Chapter 3. Experiment 1: Sound. 3.1 Introduction

Transcription

1 Chapter 3 Experiment 1: Sound 3.1 Introduction Sound is classified under the topic of mechanical waves. A mechanical wave is a term which refers to a displacement of elements in a medium from their equilibrium state; but to be a wave this displacement must then propagate through the medium. The speed at which the wave propagates is inversely related to the mass density of the propagating medium and directly 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 is characterized by a displacement from equilibrium which takes place at right angles to the direction the wave propagates; longitudinal waves have the displacement from equilibrium along the axis of propagation. Since two directions are perpendicular to the direction of propagation, transverse waves have two independent polarization directions. The form of the equations describing these two types of waves is very similar. However, transverse waves can only exist in solid media, 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 rather than continue to slide; this intermolecular connection will transmit the transverse wave from molecule to molecule. Longitudinal waves rely only on pressure and can exist in both solids and fluids. They depend on the compressibility of the media. Solids and fluids all show a resistance to compression. Sound waves are longitudinal waves that are transmitted as a result of compression displacement of molecules of the medium. We usually discuss sound in air, but sound travels in everything except empty space. A sound wave can be generated in solids, liquids, or gasses and can continue to propagate in a different medium. The equation used to describe a simple sinusoidal function that propagates in space is given by Y(x, t) = A 0 sin [ k(x vt) ]ˆp (3.1) 25

2 where Y is the time and position dependent displacement of the media from equilibrium, A 0 is the maximum displacement or amplitude of the medium s motion, v is the velocity of the wave which depends on the characteristics of the media. This particular wave travels along the x-axis... x must increase at speed v to keep up with vt. ˆp is the polarization of the wave. The case a longitudinal wave has ˆp = ˆx and a transverse wave has ˆp = p y ŷ + p z ẑ some combination of y and/or z polarization. k is a constant that is determined by both the speed of the wave and the frequency of the wave. The constant k is usually expressed as k = 2π λ, (3.2) where λ is the wavelength. The wavelength is related to the wave velocity v and the wave frequency, f, by the expression v = λf. (3.3) A periodic mechanical wave is characterized by a frequency of oscillation, f, which is determined by the source of vibration motion that creates the disturbance. Thus, the frequency and the speed of the wave in the media determines the wavelength. The source can choose to oscillate at any frequency it chooses, but the medium decides the velocity of propagation. Checkpoint What is the difference between a displacement wave and a pressure wave? Checkpoint Is 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. These equations describe either longitudinal or transverse waves. The difference lies in the interpretation 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 and is easily visualized. For a longitudinal wave, Equation (3.1) describes oscillations of the x coordinate of the particles at x in equilibrium. This results in a sinusoidal variation in the density 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 would be 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 a variation in the density of air molecules along the direction of the wave, it can be viewed as either a displacement wave or a pressure wave. The above equation may be used to describe either picture. The displacement maximum is usually 90 degrees out of phase with the 26

3 Pressure Wave (a) (b) (c) Displacement Wave Figure 3.1: An illustration of the two representations of a longitudinal wave. The displacement representation is the position of particles with respect to their average positions, but the pressure increases as the particles move toward each other (compression) and decreases as the particles move away from each other (rarefaction). The displacement wave leads the pressure wave by 90. pressure maximum as shown in Figure 3.1. A sound wave is shown with both displacement and pressure representations. The picture represents the density of the medium as the wave passes through it. Checkpoint What determines the pitch of a sound wave? The source, the medium? Which determines the speed of sound, the sound generator, the medium, or both? Which determines the wavelength of sound, the sound generator, the medium, or both? Superposition When two sound waves happen to propagate into the same region of a medium, the instantaneous displacement of the molecules of the medium is normally the algebraic sum of the displacements of the two waves as they overlap. If at one time and place each individual wave would happen to be at a maximum amplitude, say Y 1max and Y 2max the net result would be a displacement of the medium at a value equal to the sum of Y 1max and Y 2max. This is 27

4 Y 1 (a) Y 2 Y 1 + Y 2 (c) (b) Figure 3.2: An illustration of constructive interference. Y 1 and Y 2 are in phase at all times so that their sum has amplitude equal to the sum of Y 1 s and Y 2 s amplitudes. shown in Figure 3.2. If on the other hand, the second wave were at Y 2 max = Y 2max, the net displacement equal to the sum of Y 1max + Y 2 max, which in effect would be the difference Y 1max Y 2max or zero if the amplitudes are equal, as shown in Figure 3.3. Y 1 (a) Y' 2 Y 1 + Y' 2 (c) (b) Figure 3.3: An illustration of destructive interference. Y 1 and Y 2 are out of phase by 180 or half a wavelength. The sum of the two waves is zero if the two amplitudes are equal. Checkpoint What happens when two sound waves overlap in a region of space? Reflection We most often think of a reflection as occurring when a wave encounters the border of the medium 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 of the energy of the wave is redirected to the reflected wave. The amplitude and phase of the reflected wave is determined by the boundary conditions at the point of reflection. 28

5 In this lab, we will consider the effects of reflection from a solid boundary, such that the boundary condition requires that the sum of the waves have a displacement of zero at the point of reflection. Air molecules cannot be displaced from equilibrium at the wall. They cannot move into the wall and atmospheric pressure presses them into the wall; they simply have nowhere to go. This condition can only exist if we were to superpose a second wave moving in the opposite direction with exactly the same amplitude, and 180 degrees out of phase with the original wave. Hence, in order to satisfy the boundary condition, a reflection wave is generated with exactly these properties. (a) (b) (c) (d) A A A Pressure A A Displacement A A A Wall Figure 3.4: Illustrations of pressure waves and displacement waves reflected at a wall. The incident wave and reflected waves interfere to produce a series of nodes () and anti-nodes (A) 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 gas moves like the arrows between indicate. The gas sloshes back and forth between the dotted lines, but on average does not cross them. 29 The resulting superposition of incident and reflected waves in the region in front of the boundary also sets up a second null area where the amplitudes cancel at a distance of one half of a wavelength from the boundary as shown in Figure 3.4. The null areas are called nodes. If the wave didn t loose amplitude as it traveled, a null would be present at successive half wavelength intervals over the entire region. As it is, the wave looses amplitude as it propagates, and the cancellation is only partial. The same boundary does not place such restrictions on the pressure wave. The pressure at the boundary may rise and fall, as is required. The wall can easily support whatever pressure results from the superposition of any two waves. A suitable pressure wave which takes advantage of the boundary restriction (which is none) is directed in the opposite direction with an equal amplitude to conserve energy, and is in phase with the incident wave. The resulting superposed waves show a maximum or anti-node at the

6 boundary (see Figure 3.4) and a null point or node at a distance of one quarter wavelength away from the boundary. If the amplitude is sustained as the wave travels, a second null appears a half wavelength from the first, or at a point three quarters of a wavelength from the boundary. Between each node, the wave is seen to oscillate between maximum positive and maximum negative amplitudes, where the maximum amplitude is the sum of the maximum amplitudes of the waves considered separately. These areas are called anti-nodes. Such a wave is referred to as a standing wave because it stands still. In either case, successive null points 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 the wavelength of the incident and reflected traveling waves. Since we also know the frequency at 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. You notice no delay in hearing a word that is spoken by a person nearby and the movement of the speaker s mouth. That would be quite a distraction, like watching a movie with the sound track out of synch! And yet, when you sit in the outfield bleachers at a baseball game, you can sense a noticeable delay between the arrival of the light showing the ball being hit and the 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 size of a baseball field. In principle one could measure the speed of sound by timing how long after ones sees the ball hit that the sound arrives if one knew how far away they were from home plate. Instead, we will employ an oscilloscope simulation to observe the very short time delay as sound travels a distance on the order of a meter The Speed of a Sound Pulse In this experiment, a signal generator is used to produce a repeating electrical pulse to drive a speaker. The pulse causes the speaker to emit a click or pulse of sound whose speed we will measure. A small microphone is used as a sensor. Its output is connected to one input 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 to the microphone could be used to measure the speed of the click. Are the delays introduced to the process of converting the electrical signal to mechanical sound and back, and in the travel of electrical signals through the wire negligible? They probably are; however, we do not have an exact location for where the sound is produced in the speaker or sensed in the microphone. This could be a problem which we must deal with. A sound wave will be sent down a tube and be reflected off a piston head back to a microphone. We will measure the speed of the sound wave by observing the amount of time 30

7 Figure 3.5: A sketch of the apparatus we will use to measure the speed of sound in air. A wave generator and speaker will create a wave that travels down a tube and reflects from a piston. 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 (and microphone) and the piston head is increased. This way we do not need to know exactly where the sound originates or is detected. If the position of the speaker and microphone are unchanged, the only contribution to time is piston position. Helpful Tip To avoid unnecessary interference with the measurements of other lab students, and to spare the hearing and sanity of your Lab Instructor, leave your speaker on for OLY those times you are making measurements. Set-up: Familiarize yourself with the equipment as shown in Figure 3.5. Pasco s 850 Interface will be used to supply signals to the speakers from Output 1, to supply power for the microphone from Output 2, to digitize the speaker s signal, and to digitize the microphone s output in Voltage D. Check these connections. A suitable configuration for Pasco s Capstone program ( Sound 1.cap ) can be found on the lab s website at 31

8 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 graphs microphone voltage on the vertical axis and time on the horizontal axis. Moving the piston away from the speaker/microphone increases the distance the pulse must travel and simultaneously 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 about 5-20 Hz. 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 to square wave clicks. The sound reflects off of the tube ends and travels back and forth down the tube. The microphone measures each time the click passes by. The pulse generated by the speaker travels down the tube and reflects off the moveable piston back to the microphone. Drag the vertical numbers away from zero until the microphone signal occupies most of the Scope. Set the moveable piston to 80 cm from the speaker. As you move the piston note that part of the signal moves to the right; these are the clicks echoes passing the microphone as the sound bounces back and forth. Drag the 32

9 time numbers to the right until the first echo observed by the microphone is near the right side of the Scope and t = 0 is at the left side. You should see something similar to the display 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 the reflected pulse. It must move as the distance the clicks must travel changes and it must be the first one to do so. The easiest way to identify the reflected click is to move the piston around and to look for a pulse shifting around on the scope signal. As you move the piston away from the mike you are introducing a delay in the time the microphone picks up the sound. You might also see other peaks shifting as you move the piston. These may be second and 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 echo on the oscilloscope trace ( 20 cm is a good place). ote the piston position. How accurately can you determine the piston s position? Use the Smart Tool s cross-hairs icon to locate the 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 write the correct units and how accurately your time is known. ow move the piston to a new position along the tube far from the speaker and note the position again ( 70 cm is a nice choice... why?). Using the cross-hairs, determine the new time of the shifted echo peak also. Remember that the extra distance you have introduced to the sound travel is twice the change of position of the piston (going toward the piston and coming back). Calculate the speed of sound by dividing the extra distance added to the round trip of the sound pulse by the corresponding increase in travel time. The pulse travels twice as far as the piston moved but the oscilloscope measured the time (not double the time and not half of the time), v = 2 x t. (3.4) The width of the echo s leading edge can be an indicator of the uncertainty of the measurement. If the echo moves around, this will increase your measurement error estimate. Measure the latest time and subtract off the earliest time that might reasonably be assigned to the time of the pulse s echo. Let δ = latest - earliest and δt = 1 δ is a reasonable estimate 2 of the uncertainty in your time measurements. You need measure this uncertainty only once since nothing substantial changes between measurements; each time measurement will have this same uncertainty Measuring the Wavelength - Observing Standing Waves Sound incident on a barrier will interfere with its reflection, setting up a standing wave near the reflector. The distance between nodes in the standing wave is a measure of half the wavelength of the original sound wave when the wave travels at right angles to the reflector. 33

10 Because of the inefficiency of the reflector and other losses, the nodes may be only partial nodes. Additionally, the microphone has a finite size and will average the sound intensity over a range of positions where only one position is at the intensity minimum. The wavelength of sound should be expected to be on the order of meters for audible sounds. Diffraction effects are commonly observed for sound waves passing through apertures like doors and windows on the order of meters in size. For this part of the experiment adjust the acrylic tube so there is about a centimeter gap between the speaker and the end of the tube. This will release the pressure in the tube and force this end of the tube to atmospheric pressure; this end will be a pressure node (and a displacement anti-node). This part of the experiment will use Sound 2.cap that can be downloaded from the lab s website at Set the frequency of the generator to 450 Hz. Click Signal Generator at the left to access Output 1 control panel. Keep the microphone near the piston and move the piston and microphone to where the sound resonates in the tube (the microphone output goes to a maximum). Use the mike as a probe to measure the intensity of sound in the region between the speaker and the piston by noting the amplitude of the signal on the scope as you move the mike around back and forth inside the tube. This is the sound pressure level (SPL) as a function of position, P (x), for the standing wave. Place the mike near the piston and note whether the piston head is a node or an anti-node by observing the variation in the intensity of the sound as you move the mike around near the piston. ote your observation in your notebook. What would you expect for a pressure wave or a displacement wave? Is the microphone a pressure sensor or an amplitude 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? ow, move the microphone to locate the first node away from the piston (remember that right next to the piston is a displacement node) where the microphone s output goes through a maximum. Start with the mike right near the piston and move it away from the piston and toward the speaker. Measure the first position of maximum response away from the piston. Can you detect the next node? A third node? Record your observations. Record the positions of the microphone where its output is minimum. Don t forget your units and error estimates; how accurately can you position the microphone at the maxima/minima and how accurately can you read the centimeter scale? Move the microphone and remeasure one node and one anti-node several times to check your error estimates. Are nodes and anti-node 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. After all, your measurement errors will be divided as well since the denominator will be twice as 34

11 large. Calculate the speed of sound using the wavelength just determined and the frequency from the signal generator s LED display; use Equation (3.3). ote your results in your lab book. Checkpoint The distance between successive nodes in a standing wave is a measure of what? Sound Speed at Several Frequencies Set the generator to something low around Hz. Verify that the gap between the speaker and tube end is still about 1 cm. Keep the microphone near the piston and vary the position of the piston and microphone to maximize the microphone response. Place the microphone at the end of the tube nearest the speaker. Put the piston all the way into the tube so that it is close to the mike. Slowly withdraw the piston and observe the position where the piston produces a maximum sound intensity. Move the microphone to get a maximum response and see if the piston s location can increase the microphone s output. Repeat until the piston s position remains constant. This is where the tube is in resonance with that particular frequency. ote this piston position, its units, and your error. The position of the microphone will not need to change again until the frequency changes. Withdraw the piston further and note a second resonance position. How precisely can you position the piston and measure its position? If you cannot see a second resonance the wavelength of the sound may be too long and you will need to increase the signal frequency. These successive resonance positions are the positions of nodes for the standing wave at this frequency. Since the distance between nodes is half a wavelength, merely doubling this distance and multiplying it by the frequency of the sound as read off the signal generator display will determine the speed of sound. 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 suitable configuration for Ga3 can also be downloaded from the lab s website. Does the speed vary systematically with frequency? Use Ga3 to determine an average speed v and a standard deviation s v by drawing a box around the data points and by using the Analyze/Statistics feature. Compute the deviation of the mean s v using s v = s v What frequency has a speed most similar to the speed measured in Experiment 1? Your measurement s best predictor is v s = v ± s v 35

12 Checkpoint For a displacement wave reflecting off a solid wall is the boundary a node or an antinode in the resulting standing wave? For a pressure wave reflecting off a solid wall is the boundary a node or an anti-node in the resulting standing wave? Spectral Content The Fourier Transform Most of us are familiar with the link between musical pitch and sound frequency. A soprano voice has high pitch and high frequency whereas a bass voice has low pitch and low frequency. Probably you have already noticed while performing Section above that higher frequencies have higher pitch. You might also have noticed how boring a pure sine wave sounds. For contrast, select the square and triangle waveforms for Output 1 ; use a frequency of a few hundred Hz and select the different wave shapes. Which shapes have the most pleasant sound? Record your observations in your notebook. We now want to investigate the frequency content of sounds more closely. The Sound FFT.cap from the lab s website provides a suitable configuration for Pasco s Capstone program. This will perform a fast Fourier transform (FFT) on the microphone s signal and the speaker s excitation. In graduate school you will learn more thoroughly that the Fourier transform identifies the frequency or spectral content of functions of time. Study the response 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 s frequency (the fundamental frequency). The triangle wave should have only odd multiples of the fundamental. The amplitudes decrease for higher multiples or harmonics. Try singing a pure note into the microphone and noting the spectral content of your voice. 3.3 Analysis Calculate the Difference between the speed of sound found using the two different methods, = v 1 v 3. Combine the uncertainties in these two measurements using σ = (δv 1 ) 2 + s 2 v Discuss the similarities and differences between this difference and the computed errors. What other subtle sources of error can you think of that might have affected your measurements. Communicate with complete sentences. Which measurement method gives the best accuracy and how do you know? 36

13 3.4 Conclusions What physical relations have your measurements supported? Contradicted? Which were not satisfactorily tested? Communicate with complete sentences and define all symbols. What values have you measured that you might want to know in the future? Include your units and errors. 37