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Music Physics 202 Professor Vogel (Professor Carkner’s notes, ed) Lecture 9
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Music A musical instrument is a device for setting up standing waves of known frequency A standing wave oscillates with large amplitude and so is loud We shall consider an generalized instrument consisting of a pipe which may be open at one or both ends Like a pipe organ or a saxophone There will always be a node at the closed end and an anti-node at the open end Can have other nodes or antinodes in between, but this rule must be followed Closed end is like a tied end of string, open end is like a string end fixed to a freely moving ring
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Sound Waves in a Tube
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Harmonics Pipe open at both ends For resonance need a integer number of ½ wavelengths to fit in the pipe Antinode at both ends L = ½ n v = f f = nv/2L n = 1,2,3,4 … Pipe open at one end For resonance need an integer number of ¼ wavelengths to fit in the pipe Node at one end, antinode at other L = ¼ n v = f f = nv/4L n = 1,3,5,7 … (only have odd harmonics)
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Harmonics in Closed and Open Tubes
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Adding Sound Waves If two sound waves exist at the same place at the same time, the law of superposition holds. This is true generally, but two special cases give interesting results: Adding harmonics Adding waves of nearly the same frequency
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Adding Harmonics Superposition of two or more sound waves that are all harmonics of the same fundamental frequency one may be the fundamental The sum is more complicated than a sine wave but the resultant wave oscillates at the frequency of the fundamental simulation link simulation link
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Beat Frequency You generally cannot tell the difference between 2 sounds of similar frequency If you listen to them simultaneously you hear variations in the sound at a frequency equal to the difference in frequency of the original two sounds called beats f beat = |f 1 –f 2 |
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Beats
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Beats and Tuning The beat phenomenon can be used to tune instruments Compare the instrument to a standard frequency and adjust so that the frequency of the beats decrease and then disappear Orchestras generally tune from “A” (440 Hz) acquired from the lead oboe or a tuning fork
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The Doppler Effect Consider a source of sound (like a car) and a receiver of sound (like you) If there is any relative motion between the two, the frequency of sound detected will differ from the frequency of sound emitted Example: the change in frequency of a car’s engine as it passes you
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Stationary Source
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Moving Source
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How Does the Frequency Change? If the source and the detector are moving closer together the frequency increases The wavelengths are squeezed together and get smaller, so the frequency gets larger If the source and the detector are moving further apart the frequency decreases The wavelengths are stretched out and get larger so the frequency gets smaller
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Doppler Effect
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Doppler Effect and Velocity The degree to which the frequency changes depends on the velocity The greater the change the larger the velocity This is how police radar and Doppler weather radar work Let us consider separately the situations where either the source or the detector is moving and the other is not
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Stationary Source, Moving Detector In general f = v/ but if the detector is moving then the effective velocity is v+v D and the new frequency is: f’ = v+v D / but =v/f so, f’ = f (v+v D / v) If the detector is moving away from the source than the sign is negative f’ = f (v v D /v)
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Moving Source, Stationary Detector In general = v/f but if the source is moving the wavelengths are smaller by v S /f f’ = v/ ’ ’ = v/f - v S /f f’ = v / (v/f - v S /f) f’ = f (v/v-v S ) The the source is moving away from the detector then the sign is positive f’ = f (v/v v S )
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General Doppler Effect We can combine the last two equations and produce the general Doppler effect formula: f’ = f ( v±v D / v±v S ) What sign should be used? Pretend one of the two is fixed in place and determine if the other is moving towards or away from it For motion toward the sign should be chosen to increase f’ For motion away the sign should be chosen to decrease f’ Remember that the speed of sound (v) will often be 343 m/s
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The Sound Barrier A moving source of sound will produce wavefronts that are closer together than normal The wavefronts get closer and closer together as the source moves faster and faster At the speed of sound the wavefronts are all pushed together and form a shockwave called the Mach cone In 1947 Chuck Yeager flew the X-1 faster than the speed of sound (~760 mph) This is dangerous because passing through the shockwave makes the plane hard to control In 1997 the Thrust SSC broke the sound barrier on land
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Bell X-1
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Thrust SSC
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