Physics 2310 Lab #2 Speed of Sound & Resonance in Air Objective: The objectives of this experiment are a) to measure the speed of sound in air, and b) investigate resonance within air. Apparatus: Pasco Pasco resonance tube Student Function Generator Oscilloscope Speaker and microphone Cables Thermometer Figure 1: The experimental setup for measuring the speed of sound. Figure 2: The oscilloscope and function generator. Note the configuration of the function generator outputs and the oscilloscope inputs. 9/19/2016 1
Part 1: Investigating Resonance of Sound Waves in Air: Theory: As a sound wave propagates in a tube, it is reflected from each end and interferes with itself. If the wavelength and frequency of the wave are just right for the tube, the forward going waves interfere with the reflected ones and form standing waves in the tube. We say this tube is at resonance with the sound waves and the frequency resonant frequency. The condition for the existence of such standing waves is mostly determined by the physical constraint of the tube. For open end tubes (both ends are open), air molecules are free to move at both ends, allowing maximal displacements at the ends. Therefore, anti-nodes for the displacement are expected at both ends of the tube. For closed-end tubes (one end closed, the other open), the motions of air molecules are constraint at the closed-end, allowing minimal displacements. Therefore, a node for displacements is expected at the closed-end and anti-node at the open end. The first three standing wave modes of the closed case are shown in Fig. 3 which are called the first, second, and third harmonics, respectively. In this experiment, an open-end tube is used to study the standing waves in a tube produced by sound waves. The length of the tube determines the wavelengths of the sound waves that could form standing waves in the tube. As shown in Fig. 3, the distance between adjacent nodes or adjacent anti-nodes corresponds to half of the wavelength. Therefore, the wavelength of the n-th harmonic, λn, is related to the length of the tube, L, through the following equation. Figure 3: The 1-st, 2-nd, 3-rd and 4-th harmonics in a pipe with one end closed and with both ends open (standing waves). 9/19/2016 2
! n = 2L n (1) Provided that the sound speed is a constant in the tube, the frequency of the n-th harmonic, fn, is then f n =!! = # " n "! $ &' n = f 1 ' n (2) 2L % Procedure: 1. Measure the length of the resonance tube. Record the length on the datasheet. 2. With the speaker placed a few centimeters from one end of the tube, as shown in Fig. 1 Connect the speaker to the Low Ω outputs of the frequency generator, as shown in Fig. 2 and connect the High Ω outputs to CH1 of the oscilloscope. DO NOT turn on the function generator at this moment. 3. Make sure that the microphone output to CH2 of the oscilloscope with the BNC -to-clips connectors, and turn on the microphone switch. 4. Turn on the oscilloscope (switch at lower left). 5. Set up the frequency generator as following: (NOTE: it is not too hard to destroy a speaker with inappropriate voltage, so start with the amplitude on its lowest setting before you turn it up or ask for help if you are not certain what you are doing.) a. Make sure that the amplitude of the frequency generator is set to its minimum (clock-wise to the end). b. Turn on the frequency generator. c. With the default frequency 1000.0Hz, slowly increase the amplitude until you can clearly hear a weak tone from the speaker. (You should also see a sinusoidal waveform on the oscilloscope.) d. Adjust the frequency to 100.0Hz (turn knob counter-clockwise). 6. Slowly increase the Frequency from the generator, and monitor the amplitude of the waveform at CH2 on the oscilloscope. Note: use the buttons on the oscilloscope to switch between CH1, CH2 and chop which shows both at the same time. This amplitude (CH2) corresponds to the sound intensity measured by the microphone. Record the frequencies at which maximum intensities at the microphone are measured. The first frequency (should be below 200Hz) of such corresponds to the fundamental mode (n=1), and consequently the 2 nd (n=2), 3 rd (n=3), (n=4), and so on. (You might want to adjust the VOLTS/DIV and SEC/DIV settings on the oscilloscope to make the observation easier.) SA496 9/11/2016 4:58 PM Comment [1]: Do you think this should be removed? If the new oscilloscopes are going to be used it isn t necessary, but it is for the old oscilloscopes. Analysis of the Data: 1.Calculate fn for each mode with Eq.(2). Enter these values into a data table (Excel) along with n. 9/19/2016 3
2. Plot the resonance frequency, fn, as function of n. 3. Find the slope of this plot, and then use Eq.(2) to find the speed of sound in the tube. Part 2: Measuring the Speed of Sound Theory: Because the speed of sound in air is too fast to be determined from distance and time measurements, it will be determined using the standing wave method. The method consists of setting up standing waves at a known frequency f in an air column, finding the wavelength λ of the waves from the length of the air column, and computing the speed from the frequency/wavelength relationship v = fλ. The apparatus for setting up standing waves in an air column is shown in Figure 1. The apparatus consists of a long horizontal plastic tube with a movable piston that is used to vary the effective length of the air column. Sound waves at a given frequency are sent into the tube by a small speaker connected to a harmonic sound generator. When the air column is adjusted to the proper length, standing waves are set up within the column with a node at the piston (fixed end) and an antinode near the open end (free end). A microphone, placed near the antinode end, picks up the resonance vibrations and the output of the microphone is displayed on an oscilloscope. Recall that the condition for standing waves in a tube closed at one end is that an odd number of quarter wavelengths should fit into length (L) of the tube. Because the antinode at the open end does not occur exactly at the end of the tube, two adjacent resonance positions such as x 1 and x 2, shown in Figure 3 must be found. Note that the distance between the two positions, (x 2 -x 1 ), is equal to one-half wavelength of the standing waves. Procedure: 1. Use the following table, or something like it in Excel, to record your data. Node Posn. x 1 (cm) Node Posn. x 2 (cm) Wavelength λ (cm) Wave Speed (m/s) Air Temp. ( o C) 2. Set up the apparatus with the speaker close to the open end of the tube as shown in Figure 1. So that this end is truly open, position the speaker about 5 mm from the end of the tube. Attach the microphone to the apparatus, turn on the microphone amplifier, and connect it to the oscilloscope. Before the function generator is turned on make sure the amplitude is set to the lowest setting. Set the frequency to 550 Hz and place a thermometer near the apparatus. 9/19/2016 4
(a) Push the piston close to the speaker and then slowly pull it back until you detect a resonance indicating the presence of standing waves. You should be able to hear the resonance, as well as see the maximum occur on the oscilloscope. To see better the maximum on the scope, set the time/cm control so that many cycles, spaced closely together, are displayed on the screen. Move the piston back and forth through the resonance and determine the best measurement for its position. Record the position to the nearest 0.1 cm. Repeat for the next adjacent resonance position. Use your data to compute the wavelength of the waves and compute a value for wave speed using the frequency/wavelength expression v = fλ. (Remember to convert x 1 and x 2 from centimeters to meters first so that the speed of sound will be in m/s.) Record the temperature. (b) Repeat the above for frequencies of 800 and 1100 Hz. Analysis of the Data: 1. Use the following example for an analysis table. Frequency f (Hz) Measured Speed Theoretical Speed % Difference 2. Theoretical Basis A theoretical value of the speed of sound in air at temperature T is given by the expression v = 331.0 + 0.6 T m/s, (1) where T is the temperature of the air in degrees Celsius. Use this expression to calculate a theoretical value of the speed of sound for each frequency and record the results in the above analysis table. Also calculate the percent difference between the measured and theoretical values and record in the last column of the table. The percent difference between the measured and theoretical values is found from % error = (measured v - theoretical v)/(theoretical v) x 100 3. Compare your result with what you got in part 1. 9/19/2016 5
Prelab Questions: Question 1: In lab 3 what are we using a resistance for the driven harmonic oscillator? Question 2: What is required for a high quality factor in a frequency vs amplitude plot? SA496 9/11/2016 5:25 PM Comment [2]: Changed this to reflect the Electronics format, but I think the title of postlab questions is clearer to the students about when they need to answer the questions. Also, better questions will need to be made. 9/19/2016 6