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Music

Music. Physics 202 Professor Vogel (Professor Carkner’s notes, ed) Lecture 9. 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

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Music

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  1. Music Physics 202 Professor Vogel (Professor Carkner’s notes, ed) Lecture 9

  2. 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

  3. Sound Waves in a Tube

  4. 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 l n v = lf 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 = ¼l n v = lf f = nv/4L • n = 1,3,5,7 … (only have odd harmonics)

  5. Harmonics in Closed and Open Tubes

  6. 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

  7. 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

  8. 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 fbeat = |f1 –f2|

  9. Beats

  10. 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

  11. 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

  12. Stationary Source

  13. Moving Source

  14. 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

  15. Doppler Effect

  16. 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

  17. Stationary Source, Moving Detector • In general f = v/l but if the detector is moving then the effective velocity is v+vD and the new frequency is: f’ = v+vD/l • but l=v/f so, f’ = f (v+vD / v) • If the detector is moving away from the source than the sign is negative f’ = f (v  vD /v)

  18. Moving Source, Stationary Detector • In general l = v/f but if the source is moving the wavelengths are smaller by vS/f f’ = v/ l’ l’ = v/f - vS /f f’ = v / (v/f - vS/f) f’ = f (v/v-vS) • The the source is moving away from the detector then the sign is positive f’ = f (v/v vS)

  19. General Doppler Effect • We can combine the last two equations and produce the general Doppler effect formula: f’ = f ( v±vD / v±vS ) • 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

  20. 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

  21. Bell X-1

  22. Thrust SSC

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