Ch16Lectures Page 1 Ch16Lectures Thursday, April 16, 2009 12:22 PM
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Ch16Lectures Page 4 The following animation illustrates the interference of two wave pulses travelling in opposite directions: http://physics.info/interference/ The following animation illustrates the interference of two waves travelling in opposite directions to produce a standing wave: http://www.physicsclassroom.com/mmedia/waves/swf.cfm The following animation illustrates more complex examples of interference for transverse waves: http://serc.carleton.edu/nagtworkshops/deepearth/ activities/40826.html
Ch16Lectures Page 5 The following animation contrasts travelling waves and standing waves: http://physics.info/waves-standing/ The following animation illustrates interference of waves travelling in two dimensions (such as waves on the surface of water): http://phet.colorado.edu/en/simulation/wave-interference The following pages illustrate reflection of a wave from a boundary and the creation of standing waves: http://www.acs.psu.edu/drussell/demos/reflect/reflect.ht ml http://www.animations.physics.unsw.edu.au/jw/waves_s uperposition_reflection.htm#reflections http://www.animations.physics.unsw.edu.au/jw/waves_s uperposition_reflection.htm#superposition The following page illustrates standing waves: http://en.wikipedia.org/wiki/standing_wave Video clip of standing waves on a string (captions in French): http://www.youtube.com/watch?v=4boeatjk7dg
Transmission and reflection of waves at boundaries Ch16Lectures Page 6
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Fundamental frequency and higher harmonics Ch16Lectures Page 8
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Ch16Lectures Page 11 Standing sound waves in tubes The previous discussion was about standing waves on strings, which are relevant for musical instruments such as guitars or pianos. However, wind instruments make sounds via standing sound waves, and for them there are more possibilities. For simplicity, model a wind instrument (flute, trumpet, etc.) by a straight tube filled with air. The tube could be open at both ends (which approximates a flute), open at one end and closed at one end (which approximates a trumpet), or closed at both ends (no musical instrument is like this, but the textbook includes this case for completeness). The standing waves on such simple tubes can be modeled as follows, where the amplitude of the wave represents pressure:
Ch16Lectures Page 12 OMIT Sections 16.5 AND 16.7 Interference of Waves from Two Sources The basic simplifying assumption in this section is that the two sources are in phase. interference at points along the same line as the line joining the sources
Ch16Lectures Page 13 interference at other points Interference at points along the same line as the line joining the sources is illustrated in the following two figures. Note that the resulting interference pattern depends on the distance between the two sources. (Also note that the two sources have the same amplitude and are in phase.)
In two or three dimensions, the pattern is more complex, as shown by the following figures: Ch16Lectures Page 14
(Remember, we make the usual simplifying assumptions that the sources are in phase and have the same frequency.) Ch16Lectures Page 15
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Ch16Lectures Page 17 Selected problems and solutions CQ 1 Light can pass easily through air and water, but light can reflect from the surface of a lake. What does this tell you about the speed of light in air and in water? Solution: The fact that light can reflect from the boundary of air and water shows that the speed of light in the two media is different. (This is like the story of a wave travelling across the boundary between a thick string and a thin string, with some of the wave transmitted across the boundary, and some of the wave reflected.) CQ 4 A guitarist finds that the frequency of one of her strings is too low. Should she increase or decrease the tension in the string? Explain. Solution: The frequency is proportional to the speed
Ch16Lectures Page 18 of the wave on the string, but the speed of the wave is proportional to the square root of the tension. Thus, the tension should be increased. CP 6 You are holding one end of an elastic cord that is fastened to a wall 3.0 m away. You begin shaking your end of the cord at 3.5 Hz, creating a continuous sinusoidal wave of wavelength 1.0 m. How much time will pass until a standing wave fills the entire length of the cord? Solution: The standing wave will be set up once the wave makes a round trip to the wall and back to your hand. The time needed for the round trip is the distance divided by the speed. The speed of the wave is Thus, the time needed for a round trip is CP 8 The figure shows a standing wave oscillating at 100 Hz. Determine the wave speed.
Ch16Lectures Page 19 Solution: Note that 3 half-wavelengths fit into a 60 cm length; thus the wavelength of the wave is The wave speed is therefore CP 12 A 121-cm-long, 4.00 g string oscillates in its m = 3 mode with a frequency of 180 Hz and a maximum amplitude of 5.00 mm. Determine the (a) wavelength and (b) the tension in the string. Solution: (a) In the m = 3 mode, 3 half-wavelengths of the wave fit into the 121 cm length of the string. Thus, the wavelength of the wave is:
Ch16Lectures Page 20 (b) We know the length and mass of the string, so we can determine its linear density. If we could independently determine the speed of the wave, then we could determine the tension as well, using But we can independently determine the speed of the wave, because we know the frequency and wavelength of the wave: Now we'll work with the relation between the speed of the wave and the tension and linear density of the string. Let's solve for the tension:
Ch16Lectures Page 21 CP 23 A drainage pipe running under a freeway is 30.0 m long. Both ends of the pipe are open, and wind blowing across one end causes air inside to vibrate. (a) Determine the fundamental frequency of air vibration in this pipe on a day when the speed of sound is 340 m/s. (b) What is the frequency of the lowest harmonic that would be audible to a human ear? (c) What will happen to the fundamental frequency in the later afternoon as the air begins to cool? Solution: (a) The standing-wave frequencies for an openopen pipe are Therefore, the fundamental frequency is (b) The lowest frequency audible to a human ear is about 20 Hz. By calculating with various values of m, we can determine the lowest mode whose frequency exceeds 20 Hz:
Ch16Lectures Page 22 Hz: Thus, the lowest standing-wave frequency that is audible to a human ear is the third harmonic (i.e., the m = 4 mode), which has a frequency of about 23 Hz. (c) As the air temperature decreases, the speed of sound also decreases, and so all of the standing-wave frequencies in the pipe also decrease. CP 28 Two loud speakers in a 20 degrees Celsius room emit 686 Hz sound waves along the x-axis. What is the smallest distance between the speakers for which the interference along the x-axis is destructive? Solution: As you can see from Figure 16.26 on Page 523 of the textbook, the smallest distance between the speakers that results in destructive interference is equal to half of a wavelength of sound. First let's determine the wavelength of sound, using the fact that the speed of sound in air at the given temperature is 343 m/s:
Ch16Lectures Page 23 Thus, the smallest distance between the speakers that results in destructive interference is 0.5/2 = 0.25 m = 25 cm. CP 32 Two loudspeakers 2.0 m apart emit 1800 Hz, equal-intensity sound waves in phase into a room where the speed of sound is 340 m/s. Is the point 4.0 m directly in front of one of the speakers, perpendicular to the plane of the speakers, a point of maximum constructive interference, perfect destructive interference, or something more complex? Solution: First determine the wavelength of the sound waves emitted from the speakers. Then determine the difference in the distances from each speaker to the point of interest, and then express this distance in terms of the wavelength of the sound.
Ch16Lectures Page 24 The path difference is therefore In terms of wavelengths of sound, the path difference is Because the path difference differs from a whole number of wavelengths by half a wavelength, the point of interest is a point of destructive interference.