# Experiment 1: SOUND. The equation used to describe a simple sinusoidal function that propagates in space is given by Y = A o sin(k(x v t))

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1 Experiment 1: SOUND 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, which then propagates 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 and longitudinal. A transverse wave is characterized by a displacement from equilibrium which takes place at right angles to the direction of motion of the wave, and longitudinal waves have the displacement from equilibrium along the axis of propagation. 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 easily sliding past one another. This is referred to as shearing. Solids support shear forces which transmit the transverse wave from molecule to molecule. Longitudinal waves 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, usually but not exclusively air. A sound wave can be generated in solids, liquids, and gasses. The equation used to describe a simple sinusoidal function that propagates in space is given by Y = A o sin(k(x v t)) where Y is the time dependent displacement of the media from equilibrium, Ao is the maximum displacement, v is the velocity of the wave which depends on the characteristics of the media, and k is a constant that is determined by both the media (the speed of the wave) and the frequency of the wave. The constant k is usually expressed as k = 2π/λ,

2 where λ is the wavelength, and this wavelength is related to the wave velocity v and the wave frequency f by the expression λf = v. A periodic mechanical wave is characterized by a frequency f of oscillation which is determined by the source of vibration motion that creates the disturbance. Thus, the frequency f and the speed v of the wave in the media determines the wavelength λ. This equation is used to 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, the displacement is at right angles to the direction of propagation. The actual wave looks very similar to the plot of the displacement and is easily visualized. For a longitudinal wave, the displacement occurs along the same axis as the direction of propagation and 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 pressure maximum as shown in Figure 1.

3 Y Displacement Representation P Sound Wave Pressure Representation Figure 1. Sound Wave shown with both displacement and pressure representations. The picture represents the density of the medium as the wave passes through it. 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 shown in Figure 2. If on the other hand, the second wave were at Y 2max = - Y 2max the net displacement equal to the sum of Y 1max plus Y 2max which in effect would be the difference Y 1max - Y 2max or zero, as shown in Figure 3.

4 Y 1 Y + Y 1 2 Y 2 Figure 2. Constructive interference Y 1 Y + 1 Y' 2 Y' 2 Figure 3. Destructive interference 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

5 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. In this lab, we will consider the effects of reflection from a solid boundary, such that the boundary condition requires that the wave have a displacement of zero at the point of reflection. Air molecules cannot be displaced from equilibrium at the wall. They simply have nowhere to go. This condition could only be imposed 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. 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 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 it 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 boundary (see Figure 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 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

6 antinodes. Such a wave is referred to as a standing wave. In either case, successive null points or nodes occur at intervals of half of the wavelength of the traveling waves. Measuring the distance between nodes of the standing wave, we can determine the wavelength of the incident and reflected traveling waves. Nodes Wall Nodes Wall λ λ Standing Pressure Wave (Antinode at boundary) Standing Displacement Wave (Node at boundary) Figure 4. Standing Waves Microphone rod Tube Microphone power cable Wave Generator Speaker Microphone Computer/ Computer Oscilloscope Piston A B C A B C Speaker cable (BNC to banana plug) Cable from Microphone to computer input B Cable from speaker to computer input A (banana plug to BNC) Inputs 750 Interface to Computer Scope Figure 5 Experimental Set-up

7 Experiment 1A: MEASURE THE SPEED OF SOUND IN AIR From personal experience one can get a sense that the speed of sound in air is rapid. You do not notice any 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 were sitting in the outfield bleachers at a baseball game, you can sense a noticeable delay in the arrival of the sound of the crack of a bat on the baseball from the moment when you see the ball being hit. 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 the oscilloscope to observe the very short time delay as sound travels a distance in the order of a meter. 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 sine 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 is introduced in 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 a microphone. We will measure the speed of the sound wave by observing the amount of time delay is introduced to the arrival of the echo as the distance between the speaker and

8 the piston head is increased. This way we do not need to know exactly where the sound originates or is detected. CAUTION: 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 ONLY those times you are making measurements. 1. Set-up Familiarize yourself with the equipment as shown in Figure 5. The signal generator should be connected to the speaker. Check this connection Turn the signal generator on and let it warm up. Can you hear a sound coming from the speaker? If the signal generator is set to generate a sine wave and the frequency is set in the audible range with sufficient amplitude you should be able to hear the tone coming from the speaker. Be careful not to set the amplitude too high and blow the speaker. If there is no sound recheck the connections and signal generator settings or call your TA. We will be using the 750 Interface and the computer program Data Studio to take data. Follow the TA instructions for connecting the 750 Interface box to the computer. Then load up Data Studio. This program interfaces the connections to the computer. It is very graphic and intuitive. From the opening screen select New Experiment. A picture of the 750 Interface should appear. You will want to set up the interface on the computer graphic to reflect the actual physical connections that are made to the 750 Interface box. With the cursor select the input port A. A menu should appear to allow you to specify what you are connecting to port A. Choose a voltage sensor (at the very bottom of the list). This should be the voltage coming from the signal generator picked off the speaker connections with a coaxial cable. The end connected to the speaker has banana connectors and the end at the 750 interface has a BNC type connector. With the cursor select the next input, input B and choose a voltage sensor again. This would be the signal coming from the microphone, a BNC connector at the 750 Interface box.

9 The 750 Interface box also provides power to the microphone. A convenient voltage source is located on the right of the 750 Interface box and should be connected with a cable to the microphone. To activate the voltage source use the cursor to select the voltage source connection on the right side of the 750 box image on the computer screen. A menu should appear that will allow you to set the parameters of the voltage source. Select D.C. signal and choose 1.5 volts. Now you must choose the method of displaying the data. Select scope from the options on the lower left list. The scope window will appear and can be adjusted to fill the desktop. By default the signal from input A will be displayed as indicated by the information in the control box in the upper right corner. We would like to add input B as well. With your cursor drag voltage B from the list in the upper left corner over to the control box and drop it in. You should now have a dual trace scope. You are ready to take data. Activate the interface by hitting the Start button in the upper left corner of the screen. If the speaker is emitting an audible tone you should see a signal on the scope. At first the signal may be confusing with a sine wave slowly creeping across the screen. The signal needs to be synchronized with the display using the trigger feature. First with the cursor select the voltage a box. The note a small triangle at the origin of the display. Click on this triangle. It will turn the color of the voltage A source and the sine wave should now be stationary. If you double click on the small triangle a widow will appear with options. The trigger is set by default to trigger on the upward swing of the signal. Click on the option to trigger on the down swing and observe the results. You can also adjust the trigger level by dragging the triangle up or down with your cursor. Play with this feature if you like. Adjust the time scale to see the individual cycles of the signal. Each cycle of the wave shown on the screen represents a period of oscillation. As a check (you may be directed to skip this step) measure the period T of a single cycle of the wave using the calibration

10 lines of the scope and the time scale as indicated on the x-axis label, and verify that the frequency f setting of the sine wave generator is accurate. Remember f = 1/T. 2. Observe the two signals from A and B inputs displayed at the same time (again you may be directed to skip this step). Are the two signals in phase? You may have to readjust the voltage scale to see the complete sign wave of the microphone. Also, the signal generator signal may be clipped (squared of at the top and bottom) if the signal generator output is set too high. If that is the case just turn down the generator amplitude until you see a smooth sine signal. Note the relative phase relation by drawing the two signals in your notebook. 3. To measure the speed of sound we will turn the signal generator to the square wave output and adjust the frequency to a very low setting, about 5-20 Hz. You will hear the speaker output as clicks instead of a tone. The signal generator signal displayed on the scope will be a square wave. You might need to adjust the time display to see the square wave on the scope. The microphone will pick up these clicks and display them as sharp spikes on the display. Trigger the display from channel A, the square wave. Use the down option for the trigger set on the voltage A input. The pulse generated by the speaker travel s down the tube and reflects off the moveable piston back to the speaker. Set the moveable piston about 20 cm from the speaker. Adjust the time display to 1 ms/div. and the voltage range to V/div. You should see something similar to the display shown in Figure 6. Move the piston closer to the mike. Does the spike shift on the time scale? Sometimes it is hard to initially identify the reflection pulse. The easiest way is to move the piston around and 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.

11 first echo Figure 6 4. 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). Note the piston position. Use the cross hairs icon to locate the leading edge of the pulse and note the number. Be careful to write the correct units. 5. Now move the piston to a new position along the tube far from the speaker and note the position (70 cm is a nice choice, why?). Using the crosshairs feature, determine the time of the shifted echo peak. Remember that the extra distance you have introduced to the sound travel is twice the change of position of the piston (coming and going back). 6. Calculate the speed of sound by dividing the extra distance added to the round trip of the sound pulse by the corresponding time delay introduced as measured on the scope (not multiplied by two). 7. The width of the spike of the echo can be an indicator of the uncertainty of the measurement. Determine the time width of the spike on the scope trace using the same tools you used to measure the time delay. Divide this time by the time delay introduced by moving the piston. Use the resulting fraction to find a percent uncertainty in your speed. What is the frequency of the sound whose speed you just measured? Could you

12 use the time between successive echoes as an accurate measure to determine the speed of sound? Experiment 1B: MEASURE THE WAVELENGTH OF SOUND IN AIR FOR A STANDING WAVE Sound reflected off 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. Because of the inefficiency of the reflector, the nodes may only be partial nodes. The wavelength of sound should be expected to be on the order of meters. Diffraction effects are commonly observed for sound waves passing through apertures on the order of meters. 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. 1. Set the frequency of the generator to 450 Hz. Move the piston to where the sound resonates in the tube (the microphone output goes to a maximum). 2. 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. Place the mike near the piston and note whether the piston head is a node or an antinode by observing the variation in the intensity of the sound as you move the mike around near the piston. Note 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?

13 3. Now move the position of the microphone to locate the first node away from the piston (remember that right next to the piston is an node) where the sound intensity goes through a maximum. Start 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? 2. Calculate the wavelength of the sound from the distance between the first node of the standing wave and the second node. If the antinodes are close together you can measure the distance between two nodes and divide by two to get a better accuracy. 3. Calculate the speed of the sound using the wavelength just determined and the frequency of the signal generator. Note your results in your lab book. Experiment 1C: MEASURE THE SPEED OF SOUND IN AIR FOR A RESONATING WAVE OVER A RANGE OF FREQUENCIES 1. Set the generator to something low around Hz. Vary the position the piston to maximize the microphone response. 2. 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 position and observe where the piston position produces a maximum sound intensity. This is where the tube is in resonance with that particular frequency. Note this piston position. 3. Withdraw the piston further and note a second resonance position. If you cannot see a second resonance the wavelength of the sound may be too long. You may have to increase the frequency of the signal generator to produce a shorter wavelength sound. These successive resonance positions are separated by a half wavelength of the sound at this frequency. Merely doubling this distance and multiplying it by the frequency of the sound as read off the signal generator display can determine the speed of sound.

14 4. Repeat the measurements of successive resonance positions for four higher frequencies. Make a table of wavelengths, frequencies and calculated speeds for each frequency and plot your data using the GAX program. Does the speed vary systematically with frequency? Use GAX to determine an average speed and a standard deviation using the statistics feature. What frequency has a speed most similar to the speed measured in Experiment 1? Comment on any obvious trends. Following is a list of questions intended to help you prepare for this laboratory session. If you have read and understood this write up, you should be able to answer most of these questions. The TA may decide to check your degree of preparedness by asking you some of these questions: 1. Is sound a displacement wave or a pressure wave, or may it be considered as both? 2. What happens when two sound waves overlap in a region of space? 3. What determines the pitch of a sound wave? The source, the medium? 4. What is the difference between a displacement wave and a pressure wave? 5. Which determines the speed of sound, the source frequency or the medium? 6. Which determines the wavelength of sound, the source frequency, the medium, or both? 7. The distance between successive nodes in a standing wave is a measure of what? 8. For a displacement wave reflecting off a solid wall is the boundary a node or an antinode in the resulting standing wave? 9. For a pressure wave reflecting off a solid wall is the boundary a node or an antinode in the resulting standing wave?

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