Experiment 11: Addition of Waves

Transcription

1 N ame Partner(s): Experiment 11: Addition of Waves Objectives Understand the addition of waves using the superposition principle, through manifestations of two source interference, standing waves, and spectral analysis. Equipment Pre-Lab Computer w ith LoggerPro and Labview, speakers, microphone, tubes, meter stick A list of Activities is to be completed before this Lab. Completing these Pre-Lab assignments w ill expedite your progress during the lab period. A s part of your preparation you are also expected to read through the lab manual before the lab. You will be pressed for time during the lab. Since successful completion of all lab activities counts towards your final lab grade it will be important to be well prepared by doing Pre-Lab assignments and reading the entire lab before attending the lab. Points earned today Lab Challenge Total Instructor Initials XI-1

2 Pre-Lab for LAB#11 Complete the following before you attend class: Under most conditions, the addition of waves follows the principle of linear superposition, w hich states that the sum of tw o or more w aves at a particular point in space and at a particular time is equal to the sum of the individual wave amplitudes. This principle is general to all types of w aves; in this lab, you w ill study several aspects related to the addition of sound w aves. For example, standing w aves result from the addition of two w aves traveling in opposite directions. This situation often arises w hen w aves reflect from boundaries, such as the closed or open ends of a tube. You can visit this website for an applet that will help you visualize this process: / ph14e/ stwaverefl.htm Consider a pipe that is closed at one end. Sketch the standing wave pattern in each of the following situations; showing the regions of high and low air pressure variations (pressure antinodes and pressure nodes). Then formulate equations that relate the w avelength λ and frequency f to the length L of the pipe. a) Tube with one open end: fundamental. λ 1 = f 1 = b) Tube w ith one open end: first overtone (3 rd harmonic). λ 3 = f 3 = c) Find the ratio of the first overtone and fundamental frequencies: f 3 / f 1 = XI-2

3 d) Tube with both ends open: fundamental. λ 1 = f 1 = e) Tube w ith both ends open: first overtone (2 nd harmonic). λ 2 = f 2 = f) Find the ratio between fundamental and first overtone frequencies: f 2 / f 1 = XI-3

4 List of Today s Activities Standing waves Problem Solving Lab Activity Group Work Problem Measuring standing wave resonances in a tube Spectral analysis (aka Fourier analysis) Problem Solving D emonstration Lab Activity Problem Solving Discuss concept questions M easuring the Fast Fourier Transform (FFT) of a single tone Comparison of piano and flute tones musical timbre Beats - A dding w aves w ith different frequency FFT of a snap FFT of tube resonances Group Work Problem XI-4

5 Activity 1 Laboratory Group Work Problem Schumann resonances (Earth s standing waves) Radio w aves can reflect off a layer of the atmosphere called the ionosphere, w here a large portion of the atoms and molecules have been ionized by solar UV. Radio waves can also reflect off the Earth s surface. In effect, radio waves bounce back and forth off the surface and ionosphere, just as visible light can bounce between two metal mirrors. Lightning and other natural phenomena generate radio w aves w ith a range of frequencies. Those frequencies that are just right will travel around the earth, meet themselves in phase, and form standing w aves. The set of frequencies that w ill do this are known as Schumann resonances, in honor of Winfried Otto Schumann ( , Germany), w ho predicted their existence in In this exercise you will estimate the first 5 Schumann resonances using what you know about standing w aves. The picture below illustrates the standing wave pattern for one of the resonances. Keep in mind that the atmosphere is really just a thin skin surrounding the Earth, so that the w aves really circle the Earth at close to the Earth s radius, w hich is r e ~ 6378 km. 1) How many wavelengths fit around the loop in this picture? 2) What is the wavelength? λ = ( ) units? 3) Write a general equation for the standing w aves: nλ = Using this equation, fill in the following table. Compare your calculated frequencies (f calc ) with the observations (f obs ) at: / monitoring-system/ earth-rhythms.html resonance wavelength (km) f calc (Hz) f obs (Hz) XI-5

6 Activity 2 Standing waves in a tube In this activity you will use a microphone to directly measure the standing wave resonances in an open ended tube and compare this to calculated values. Set the microphone near the speaker, and open the LoggerPro template microphone.cmbl. This program allows you to directly monitor a microphone s output as a function of time. First, run the program by clicking on the green Collect button. You should see the microphone s output on the screen as it picks up background noise from the room. Now open the Labview program output sound.vi, which will allow you to generate a single tone from the laptop s speakers. To run the program, hit Ctrl-R, or the right arrow button at the top of the window. To stop, hit Ctrl-. or one of the stop buttons at the top of the window. You can adjust the sound frequency by clicking on the slider control, or by using the digital indicator. Verify that the microphone output is a clear sinusoidal signal. Place your speaker at one end of the plastic tube, and the microphone at the other end. Adjust the volume to be as low as possible, while still producing a clear signal on the computer. This will minimize interference with neighboring groups (and prevent headaches). You can adjust the horizontal and vertical axes as needed. Vary the frequency between Hz as you monitor the microphone s output on the computer. You can precisely adjust the frequency by using the digital indicator on the tone generator program. A t one or more resonant frequencies w ithin this range, standing w aves are formed, and the microphone output should be particularly large. The process is similar to blowing gently over a soda bottle, where for certain conditions, a pronounced sound is heard. If you don t notice any values where the microphone output is larger, try varying the frequency over this range more slow ly. The low est frequency resonance you can pick out is likely to be the tube s fundamental frequency. Estimate how precisely you can locate the fundamental frequency, by having your partner(s) repeat the process. Fundamental frequency = Error estimate = ± When you have precisely identified the fundamental frequency, record this value in Table 1 and repeat this process for the next 3-4 resonances. The higher resonances may be more difficult to identify. To accurately predict the resonance frequencies for comparison, it is necessary to make an end correction, which compensates for the fact that the vibrating air column extends slightly beyond the tw o ends of the tube. A s a result, the tube has an effective length, L eff, given by: XI-6

7 L eff = L d, w here L is the measured length of the tube, and d is the diameter. Record these values in Table 1, and calculate L eff. N ow calculate the standing w ave frequencies for your tube (f n calc in Table 1). Use the equations you derived in the PreLab, and take L eff as the length of the tube. Use the speed of sound that you measured in class last w eek. Enter these values in Table 1. TA BLE 1 Resonance # f n (Hz) Ratio: f n / f 1 f n calc (Hz) Percent error = f n - f n calc / f n calc n=1 f 1 /f 1 = 1 2 f 2 /f 1 = 3 f 3 /f 1 = 4 f 4 /f 1 = 5 f 5 /f 1 = L = d = L eff = Compute the ratio of measured resonance frequency, f n, to the measured fundamental frequency, f 1. Does the ratio f n /f 1 follow the pattern you expect from the PreLab? Why or why not? Were your measured frequencies close to your calculated values? Why or why not? XI-7

8 Have your instructor check your work before you proceed. XI-8

9 Activity 3 Spectral Analysis The human brain/ ear system is a remarkable instrument. Sounds with intensity spanning 12 orders of magnitude can be detected, allowing us to hear soft whispers one minute, and jet engines the next. The brain/ ear system also performs spectral analysis of sounds in real-time. This ability allows us to pick out individual instruments in a symphony, and eavesdrop on a conversation in a crowded room. This activity will introduce you to some concepts related to spectral analysis. The Fast Fourier Transform (FFT): The FFT of a signal produces a spectrum, which is a plot of ampl i t ude ver sus fr equency. Any time-domain signal f (t) can be represented as the sum of sine w aves w ith different frequency (cosines are simply shifted sine w aves): The more complicated the signal, the more sine w aves are needed. It is often convenient to view the frequency spectrum of complicated signals, w hich essentially is a plot of the coefficients a n, b n as a function of frequency. The process of calculating these coefficients from f (t) is known as a Fourier transform. The fast Fourier transform is a convenient algorithm for performing this calculation using your computer. You will use Logger Pro to calculate the FFT frequency spectrum of familiar sound w aves. FFT of a single tone: A pure sine w ave tone has the simplest FFT, because only one sine w ave in the Fourier series is required. Load the LoggerPro template FFT.cmbl. You w ill see tw o graphs: one is the time domain signal picked up by your microphone, the second is the calculated FFT frequency spectrum. Position your microphone near the speaker (the tube is not needed for this part), and set your computer to output a tone w ith frequency between Hz. Run the LoggerPro program by clicking on the Collect button. Sket ch w hat you see in the tw o panels: Time domain Frequency domain Check that the FFT signal corresponds to the frequency output: Output frequency = ( ) FFT peak = ( ) XI-9

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11 Activity: Beats (adding waves with different frequency) Activate the second output in the Labview program output sound.vi. You can now output two sine w aves at the same time. Set one of the frequencies and adjust the other to match. A s the tw o frequencies begin to match, you w ill hear a slow modulation in amplitude. This is called a beat phenomenon. Measure the spectrum using the LoggerPro FFT program, and sketch the time and frequency domain signals you observe. For clarity, the two frequencies should differ by more than 4 H z. Be sure to adjust the axes in the graphs to clearly observe the beats in both domains. Time domain Frequency domain M easure the beat period in the time domain signal: T beat = ( ) Record the frequencies: f 1 = ( ), f 2 ( ) N ow compute the expected beat period: T 1 = = calc f f ( ) Do the numbers match? Beat phenomena provide a useful w ay to measure the frequency of unknow n signals. With the Labview program running, have a partner set one of the output frequencies to an unknown value, and toggle the sw itch to hide the slider. You can now indirectly measure this frequency by tuning the frequency of the first output until you hear very slow beats. How close did you get? Matching frequency = ( ) Unknow n frequency = ( ) Switch roles with your partners until everyone has had a chance. This is an example of heterodyne detection, which is widely used to detect high frequency signals in electronics. M usical timbre (comparing a piano and a flute): H ow do w e tell the difference between a piano and a flute? Musical notes played on real instruments are not pure sine wave tones like you ve been using today: they also contain additional harmonics. You will find two mp3 files on your desktop that contain examples of the same note played on a piano and on a flute. 2 1 XI-11

12 To compare the spectrum of each instrument using the LoggerPro FFT program, you first need to activate the trigger function by clicking on the Data Collection button just to the left of the green Collect button. On the Collection tab, set the length to 0.06 s, and on the Triggering tab, enable triggering on a sensor value that is ~ 0.03 au l arger than the background noise level picked up by the microphone (this number appears in the bottom left corner of the window). A fter setting these values, exit the dialogue box. N ow w hen you click the green Collect button, the program will wait for a trigger. Play an mp3 file to trigger the data collection, and sketch the FFT spectra from 100-2kH z: Piano Flute What is different between the tw o sets of spectra? XI-12

13 Which note is being played? (you may w ant to refer to Lab 10) N ote = XI-13

14 FFT of a complicated signal: Now trigger data collection with a sharp impulse of sound, such as that produced when you snap your fingers or clap your hands. Many more sine wave components are present in such a complicated signal. A s a result, the FFT show s a spectrum that extends over the whole range of human hearing (and beyond!). Sketch what you see in the two panels from kHz: Time domain Frequency domain Activity: FFT of tube resonances N ow that you have successfully measured the standing w ave resonances of a tube the hard w ay, you can use the FFT to do the same measurement much more quickly. A s you ve seen, a continuum of sine waves is generated when you snap your fingers. Frequencies within this continuum that correspond to standing w ave resonances w ill be preferentially transmitted through the tube. Position your microphone at one end of your tube. Trigger data collection by snapping your fingers (or clapping your hands) at the other end of the tube. The FFT should show a series of peaks at the tube s resonant frequencies. You may need to repeat this a few times to get clear peaks in the FFT. Record the frequencies f n FFT in Table 2, and compare these values w ith those you measured (f n ) and calculated (f n calc ) earlier (Table 1). Table 2 Resonance # f (Hz) n FFT f (Hz) n f (Hz) n calc n = XI-14

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16 Do your values agree? If not, speculate on w hat could cause any discrepancies. This is an example of Fourier Transform spectroscopy, which is widely used in modern infrared spectrometers (FT-IR) and nuclear magnetic resonance (N M R and M RI). These techniques have applications throughout the physical and life sciences. End of Lab 11 When you are finished, close both LoggerPro and Labview. D o not save any changes. Handy links: Fourier synthesis applet: / fourier/ End corrections in wind instruments: / jw/ musfaq.html#end Flute acoustics: / jw/ fluteacoustics.html Basics of digital filtering: / dfilter/ XI-16