1. Sound
Unit 8

2. How do we perceive sound?
What is sound?
How do we perceive sound?

3. Sound Sound is a longitudinal wave.
We most commonly experience sound waves traveling through air.
However, sound can travel through any type of matter.
We detect sound using our ears or a microphone.

4. Speed of Sound
Because sound is the longitudinal vibration of matter, sound waves require matter to travel.
Sound waves cannot travel through space.
The speed of the sound wave depends on the material the wave is traveling through (much the same way the speed of mechanical waves depended on the properties of the string).

5. Speed of Sound The speed of sound also depends on temperature.
For our purposes, we will assume that our sound waves are traveling through air at 20° C.
In this case, the speed of sound is 343 m/s.

6. Vocabulary of Sound
There are many common terms associated with sound in our every day lives.
The loudness of sound is related to the intensity of the sound wave (more on this in a minute.
The pitch of a sound refers to how high (like a piccolo) or low (like a bass) a sound is.

7. Vocabulary of Sound
The pitch is determined by the frequency of the sound wave.
A human ear can detect frequencies ranging from 20 Hz to 20,000 Hz. This is known as the audible range.
Sounds with frequencies above the audible range are called ultrasonic.
Sounds will frequencies below the audible range are called infrasonic.

8. Pressure Waves
In the last unit, we described waves in terms of the oscillations of the particles making up the medium.
While sound waves can be described this way, it’s often difficult to measure the displacement of air molecules.

9. Pressure Waves Instead, we analyze sound waves by looking at pressure.
When the molecules are closer together, the pressure is greater.

10. Intensity of Sound
Decibels

11. Intensity of Sound
We experience the intensity of sound through loudness.
Recall that intensity is a measure of the power delivered by a wave over a given area.
Intensity is measured in W/m2.

12. Intensity of Sound
The human ear can detect a wide range of intensities.
However, what we perceive as loudness is not actually proportional to the intensity of the sound.
To produce a sound that is twice as loud requires a wave that has about 10 times the intensity.

13. Decibels
Because of this relationship between loudness and intensity, sound intensity levels are described using a logarithmic scale.
The unit of this scale is a bel, though the related decibel is much more commonly used.

14. Decibels
The sound level (loudness) of any sound is defined in terms of the intensity by:
Here, I0 is the intensity of a chosen reference level.

15. Decibels
Generally, we pick I0 as the threshold of hearing for a good ear:
So, for example, if we had a sound with intensity I = 1.0 x

16. Example: Sound on the Street
At a busy street corner, the sound level is 70 dB. What is the intensity of sound there?

17. Example: Loudspeaker

18. Homework
Read 12-1 and 12-2.
Do problems 3, 5, 9, and 11 on page 347.

19. Announcements
If you own a (portable) musical instrument, please come talk to me after class today.
There will be a daily exercise quiz on Friday

20. Sources of Sound

21. Sources of Sound
Sound is generated by any object that is vibrating in a medium (such as air).
While almost any object can be a source of sound, most are very difficult to analyze.
Today, we will be looking at one of the most common sources of sound: musical instruments.

22. Sources of Sound There are two main types of musical instruments:
For a string instrument, standing waves are produced when the string is plucked or bowed. This generates sound waves of the same frequency.
For a wind instrument, vibrating columns of air produce standing waves within the instrument.

23. String Instruments
We saw last unit how standing waves can be established on a string.
When a string is plucked the wave that results is actually a superposition of standing waves.
The dominant pitch that is heard corresponds to the fundamental frequency.
However, there are overtones corresponding to higher frequencies as well.

24. String Instruments
Recall from last unit that the fundamental frequency is given by:
Where v is found using

25. String Instruments
The pitch of many string instruments can be controlled by placing a finger on the string.
When this happens the effective length of the string is shortened.
This results in a change in the frequency of vibration (since L has changed but v has not).

26. Example: Violin
A 0.32 m long violin is tuned to play the A above middle C at 440 Hz. a) What is the wavelength of the standing wave on the string that corresponds to this note? b) What are the frequency and wavelength of the resulting sound wave? c) Why is there a difference?

27. Example: Piano Strings
The highest key on a piano corresponds to a frequency that is 150 times the frequency of the lowest key. If the string for the highest note is 5 cm long, how long would the string for the lowest note have to be if it had the same mass per unit length and was under the same tension? Based on your answer, can you explain why the strings for the low notes of a piano are heavier than those for the high notes.

28. Wind Instruments
Wind instruments produce sound from standing compression waves in a column of air.
This standing wave is initiated by a vibration in a reed on a person’s lips.
Sometimes the vibration is generated by directing a stream of air against one edge of the opening of the tube.

29. Wind Instruments
A string vibrating at the fundamental frequency has a node at either end.
The result is similar for wind instruments.
However, now it is the air itself that is vibrating to cause the sound.

30. Wind Instruments There are two types of wind instruments.
An open tube, where there is an opening at either end of the tube.
A closed tube, where one end of the tube is blocked.
In both cases, we can describe the wave in terms of either displacement or pressure.

31. Open Tubes In an open tube, the air molecules vibrate horizontally.
Since the air is free to move at both ends, there will always be displacement antinodes at either end of the tube.

32. Open Tubes
Since there must be at least one node to form a standing wave, the fundamental frequency corresponds to one displacement node.
This corresponds to two nodes and one antinode for the pressure wave (just like the string).

33. Open Tubes
Looking at the pressure wave, we can see that λ1 = 2L. So,

34. Closed Tubes
In an closed tube, the air molecules also vibrate horizontally.
However, now air is only free to vibrate at one end. This creates a displacement node at the closed end. And an antinode at the other.

35. Closed Tubes Based on this, we can see that λ1 = 2L. So,
Which is half the result for the open tube.

36. Closed Tubes
However, there is a second difference. Because the open end must have an antinode, only odd harmonics are allowed. Thus

37. Example: Organ Pipes
What will be the fundamental frequency and first three overtones for a 26 cm long organ pipe if the pipe is a) open, and b) closed?

38. Homework Read 12-4. Do problems 25, 28, 29 on page 348.
For problem 29, refer to the table at the beginning of section 12-4.

39. Problem Day Do problems 26, 27, 33, and 34 on page 348.
We will whiteboard these at 3:30.

40. Homework
Do problems 35, 36, and 37 on page 348.

41. Whiteboarding Groups Group Members Problem 1 25 2 26 3 27 4 28 5 29 633 7 34 8 35 9 36 10 37

42. Quality of Sound

43. Quality of Sound
Whenever we hear a sound, we are always aware of its loudness and its pitch.
However, we are also aware of a third aspect called the “quality” of the sound.
An example of this can be found in instruments: a piano, a clarinet, and a human voice can all produce the same note. However, the sounds are clearly different.

44. Quality of Sound
The quality of a sound is that which allows us to tell the difference between instruments or other sources of sound.
In music, the terms timbre and tone color are also used.
Like loudness and pitch, quality can be related to a physical characteristic of the sound wave.

45. Quality of Sound
The quality of a sound depends on the number of overtones that are present and the amplitude of each overtone.
When a note is played on an instrument, the fundamental frequency and several overtones are present simultaneously.

46. Quality of Sound
These waves all get added together through the principle of superposition to produce the sound we actually hear.
However, each overtone has a different amplitude (usually smaller than that of the fundamental).

47. Quality of Sound
The relative amplitudes for the overtones are different for each instrument.
The differences in amplitude produce different composite waveforms for each instrument, giving each a unique sound.

48. Sound Spectrum
It is possible to identify the different frequencies that make up a waveform along with the relative amplitude of each frequency.
This process is called a Fourier transform. (you won’t be required to know how to do this)

49. Sound Spectrum
Through a Fourier transform, a bar graph showing the amplitude of each frequency that makes up a sound can be generated.
This graph is called the sound spectrum.
The fundamental frequency is generally the loudest (has the greatest amplitude) and is therefore what is heard as the pitch.

50. Sound Spectrum
Sound spectra for different instruments. The spectrum changes depending on the note being played.
The clarinet acts as closed tube (odd harmonics only) at low frequencies and an open tube (all harmonics) at high frequencies.

51. Sound Spectrum
An ordinary sound, like a clap, has a definite loudness but no clear pitch.
A noise like this is a jumble of frequencies that bear little relation to each other.
The sound spectrum would not have a single dominant frequency.

52. Homework
Read section 12-5.
Work on your paper.

53. Announcements We will be having our next test on Friday.
We will have a review day on Thursday.
There will be no daily exercise quiz.

54. Interference

55. Interference
We saw in the last unit that when two waves pass through the same point in space, they interfere with each other.
We have learned that sound is a wave.
Therefore, the phenomenon of interference also occurs with sound waves.

56. Interference in Space
Imagine we have two speakers separated by a distance d.
The speakers are oscillating at the same frequency and are in phase (so they form wave fronts at the same time).

57. Interference in Space
The curved lines represent crests of sound waves at an instant in time.
Recall that a sound wave is longitudinal.
A crest corresponds to a compression of air. A trough corresponds to a rarefaction (expansion).

58. Interference in Space
A person standing at point C will experience a loud sound, because the crests of two waves are meeting and constructive interference is occurring.
At point D, little sound is heard because destructive interference is occurring.

59. Interference in Space
The question naturally arises, how do we predict what type of interference will occur at a point?
To answer this question, let’s redraw this situation using the waveforms emitted by the speakers instead of the wave fronts themselves.
NOTE: Pay close attention here. We will look at a nearly identical situation when we study optics.

60. Interference in Space First, let’s consider the constructive case.
When the two waves arrive at point C, they both have have crests or troughs.
These add together to produce constructive interference.

61. Interference in Space
In the destructive case, the wave from speaker B has traveled a longer physical distance than the wave from speaker A.
Since they both started in phase, wave B will be at a different point than wave A.

62. Interference in Space
As a result, the trough from A adds with the crest from B, producing destructive interference.
To figure out where these points of destructive interference are going to be, let’s draw in an imaginary line from A to point E.

63. Interference in Space
The point E is chosen so that segment AD is equal to ED.
This distance BE is the extra distance wave B has to travel.
Notice that if BE is equal to ½ a wavelength, the waves will arrive at point D out of phase.

64. Interference in Space
Also, notice that interference does not depend on which crest or trough of the wave is at point C or D.
So, if the distance BE had been 1.5 or 2.5 wavelengths, we would still have seen destructive interference.

65. Interference in Space
This tells us the condition for destructive interference:
In other words, if the path length difference is a multiple of a half wavelength.

66. Interference in Space
We can also establish the condition for constructive interference:
In other words, if the path length difference is a multiple of a full wavelength.

67. Example
Two speakers are 1 m apart. A person stands 4 m from one of the speakers. If each speaker is producing waves with a frequency of 1150 Hz, how far from the second speaker should the person be so that he hears no sound?

68. Interference in Time
We have been discussing interference of sound waves that takes place at different points in space.
However, an interesting phenomenon occurs when two sounds are close in frequency, but not identical.
In this case, we get interference in time.

69. Interference in Time
When two sources are close in frequency, the sound level at a given point rises and falls in time.
This phenomenon is known as beats.
This occurs because the two waves are sometimes in phase and sometimes out of phase.

70. Interference in Time
Let’s consider two waves of equal amplitudes with frequencies of 50 Hz (red) and 60 Hz (blue).
The graphs below represent the waves at a point equidistant from both sources as a function of time.

71. Interference in Time
Because the two waves have different frequencies, they also have different wavelengths.
This means they are sometimes in phase and sometimes out of phase.

72. Interference in Time At t = 0.05 s, they are completely out of phase.
By t = 0.1 s, they are back in phase.
This rising and falling of the sound intensity is what is heard as beats.

73. Interference in Time
Since the beats in this case are 0.1 seconds apart, ten beats pass that point every second.
The number of beats passing a given point in a second is known as the beat frequency.

74. Interference in Time
Notice that the beat frequency (10 Hz) is just the difference between the two individual frequencies.
This is a general result and can be used to determine the frequencies of individual waves.

75. Example
A vibrating guitar string produces a sound of 400 Hz. When the same string from a second guitar is plucked, 20 beats are observed over 5 seconds. What are the possible frequencies of the second guitar string?

76. You Try
In the last example, what would the beat frequency be if the guitar strings are vibrating at 500 Hz and 506 Hz?

77. Homework Read section 12-6.
Do problems 39, 40, 41, 43, and 46 on page 348.

78. The Doppler Effect

79. Doppler Effect
When an ambulance or fire truck passes by you at high speed, what happens to the sound of the siren?
When there is relative motion between the source of a sound and an observer, a change in pitch is observed.
This is known as the Doppler effect.

80. Doppler Effect
Consider a fire truck that emits a sound at a single frequency f in all directions.
The sound waves travel at the speed of sound, which does not depend on the velocity of the truck.

81. Doppler Effect
When the truck is stationary, observers on either side of the truck will measure the same pitch (frequency) for the siren.
However, if the truck starts moving to the right, the observer on the right will hear a higher pitch than the observer on the left.
Why is this?

82. Doppler Effect
When the fire truck is moving, it emits the sound at the same frequency.
But, since it is moving, the wave fronts in the forward direction are closer together than the wave fronts behind it.

83. Doppler Effect
The observer on the right hears more crest passing his ear per second than does the observer on the left.
As a result, the observer on the right perceives a higher pitch.

84. Doppler Effect for a Stationary Observer

85. Stationary Observer
To quantify the effect, consider first a stationary observer listening to a moving source of sound.
The source emits sounds waves with wavelength λ and frequency f.

86. Stationary Observer
If the source is stationary, the time between successive wave crests is given by
Based on this, we can conclude that in a time T the first wave crest moves a distance of

87. Stationary Observer
But, if the source is moving to the right, then the source has moved a distance
As a result, the effective wavelength is shorter because the the source has moved closer to the first wave front.

88. Stationary Observer
Based on this, the effective wavelength is given by
Plugging in, we get

89. Stationary Observer
To find the effective frequency, we take
This is

90. Stationary Observer Thus, the new frequency is
This is true if the source is moving toward the observer.

91. Stationary Observer
If the source is moving away
Which means

92. Moving Observer
If the observer is moving and the source is stationary, the frequency and wavelength of the waves do not change.
However, the observer sees the waves traveling at a velocity that is different from the speed of sound.

93. Moving Observer
To find the effective velocity of the waves, just add velocities. If you are moving toward the source:

94. Moving Observer
The frequency is then

95. Moving Observer
Plugging in for λ, we get

96. Moving Observer
If the observer is moving away, then

97. Example: Police Siren
The siren of a police car at rest emits a sound with a frequency of 1600 Hz. What frequency will you hear if you are at rest and the car moves at 25 m/s a) toward you? b) away from you?

98. Homework
Read 12-7.
Do problems 49, 50, 51, and 55 on pages

99. Problem Day Do problems 52, and 54 on pages 348-349.
We will whiteboard in a few minutes.

100. Homework
Do problems 80, 81, and 83 on page 350.