amplitudes.

shown in Figure 3.2. If on the other hand, the second wave were at Y ′2max = −Y2max , thenet displacement equal to the sum of Y1max + Y ′2max , which in effect would be the differenceY1max − Y2max or zero if the amplitudes are equal, as shown in Figure 3.3.

(a)

(b)

(c)

Y1

Y' 2

Y1 +

Y' 2

Figure 3.3: An illustration of destructive interference. Y1 and Y2’ are out of phase by 180◦or half a wavelength. The sum of the two waves is zero if the two amplitudes are equal.

CheckpointWhat happens when two sound waves overlap in a region of space?

3.1.2 Reflection

We most often think of a reflection as occurring when a wave encounters the border of themedium in which it is traveling. Anytime a wave encounters a sharp change in wave velocity,due to a change in the nature of the medium, a reflection is generated and some or all ofthe energy of the wave is redirected to the reflected wave. The amplitude and phase of thereflected wave is determined by the boundary conditions at the point of reflection.

28


In this lab, we will consider the effects of reflection from a solid boundary, such that theboundary condition requires that the sum of the waves have a displacement of zero at thepoint of reflection. Air molecules cannot be displaced from equilibrium at the wall. Theycannot move into the wall and atmospheric pressure presses them into the wall; they simplyhave nowhere to go. This condition can only exist if we were to superpose a second wavemoving in the opposite direction with exactly the same amplitude, and 180 degrees out ofphase with the original wave. Hence, in order to satisfy the boundary condition, a reflectionwave is generated with exactly these properties.

-1.03

-0.53

-0.03

0.47

0.97

-1.03

-0.53

-0.03

0.47

0.97

(a)

(b)

(c)

(d)AN

NAN

NAN

NAN

N

ANN

ANN

ANN

ANN

Displacement

Pressure

Wall

Figure 3.4: Illustrations of pressure waves and displace-ment waves reflected at a wall. The incident wave andreflected waves interfere to produce a series of nodes (N) andanti-nodes (AN) spaced every half wavelength. (a), (b), and(c) are pressure standing waves and (d) is the displacement.The pressure in (b) becomes the pressure in (c) after the gasmoves like the arrows between indicate. The gas ‘sloshes’back and forth between the dotted lines, but on average doesnot cross them.

The resulting superposi-tion of incident and reflectedwaves in the region in front ofthe boundary also sets up asecond null area where the am-plitudes cancel at a distanceof one half of a wavelengthfrom the boundary as shownin Figure 3.4. The null ar-eas are called nodes. If thewave didn’t loose amplitudeas it traveled, a null wouldbe present at successive halfwavelength intervals over theentire region. As it is, thewave looses amplitude as itpropagates, and the cancella-tion is only partial.

The same boundary doesnot place such restrictions onthe pressure wave. The pres-sure at the boundary mayrise and fall, as is required.The wall can easily supportwhatever pressure results fromthe superposition of any twowaves. A suitable pressurewave which takes advantageof the boundary restriction(which is none) is directed inthe opposite direction with anequal amplitude to conserveenergy, and is in phase withthe incident wave. The result-ing superposed waves show amaximum or anti-node at the

29


boundary (see Figure 3.4) and a null point or node at a distance of one quarter wavelengthaway from the boundary. If the amplitude is sustained as the wave travels, a second nullappears a half wavelength from the first, or at a point three quarters of a wavelength from theboundary. Between each node, the wave is seen to oscillate between maximum positive andmaximum negative amplitudes, where the maximum amplitude is the sum of the maximumamplitudes of the waves considered separately. These areas are called anti-nodes. Such awave is referred to as a standing wave because it stands still. In either case, successive nullpoints or nodes occur at intervals of half of the wavelength of the traveling waves.

By measuring the distance between nodes of the standing wave, we can determine thewavelength of the incident and reflected traveling waves. Since we also know the frequencyat which we excited the wave, we can find the speed of the wave.

3.2 Measuring the Speed of Sound in Air

From personal experience one can get a sense that the speed of sound in air is rapid. Younotice no delay in hearing a word that is spoken by a person nearby and the movement of thespeaker’s mouth. That would be quite a distraction, like watching a movie with the soundtrack out of synch! And yet, when you sit in the outfield bleachers at a baseball game, youcan sense a noticeable delay between the arrival of the light showing the ball being hit andthe sound of the crack of the bat on the baseball.

The speed of sound is noticeably slower than the speed of light over distances the sizeof a baseball field. In principle one could measure the speed of sound by timing how longafter ones sees the ball hit that the sound arrives if one knew how far away they were fromhome plate. Instead, we will employ an oscilloscope simulation to observe the very shorttime delay as sound travels a distance on the order of a meter.

3.2.1 The Speed of a Sound Pulse

In this experiment, a signal generator is used to produce a repeating electrical pulse to drivea speaker. The pulse causes the speaker to emit a ‘click’ or pulse of sound whose speedwe will measure. A small microphone is used as a sensor. Its output is connected to oneinput of the oscilloscope. The wave generator signal is also fed directly into the oscilloscope.This signal will serve as a time reference against which to compare the microphone signal.The delay introduced due to the distance in air that the sound travels from the speaker tothe microphone could be used to measure the speed of the click. Are the delays introducedto the process of converting the electrical signal to mechanical sound and back, and in thetravel of electrical signals through the wire negligible? They probably are; however, we donot have an exact location for where the sound is produced in the speaker or sensed in themicrophone. This could be a problem which we must deal with.

A sound wave will be sent down a tube and will be reflected off a piston head back to amicrophone. We will measure the speed of the sound wave by observing the amount of time

30


Figure 3.5: A sketch of the apparatus we will use to measure the speed of sound in air. Awave generator and speaker will create a wave that travels down a tube and reflects from apiston. A computer senses the travel time for the wave.

delay that is introduced to the arrival of the echo as the distance between the speaker (andmicrophone) and the piston head is increased. This way we do not need to know exactlywhere the sound originates or is detected. If the position of the speaker and microphone areunchanged, the only contribution to time is piston position.

Helpful TipTo avoid unnecessary interference with the measurements of other lab students, and tospare the hearing and sanity of your Lab Instructor, leave your speaker on for ONLYthose times you are making measurements.

Set-up:

Familiarize yourself with the equipment as shown in Figure 3.5. Pasco’s 850 Interface will beused to supply signals to the speakers from “Output 1”, to supply power for the microphonefrom “Output 2”, to digitize the speaker’s signal, and to digitize the microphone’s outputin “Voltage D”. Check these connections. A suitable configuration for Pasco’s Capstoneprogram (“Sound 1.cap”) can be found on the lab’s website at

http://groups.physics.northwestern.edu/lab/sound.html

31


Click the “Monitor” button at the bottom left to start taking data.Observe the two signals from A and B inputs displayed at the same time. Without

disturbing the microphone, move the piston and watch the computer’s oscilloscope display.Can you see the returned pulse(s) move as you move the piston? The oscilloscope graphsmicrophone voltage on the vertical axis and time on the horizontal axis. Moving thepiston away from the speaker/microphone increases the distance the pulse must travel andsimultaneously increases the time needed.

Measure the Speed of Sound

To measure the speed of sound we want a square wave output and a frequency of about5 - 20Hz. To adjust the signal, click “Hardware” at the left and change only the settings for“Output 1”. It is possible that the default signal needs no adjustments.

Figure 3.6: A sketch of the oscilloscope display showing the microphone’s response tosquare wave ‘clicks’. The sound reflects off of the tube ends and travels back and forth downthe tube. The microphone measures each time the click passes by.

The pulse generated by the speaker travels down the tube and reflects off the moveablepiston back to the microphone. Drag the vertical numbers away from zero until the mi-crophone signal occupies most of the Scope. Set the moveable piston to ∼80 cm from thespeaker. As you move the piston note that part of the signal moves to the right; these arethe clicks’ echoes passing the microphone as the sound bounces back and forth. Drag the

32


time numbers to the right until the first echo observed by the microphone is near the rightside of the Scope and t = 0 is at the left side. You should see something similar to thedisplay shown in Figure 3.6. Move the piston closer to the mike.

Does the spike shift on the time scale? Sometimes it is hard initially to identify thereflected pulse. It must move as the distance the clicks must travel changes and it must bethe first one to do so. The easiest way to identify the reflected click is to move the pistonaround and to look for a pulse shifting around on the scope signal. As you move the pistonaway from the mike you are introducing a delay in the time the microphone picks up thesound. You might also see other peaks shifting as you move the piston. These may be secondand third echoes of the pulse bouncing off the speaker end of the tube.

Set the piston at some minimum distance for which you can readily observe the first echoon the oscilloscope trace (∼20 cm is a good place). Note the piston position. How accuratelycan you determine the piston’s position? Use the Smart Tool’s ( ) cross-hairs’ icon to locatethe leading edge of the pulse and note the time. Right-click the center of the SmartTool,choose Properties, and increase the number of significant digits to 5-6. Be careful to writethe correct units and how accurately your time is known.

Now move the piston to a new position along the tube far from the speaker and note theposition again (∼70 cm is a nice choice. . . why?). Using the cross-hairs, determine the newtime of the shifted echo peak also. Remember that the extra distance you have introducedto the sound travel is twice the change of position of the piston (going toward the pistonand coming back).

Calculate the speed of sound by dividing the extra distance added to the round trip ofthe sound pulse by the corresponding increase in travel time. The pulse travels twice as faras the piston moved but the oscilloscope measured the time (not double the time and nothalf of the time),

v = 2∆x∆t . (3.4)

The appendix shows how to propagate the uncertainties δxi and δti to a commensurate δv.The width of the echo’s leading edge can be an indicator of the uncertainty of each

time measurement. Another measurement strategy is to measure the two times severaltimes and then to compute the statistics. If the echo moves around, this will increaseyour measurement error estimate and it will increase the standard deviation. Remember todisturb the instrument (except the piston) as little as possible. Measure the latest time andsubtract off the earliest time that might reasonably be assigned to the time of the pulse’secho. Let δ = latest - earliest and δt = 1

2δ is a reasonable estimate of the uncertainty in yourtime measurements. You need measure this uncertainty only once since nothing substantialchanges between measurements; each time measurement will have this same uncertainty.

33


3.2.2 Measuring the Wavelength –Observing Standing Waves

Sound incident on a barrier will interfere with its reflection, setting up a standing wave nearthe reflector. The distance between nodes in the standing wave is a measure of half thewavelength of the original sound wave when the wave travels at right angles to the reflector.Because of the inefficiency of the reflector and other losses, the nodes may be only partialnodes. Additionally, the microphone has a finite size and will average the sound intensity overa range of positions where only one position is at the intensity minimum. The wavelength ofsound should be expected to be on the order of meters for audible sounds. Diffraction effectsare commonly observed for sound waves passing through apertures like doors and windowson the order of meters in size.

For this part of the experiment adjust the acrylic tube so there is about a 5–10mm gapbetween the speaker and the end of the tube. This will release the pressure in the tube andforce this end of the tube to atmospheric pressure; this end will be a pressure node (and adisplacement anti-node). This part of the experiment will use “Sound 2.cap” that can bedownloaded from the lab’s website also

Set the frequency of the generator to 450Hz. Click “Signal Generator” at the left toaccess “Output 1” control panel. Keep the microphone near the piston and move the pistonand microphone to where the sound resonates in the tube (the microphone output goes to amaximum and the loudness increases significantly).

Use the mike as a probe to measure the intensity of sound in the region between thespeaker and the piston by noting the amplitude of the signal on the scope as you move themike around back and forth inside the tube. This is the sound’s intensity as a function ofposition, P (x), for the standing wave.

Place the mike near the piston while the tube is in resonance and note whether the pistonhead is a node or an anti-node by observing the variation in the intensity of the sound asyou move the mike around near the piston. Note your observation in your notebook. Whatwould you expect for a pressure wave or a displacement wave? Is the microphone a pressuresensor or a displacement sensor? Place the microphone near the speaker end of the tube,and note whether this is a maximum or minimum. Explain this result in your notebook.You would think that near the speaker you would get a large response from the microphone.Is that what you see?

Now, move the microphone to locate the first node away from the piston (remember thatright next to the piston is a displacement node) where the microphone’s output goes througha maximum. Start with the mike right near the piston and move it away from the pistonand toward the speaker. Measure the first position of maximum response away from thepiston. Can you detect the next node? A third node? Record your observations. Recordthe positions of the microphone where its output is minimum. Don’t forget your units anderror estimates; how accurately can you position the microphone at the maxima/minima andhow accurately can you read the centimeter scale? Move the microphone and remeasure onenode and one anti-node several times to check your error estimates. Are nodes and anti-node

34


measurements equally precise?Calculate the wavelength of the sound from the distance between the first node of the

standing wave and the second node. If the nodes are close together, you can skip a node,measure the distance between two nodes, and divide by two to get a better accuracy. Afterall, your measurement errors will be divided as well since the denominator will be twice aslarge.

Calculate the speed of sound using the wavelength just determined and the frequencyfrom the signal generator’s display; use Equation (3.3). Note your results in your lab book.

CheckpointThe distance between successive nodes in a standing wave is a measure of what?

3.2.3 Sound Speed at Several Frequencies

Set the generator to something low around 450 - 500Hz. Verify that the gap between thespeaker and tube end is still about 5–10mm. Keep the microphone near the piston and varythe position of the piston and microphone to maximize the microphone response and thetone’s loudness.

Place the microphone at the end of the tube nearest the speaker. Pull the piston all theway out to the end. Slowly insert the piston and find the first position where the pistonproduces a maximum sound intensity and is loudest. Move the microphone to get a maximumresponse and see if the piston’s location can increase the microphone’s output. Repeat untilthe piston’s position remains constant. This is where the tube is in resonance with thatparticular frequency. Note this piston position, its units, and your uncertainty in position.The position of the microphone will not need to change again until the frequency changes.

Insert the piston further and note a second resonance position. How precisely can youposition the piston and measure its position? If you cannot see a second resonance thewavelength of the sound may be too long for the tube and you will need to increase thesignal frequency.

These successive resonance positions are the positions of nodes for the standing wave atthis frequency. Since the distance between nodes is half a wavelength, merely doubling thisdistance and multiplying it by the frequency of the sound as read off the signal generatordisplay will determine the speed of sound. (See Equation (3.3).)

Repeat the measurements of successive resonance positions for four higher frequencies.Make a table of wavelengths, frequencies, and calculated sound speeds for each frequency.Plot your data using Vernier Software’s Graphical Analysis 3.4 (Ga3) program. A suitableconfiguration for Ga3 can also be downloaded from the lab’s website. Does the speed varysystematically with frequency? Use Ga3 to determine an average speed v and a standarddeviation sv by drawing a box around the data points and by using the Analyze/Statistics

35


feature. Compute the deviation of the mean sv using

sv = sv√N

What frequency has a speed most similar to the speed measured in Experiment 1? Yourmeasurement’s best predictor is vs = v ± sv

CheckpointFor a displacement wave reflecting off a solid wall is the boundary a node or an anti-node in the resulting standing wave? For a pressure wave reflecting off a solid wall isthe boundary a node or an anti-node in the resulting standing wave?

3.2.4 (optional) Spectral ContentThe Fourier Transform

Most of us are familiar with the link between musical pitch and sound frequency. Asoprano voice has high pitch and high frequency whereas a bass voice has low pitch andlow frequency. Probably you have already noticed while performing Section 3.2.3 above thathigher frequencies have higher pitch. You might also have noticed how ‘boring’ a pure sinewave sounds. For contrast, select the square and triangle waveforms for “Output 1”; use afrequency of a few hundred Hz and select the different wave shapes. Which shapes have themost pleasant sound? Record your observations in your notebook.

We now want to investigate the frequency content of sounds more closely. The “SoundFFT.cap” from the lab’s website provides a suitable configuration for Pasco’s Capstoneprogram. This will perform a ‘fast Fourier transform’ (FFT) on the microphone’s signaland the speaker’s excitation. In graduate school you will learn more thoroughly that theFourier transform identifies the frequency or spectral content of functions of time. Study theresponse vs. frequency for various waveform shapes and note your observations in your data.

The square wave should have responses at all integer multiples of the signal generator’sfrequency (the fundamental frequency). The triangle wave should have only odd multiples ofthe fundamental. The amplitudes decrease for higher multiples or harmonics. Try singing apure note into the microphone and noting the spectral content of your voice. The sine waveshould have only a fundamental response and should be absent of harmonics.

3.3 Analysis

Calculate the Difference between the speed of sound found using the two different methods,

∆ = |v1 − v| .

36


Combine the uncertainties in these two measurements using

σ =√

(δv1)2 + s2v

Discuss the similarities and differences between this difference and the computed expectederror. What other subtle sources of error can you think of that might have affected yourmeasurements. Communicate with complete sentences. Which measurement method givesthe best accuracy and how do you know?

3.4 Conclusions

What physical relations have your measurements supported? Contradicted? Which were notsatisfactorily tested? Communicate with complete sentences and define all symbols. Whatvalues have you measured that you might want to know in the future? Include your unitsand errors. What improvements might you suggest?

37


3.5 APPENDIX

Given that the uncertainties in two positions are δx1 and δx2 Equation (2.6) can be used tofind

δ(∆x) =√

(δx1)2 + (δx2)2.

Similarly, the uncertainties in two times yields

δ(∆t) =√

(δt1)2 + (δt2)2.

Then Equation (2.7) allows us to find

δv = v

√(δ(∆x)

∆x

)2+(δ(∆t)

∆t

)2.

38

chapter3 experiment1: sound - northwestern...

Documents