Speed of Sound in Air OBJECTIVE To explain the condition(s) necessary to achieve resonance in an open tube. To understand how the velocity of sound is affected by air temperature. To determine the speed of sound in air for different frequencies. To compare the experimental measure of the speed of sound in air to the accepted value. INTRODUCTION Like that of the standing wave in a string, standing waves can also be produced in a column of air. A variety of these standing waves can be produced, each with a different frequency, depending on the number of standing waves that are able to fit into a particular column length. These standing waves can be observed in the form of sound that resonates from the column. This resonance is observed as a maximum vibration in the amplitude of the sound. For this experiment, the column of air analyzed via a change in position of a microphone in the tube. The sound source will be provided by a vibrating speaker. From the source frequency, air column length at resonance, and the standing wave number, the velocity of sound can be determined. APPARATUS (1) resonance tube with attached speaker and BNC connection, (1) oscilloscope, (1) function generator, (1) 2.5 VDC power supply, (1) microphone with guide rod, power connection and BNC connector, (1) thermometer {for the class} and (1) meter stick. THEORY The velocity, v [m/s], of a traveling wave can be determined from the relationship between the frequency, f [Hz], and the wavelength, λ [m], of the wave. The relationship between these quantities is given by: v = f Equation 1 When a sound wave is sent down the opening of an open tube is not reflected at the open end of the tube. Within the tube, a standing wave is produced. This standing wave is characterized by places of constructive interference (locations of maximum amplitude = antinodes) and places of destructive interference (locations of zero amplitude = nodes). These traveling waves, in the air column, are longitudinal waves. They are produced by a disturbance that is in the same direction as the velocity of the wave. Speed of Sound in Air - Page 1
More precisely, series of condensations (denser packets of air molecules) and rarefactions (less dense packets of air molecules) propagating in the air form the traveling wave. Figure 1 illustrates a tuning fork as it vibrates in the air. Non-vibrating Tuning Fork -- Evenly spaced air molecules Figure 1 t & λ λ Time & Position Rarefractions T Condensations Vibrating Tuning Fork Where, λ [m] is the wavelength of the vibration, measured from the beginning of one rarefaction/condensation to the beginning of the next rarefaction / condensation, and T [s] is the period (the time it takes to complete one vibration). A similar illustration is used for any device that is capable of generating a set frequency or sets of frequencies. Resonance occurs in the tube when the length of the tube is such that an antinode (the maximum vibration of the standing wave) is produced at the tube's open end. Likewise, there must be an antinode at the opposite, open end of the tube. The resonance condition can only be satisfied for specific lengths of the open tube. Figure 2 details several open tube standing wave cases. λ/2 λ 3λ/2 Figure 2 Where, λ [m] is the wavelength of the air column and L [m] is the length of the tube. Note that the figure shows the standing wave as a transverse plot. This shows the relationship between the antinodes and the nodes of an air column resonance compared to a transverse wave. Speed of Sound in Air - Page 2
Only a certain number of wavelengths can be fit into the tube's lengths that satisfy the antinode at both open ends of the tube. It can be seen from Figure 2 that only standing waves with an even number of half wavelengths will give the conditions for resonance in the tube. The first standing wave case corresponds to standing wave number 1 (n = 1) where L = λ/2. Standing wave number two corresponds to two half waves added together; L = λ/2 + λ/2 = λ. In general, this relationship can be expressed by the equation: L= n 2 Where, n [#] is the standing wave number. ; n=1,2,3, Equation 2 Combining Equations 1 and 2 gives: v v f = = 2 L n n v = ; n= 1,2,3, 2 L Equation 3 Where, the driving frequency, f [Hz], of the vibrations corresponds to the relation between the velocity of sound in the air, the length of the tube, and the standing wave number. To study the resonance condition, it is easier to probe the length of the tube than it is to change the driving frequency for a set tube length. As the sound within the tube is explored, node and antinode locations are noted along the length of the tube. The difference between adjacent nodes or antinodes (ΔL) varies by one-half the wavelength of the standing wave in the tube. An analysis of Figure 2 reveals this relationship. L= L2 - L1= 2 and L= L 3 - L Equation 4 2 = 2 Similar equations/differences can be done for each successive resonance position. Thus, if the driving frequency is known, and the wavelength is determined from Equation 4, the speed of sound in the air can be determined from Equation 1. Note Equation 4 always gives λ/2 as the difference between successive node or antinode locations. Speed of Sound in Air - Page 3
The speed of sound in air varies as the temperature changes. Thus the known speed of sound in air must be calculated with the aid of the temperature dependent velocity equation, given below. v air = 331. 5 m/s 1 + T C 273. 15 o C Equation 5 Where, vair [m/s] is the velocity of sound in air at the temperature TC [ o C]. Thus, for each degree of temperature change, the speed of sound in air changes by about 0.6 m/s. The Oscilloscope: It is necessary to read the period of the sine wave generated from the speaker in order to determine the exact frequency used. Notice the grid across the face of the oscilloscope screen. It is divided into large grid boxes (divisions) with hash marks (each is 0.2 of a division) along the horizontal and vertical axes. When a sine wave is observed on the screen, the number of horizontal divisions and sub divisions between two successive crests is proportional to the period of the wave. Figure 3 Speed of Sound in Air - Page 4
The oscilloscope has a dial labeled "Time/Div." This dial number times the number of horizontal divisions and sub divisions calculates the period (T [s]) of the wave. From this information, the frequency (f [Hz]) can easily be calculated; recalling that: f 1 = T Equation 6 Speed of Sound in Air - Page 5
EXPERIMENTAL SETUP The resonance tube apparatus is illustrated below. Figure 4 The speaker is attached to the output of the function generator using the attached coaxial cable (BNC connection). The coaxial cable (BNC connection) from the microphone is attached to the channel one input of the oscilloscope. The two remaining wires from the microphone are attached to the positive (black with white stripe wire) and negative (black wire) terminals of the power supply, as indicated on each wire. Speed of Sound in Air - Page 6
EXPERIMENTAL PROCEDURE a) Assemble the experimental setup as illustrated and described by Figure 4. In the Questions section of the laboratory report, be sure to write a brief description of how each of the illustrated items is to be used; i.e. what are their individual functions as they pertain to the experiment? b) Turn on the oscilloscope and set the function generator for a sine wave that has a frequency between 500 Hz and 1000 Hz. c) Turn the voltage and current knobs of the power supply off (rotated completely counterclockwise) and then turn on the power supply. The power supply should be adjusted up to a 2.0 VDC. Do not increase the voltage beyond 2.5 VDC as it may do damage to the microphone. Set the microphone (without the guide rod) into the end of the resonance tube. Slowly push or pull the microphone until a sine wave of maximum possible amplitude is observed on the oscilloscope screen. d) As indicated in the "The Oscilloscope " section, it is necessary, at this point, to determine the exact frequency being used by analyzing the sine wave pattern from the scope screen. Record the number of divisions and the Time/Div knob setting in your data table. Speed of Sound in Air - Page 7
e) Attach the guide rod to the microphone and slowly begin to insert the microphone into the resonance tube. In the Questions section of the laboratory report, indicate what led you to know that you were sensing the antinode of the standing wave? In the Questions section of the laboratory report, indicate what led you to know that you were sensing the node of the standing wave? You will note that it is easier to locate the node locations with the microphone as the locations of the antinodes seem to span several centimeters. Thus, for the purpose of our investigations, we will be using the node positions for measurement data. f) Begin by locating the position of a node, to the nearest millimeter, by marking the guide rod with a pencil. It may be necessary to insert and remove the guide rod several times to determine the exact position of each node. Continue recording node positions until a total of four successive node positions are located; these may be before or after the one you marked initially as long as there are four consecutive identified. Remove the guide rod and measure from the end of the rod to each of the 4 successive node points...not between each node mark! Record these values in your data table. In the Questions section of the laboratory report, indicate what the general relationship between the location of each of the nodes respect to each other is; i.e. the significance of the distance between the nodes? Illustrate a diagram of a standing wave in a tube open at both ends to justify your conclusion. g) Repeat the data gathering procedure for a second frequency between 1000 Hz and 1500 Hz. h) Repeat the data gathering procedure for a second frequency between 1500 Hz and 2000 Hz. Speed of Sound in Air - Page 8
REPORT ITEMS (To be submitted and stapled in the order indicated below) (-5 points if this is not done properly) COVER PAGE Completed Laboratory Responsibility and Cover Sheet DATA (worth up to 20 points) Data tables are available as a downloadable Excel file DATA ANALYSIS (worth up to 20 points) The SIX required sample calculations, to be shown on a separate sheet of paper in your laboratory report (NOT on the data table sheets themselves), are highlighted in yellow on the downloadable Excel data table spreadsheet Some useful calculation information: Percentage Error: In several laboratory settings the true value of a quantity being measured is known. This comparison is normally done by a percentage error calculation, where the accepted value of the quantity is compared to the experimental result. The formula for achieving this comparison is given below: % Error = Experimental Accepted Accepted *100 Where, the parallel lines in the numerator indicate an absolute value. GRAPHS (None Required) GRAPH ANALYSIS (None Required) Speed of Sound in Air - Page 9
CONCLUSION (worth up to 40 points) See the Physics Laboratory Report Expectations document for detailed information related to each of the four questions indicated below. 1. What was the lab designed to show? 2. What were your results? 3. How do the results support (or not support) what the lab was supposed to show? o For example, you might want to say something like: "From equation # I'd expect that the will because. Based on the data I collected I found that ; thereby." 4. What are some reasons that the results were not perfect? QUESTIONS (worth up to 10 points) DO NOT forget to include the answers to the FOUR questions that were asked within the experimental procedure 1) Suppose that the temperature in the laboratory would have been 10 o C different (up or down) than it was; would the experimental results have been significantly different? Again, be specific and quantitative in your conclusion. 2) It is possible that a different speaker frequency would have produced a node (not every node, just some of them) in the same location as a previous frequency. Meaning, is it possible to have a node located where you would have found a node before if the frequency of the speaker were different? Justify this statement. You will need to draw an illustration of an open tube to assist with this answer. 3) Would the experiment differ, procedurally or in results, if the sound coming from the speaker were louder/softer? Speed of Sound in Air - Page 10