Review of Standing Waves on a String

Transcription

1 Review of Standing Waves on a String

2 <ct > Below is a picture of a standing wave on a 30 meter long string. What is the wavelength of the running waves that the standing wave is made from? 30 m A. 30 m B. 60 m C. 10 m D. 20 m E. Impossible to tell

3 Ct Could you observe standing waves made from running waves with a wavelength of 2/3 m on a string of length 1 m? (If so, what mode would that be? ) A. Yes, n = 1 B. Yes, n = 2 C. Yes, n = 3 D.Yes, n = 4 E. No

4 A string vibrates with a fundamental frequency of 220 Hz. Besides 220 Hz, which of the following are "resonant frequencies" you might also observe? i) 110 Hz ii) 330 Hz iii) 440 Hz <ct > A: i only B: ii only C: iii only D: i and ii E: all three

5 <ct b> If you want to lower the pitch of a string by two octaves, what must be done to its tension? A. Raise it by a factor of 4 B. Lower it by a factor of 4 C. Lower it by a factor of 2 D. Lower it by a factor of 16 E. None of these / I don t know

6 ct a A string on an instrument plays an A (440 Hz) when plucked. If you lightly touch the string ½ way from one end, and then pluck, you are mostly likely to hear A: Still 440 Hz B: 220 Hz C: 880 Hz D: Something entirely different

7 v T <ct b> = T µ If the tension is increased by a factor of 9 what happens to the speed of waves on a string? A. Goes up by a factor of 3 B. Goes up by a factor of 4.5 C. Goes up by a factor of 9 D. Goes up by a factor of 81 E. None of these / I don t know What happens to the frequency of the fundamental?

8 i Ct b ii L Which of the two points on the string oscillates with the LARGER (higher) frequency? A) Left point (i) B) Right point (ii) C) They both have the same frequency

9 <ct > Below is a picture of a standing wave on a 30 meter long string. What is the wavelength of running waves that the standing wave is made from? A. 30 m B. 60 m 30 m C. 15 m D. Impossible to tell

10 L5 Sound Waves Frequency, Harmonics, Tone Quality (spectral content), Pitch

11 The Sonic Spectrum Infrasound: < 20 Hz Sound: 20 Hz 20,000 Hz (20kHz) Ultrasound: > 20 khz (~10 13 Hz maximum)

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14 Sound- a Pressure Wave PhET Simulation Wave Interference (Sound, Particles)

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16 Components of Sound 1) Longitudinal (along direction of propagation) vibrations, e.g. speaker cone. 2) Material medium capable of transmission of these vibrations, e.g. air. 3) Detector of the sound wave e.g.ear.

17 Longitudinal wave representation

18 Longitudinal wave propagation

19 Pressure wave amplitude about 10-5 atmosphere Displacement wave amplitude about m (smaller than an atom!)

20 In air at atmospheric pressure and at 20 degrees Celsius, the speed of sound, v, is 344 m/s v is temperature dependent, V = 331 m/s +0.6 T, where T is the temperature in degrees Celsius above freezing, i.e. above 0 degrees Celsius

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22 Standing Sound Waves

23 Cylindrical tube Open at both ends (a flute, more or less) L Easy to get overpressure in middle (Ends are just open atmosphere )

24 overpressure L

25 open tube overpressure L n=2, the 2nd mode of the tube

26 Displacement (not pressure) graphs. open tube displacement L

27 Displacement is longitudinal (despite the graph going up ) Pressure nodes <=> displacement antinodes (and vice versa)

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29 Open Tube Frequencies and Wavelengths f = v / λ N = 1 λ = 2L o f = v/2l o = f o N = 2 λ = L o f = v/l o = 2f o N = 3 N = 4 N = 5 N = 6 λ = 2L o /3 f = 3v/2L o = 3f o λ = 2L o /4 f = 4v/2L o = 4f o λ = 2L o /5 f = 5v/2L o = 5f o λ = 2L o /6 f = 6v/2L o = 6f o

30 Pressure waves fit in the open tube n (λ/2) = L Since f λ = v, f n = n (v/2l) Same modes as a string!

31 CT How will the normal mode frequencies of an open tube compare with those of a string (with the same fundamental frequency)? A) All different frequencies (except fundamental) B) All the same frequencies C) Some of the overtones will be the same and some different

32 CT c The air in an open pipe is in the n=2 mode (shown above). A small speck of dust is located 1/2 of the way down the pipe. What does the dust do? A) Wiggles up and down (towards /away from wall of tube) B) Wiggles back and forth (left/right, along the tube ) C) Sit still at center of the pipe D) Something else

33 CT b What is v in the formula f λ = v for an open pipe? A) The speed of sound, 344 m/s B) Speed of vibrations of the pipe wall C) Related to speed of sound, but depends of pipe diameter

34 CT d The speed of sound in helium gas is considerably higher than 344 m/s. If I fill a tube with helium, what will happen to the fundamental tone produced by that tube? A) Goes up in pitch B) Goes down in pitch C) Stays about the same

35 CT If you double the diameter of a long organ pipe, what happens to the fundamental you hear? A) Goes down by an octave B) Goes down, but way less than an octave C) Goes up, by an octave D) Goes up, but way less than an octave E) Something else

36 real tubes - end effect overpressure L Outer node is a bit outside tube (about 0.3 * diameter)

37 Tube closed at one end, open at the other

38 overpressure Closed tubes (closed on one end) L Closed end: pressure antinode open end: pressure node

39 overpressure Closed tubes (closed on one end) L Closed end: antinode open end:node

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41 CT L What is the wavelength of the fundamental (shown above) in a closed tube? A) λ=l B) λ=2l C) λ=4l D) λ=l/2 E) λ=l/4

42 Draw the next higher mode (zero at right end, antinode at left, one extra node in middle) overpressure L Closed end: antinode open end:node

43 Draw the next higher mode (zero at right end, antinode at left, one extra node in middle) overpressure L Closed end: antinode open end:node

44 CT L What is the wavelength of the standing wave (shown above) in a closed tube? A) λ=l B) λ=3l/2 C) λ=3l/4 D) λ=4l/3 E) Something else

45 CT L What is the wavelength of the fundamental (shown above) in a closed tube? A) λ=l B) λ=2l C) λ=4l D) λ=l/2 E) λ=l/4

46 Pressure waves fit in the closed tube differently: (odd n) (λ/4) = L Since f λ = v, f n = (odd n) (v/4l) Lower fundamental Missing harmonics

47 CT c A panpipe tube is sealed at one end but open at the other. A flute is open at both ends. If you have a panpipe and flute of equal lengths, and play the fundamental A) The flute will sound lower B) The panpipe will sound lower C) They will have identical pitch

48 CT12.1.4b A flute playing its lowest note is shown in spectrum A. f 1 refers to the fundamental of the flute. Which spectrum best matches a panpipe (of equal length) playing its lowest note? amplitude A amplitude B f 1,flute 6f 1 f 1,flute 6f 1 amplitude C amplitude D f 1.flute 6f 1 f 1,flute 6f 1 E) None of these looks right.

49 CT12.1.4b A flute playing its lowest note is shown in spectrum A. f 1 refers to the fundamental of the flute. Which spectrum best matches a panpipe half as long as the flute? amplitude A amplitude B f 1,flute 6f 1 f 1,flute 6f 1 amplitude C amplitude D f 1.flute 6f 1 f 1,flute 6f 1 E) None of these looks right.

50 CT How will the harmonics of an open tube compare with those of a stringed instrument (with the same fundamental?) A) Totally different frequencies B) The same frequencies

51 Fourier Synthesis Any periodic complex wave can be synthesized by addition of its harmonics, each with the proper amplitude and phase

52 First and second harmonics with equal amplitudes

53 First and Second Harmonics

54 First and third harmonics with equal amplitudes

55 First and Third Harmonics

56 First and second harmonics with unequal amplitudes

57 First and Second Harmonics

58 2 nd and 3 rd harmonics Synthesis with missing fundamenta l 3 rd and 4 th harmonics

59 Triangular Wave A N = 1, 0, 1/9, 0, 1/25, 0, 1/49,. Odd N only

60 Square Wave A N = 1, 0, 1/3, 0, 1/5, 0, 1/7,. Odd N only

61 Sawtooth Wave A N = 1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7,. All N

62 White Light and White Noise

63 Noise Blowing gently across a microphone