1. Lecture 20
Waves and Sound

2. Reading and Review

3. Wave Speed III
A length of rope L and mass M hangs from a ceiling. If the bottom of the rope is jerked sharply, a wave pulse will travel up the rope. As the wave travels upward, what happens to its speed? Keep in mind that the rope is not massless.
a) speed increases
b) speed does not change
c) speed decreases
Answer: a

4. Wave Speed III
A length of rope L and mass M hangs from a ceiling. If the bottom of the rope is jerked sharply, a wave pulse will travel up the rope. As the wave travels upward, what happens to its speed? Keep in mind that the rope is not massless.
a) speed increases
b) speed does not change
c) speed decreases
The tension in the rope is not constant in the case of a massive rope! The tension increases as you move up higher along the rope, because that part of the rope has to support all of the mass below it! Because the tension increases as you go up, so does the wave speed.

5. Sound Intensity
The intensity of a sound is the amount of energy that passes through a given area in a given time.
Expressed in terms of power,

6. R1 R2
If power output is constant in time, then intensity compared to distance from source:
- at R1 must be P / (4πR12)
- at R2 must be P / (4πR22)
Sound intensity from a point source will decrease as the square of the distance.
Surface area of a sphere = 4π r2

7. Sound Intensity

8. Intensity Level
When you listen to a variety of sounds, a sound that seems twice as loud as another is ten times more intense. Therefore, we use a logarithmic scale to define intensity values.
[dB ]
Here, I0 is the faintest sound that can be heard:

9. Sound Intensity Level The common unit is the decibel, dB
The loudness of sound doubles with each increase in intensity level of 10 dB.

10. Decibel Level II a) about the same b) about 10 times
c) about 100 times
d) about 1000 times
e) about 10,000 times
A quiet radio has an intensity level of about 40 dB. Busy street traffic has a level of about 70 dB. How much greater is the intensity of the street traffic compared to the radio?
Answer: d

11. Follow-up: How many radios equal the street sound intensity?
Decibel Level II
a) about the same
b) about 10 times
c) about 100 times
d) about 1000 times
e) about 10,000 times
A quiet radio has an intensity level of about 40 dB. Busy street traffic has a level of about 70 dB. How much greater is the intensity of the street traffic compared to the radio?
increase by 10 dB →→ increase intensity by factor of (10)
increase by 20 dB →→ increase intensity by factor of (100)
increase by 30 dB →→ increase intensity by factor of 103 (1000)
Follow-up: How many radios equal the street sound intensity?

12. Decibel Level I
When Mary talks, she creates an intensity level of 60 dB at your location. Alice talks with the same volume, also giving 60 dB at your location. If both Mary and Alice talk simultaneously from the same spot, what would be the new intensity level that you hear?
a) more than 120 dB
b) 120 dB
c) between 60 dB and 120 dB
d) 60 dB
e) less than 60 dB
Answer: c

13. Decibel Level I
When Mary talks, she creates an intensity level of 60 dB at your location. Alice talks with the same volume, also giving 60 dB at your location. If both Mary and Alice talk simultaneously from the same spot, what would be the new intensity level that you hear?
a) more than 120 dB
b) 120 dB
c) between 60 dB and 120 dB
d) 60 dB
e) less than 60 dB
Recall that a difference of 10 dB in intensity level  corresponds to a factor of 101 in intensity. Similarly, a difference of 60 dB in  corresponds to a factor of 106 in intensity!! In this case, with two voices adding up, the intensity increases by only a factor of 2, meaning that the intensity level is higher by an amount equal to   = 10 log(2) = 3 dB. The new intensity level is  = 63 dB.

14. Human perception (loudness) is logarithmic, frequency dependent

15. The Doppler Effect
The Doppler effect is the change in pitch of a sound when the source and observer are moving with respect to each other.
When an observer moves toward a source, the wave speed appears to be higher. Since the wavelength is fixed, the frequency appears to be higher as well.

16. The Doppler Effect, moving Observer
The distance between peaks is the wavelength
The time between peaks is:
(stationary)
(moving)
Since the distance between peaks is the same:
The new observed frequency f’ is:
If the observer were moving away from the source, only the sign of the observer’s speed would change

17. The Doppler Effect, moving Source
The Doppler effect from a moving source can be analyzed similarly. Now it is the wavelength that appears to change:
In one period, how far does the wave move?
So how far apart are the peaks?

18. The Doppler Effect, moving Source
Given the speed of sound, this new wavelength corresponds to a specific frequency:
minus for source moving toward observer
plus for source moving away from observer

19. The Doppler Effect
The Doppler shift for a moving source compared to that for for a moving observer.
The two are similar for low speeds but then diverge.
If the source moves faster then the speed of sound, a sonic boom is created.
What if the observer is moving away at the speed of sound?
What if the source is moving away at the speed of sound?

20. Under the right conditions, the shock wavefront as a jet goes supersonic will condense water vapor and become visible

21. The Doppler Effect
These results can be combined for the case where both observer and source are moving:
A police car moving at 78 km/hr sounds a siren with a frequency of 5.0 kHz. Standing at the side of the road, what frequency do you hear as the car a) approaches, b) passes, and c) recedes from you?

22. Doppler effect can measure velocity.
Doppler radar showing the “hook echo” characteristic of tornado formation.
Doppler blood flow meter

23. Here a Doppler ultrasound measurement is used to verify sufficient umbilical blood flow in early pregnancy

24. Doppler Effect
You are heading toward an island in a speedboat and you see your friend standing on the shore, at the base of a cliff. You sound the boat’s horn to alert your friend of your arrival. If the horn has a rest frequency of f0, what frequency does your friend hear ?
a) lower than f0
b) equal to f0
c) higher than f0
Answer: c

25. Doppler Effect
You are heading toward an island in a speedboat and you see your friend standing on the shore, at the base of a cliff. You sound the boat’s horn to alert your friend of your arrival. If the horn has a rest frequency of f0, what frequency does your friend hear ?
a) lower than f0
b) equal to f0
c) higher than f0
Due to the approach of the source toward the stationary observer, the frequency is shifted higher.

26. Doppler Effect
In the previous question, the horn had a rest frequency of f0, and we found that your friend heard a higher frequency f1 due to the Doppler shift. The sound from the boat hits the cliff behind your friend and returns to you as an echo. What is the frequency of the echo that you hear?
a) lower than f0
b) equal to f0
c) higher than f0 but lower than f1
d) equal to f1
e) higher than f1
Answer: e

27. Doppler Effect
In the previous question, the horn had a rest frequency of f0, and we found that your friend heard a higher frequency f1 due to the Doppler shift. The sound from the boat hits the cliff behind your friend and returns to you as an echo. What is the frequency of the echo that you hear?
a) lower than f0
b) equal to f0
c) higher than f0 but lower than f1
d) equal to f1
e) higher than f1
The sound wave bouncing off the cliff has the same frequency f1 as the one hitting the cliff (what your friend hears). For the echo, you are now a moving observer approaching the sound wave of frequency f1 so you will hear an even higher frequency.

28. Superposition and Interference
Waves of small amplitude traveling through the same medium combine, or superpose, by simple addition.
If two pulses combine to give a larger pulse, this is constructive interference (left). If they combine to give a smaller pulse, this is destructive interference (right).
constructive
destructive

29. constructive destructive
Two waves with distance to the source different by whole integer wavelengths Nλ
Two waves with distance to the source different by half-integer wavelengths Nλ

30. Two-dimensional waves exhibit interference as well
Two-dimensional waves exhibit interference as well. This is an example of an interference pattern.
A: Constructive
B: Destructive

31. Superposition and Interference
If the sources are in phase, points where the distance to the sources differs by an equal number of wavelengths will interfere constructively; in between the interference will be destructive.

32. Constructive: L = n : , 2 , 3 ....
Destructive: L = (n+1/2)  :  /2, 3  /2, 5  /2...

33. Interference
Speakers A and B emit sound waves of  = 1 m, which interfere constructively at a donkey located far away (say, 200 m). What happens to the sound intensity if speaker A steps back 2.5 m?
a) intensity increases
b) intensity stays the same
c) intensity goes to zero
d) impossible to tell
Answer: c L A B

34. Follow-up: What if you move back by 4 m?
Interference
Speakers A and B emit sound waves of  = 1 m, which interfere constructively at a donkey located far away (say, 200 m). What happens to the sound intensity if speaker A steps back 2.5 m?
a) intensity increases
b) intensity stays the same
c) intensity goes to zero
d) impossible to tell
If  = 1 m, then a shift of 2.5 m corresponds to 2.5 , which puts the two waves out of phase, leading to destructive interference. The sound intensity will therefore go to zero. L A B
Follow-up: What if you move back by 4 m?

35. A standing wave is fixed in location, but oscillates with time.
Standing Waves
A standing wave is fixed in location, but oscillates with time.
These waves are found on strings with both ends fixed, or vibrating columns of air, such as in a musical instrument.
The fundamental, or lowest, frequency on a fixed string has a wavelength twice the length of the string.
Higher frequencies are called harmonics.

36. Standing Waves on a String
Points on the string which never move are called nodes; those which have the maximum movement are called antinodes.
There must be an integral number of half-wavelengths on the string (must have nodes at the fixed ends).
This means that only certain frequencies (for fixed tension, mass density, and length) are possible.
First Harmonic
Second Harmonic
Third Harmonic

37. First Harmonic
Second Harmonic
Third Harmonic

38. Musical Strings
Musical instruments are usually designed so that the variation in tension between the different strings is small; this helps prevent warping and other damage.
A guitar has strings that are all the same length, but the density varies.
In a piano, the strings vary in both length and density. This gives the sound box of a grand piano its characteristic shape.

39. Standing Waves in Air Tubes
Standing waves can also be excited in columns of air, such as soda bottles, woodwind instruments, or organ pipes.
A sealed end must be at a NODE (N),
an open end must be an ANTINODE (A).

41. Standing Waves With one end closed and one open:
the fundamental wavelength is four times the length of the pipe, and only odd-numbered harmonics appear.

42. Standing Waves If the tube is open at both ends:
both ends are antinodes, and the sequence of harmonics is the same as that on a string.

43. Resonance in the ear canal

44. Musical Tones
Human Perception: equal steps in pitch are not additive steps, but rather equal multiplicative factors
Frequency doubles for octave steps of the same note
The frets on a guitar are used to shorten the string.
Each fret must shorten the string (relative to the previous fret) by the same fraction, to make equal spaced notes.

45. Standing Waves I
A string is clamped at both ends and plucked so it vibrates in a standing mode between two extreme positions a and b. Let upward motion correspond to positive velocities. When the string is in position b, the instantaneous velocity of points on the string:
a) is zero everywhere
b) is positive everywhere
c) is negative everywhere
d) depends on the position along the string
Answer: a a b

46. Standing Waves I
A string is clamped at both ends and plucked so it vibrates in a standing mode between two extreme positions a and b. Let upward motion correspond to positive velocities. When the string is in position b, the instantaneous velocity of points on the string:
a) is zero everywhere
b) is positive everywhere
c) is negative everywhere
d) depends on the position along the string
Observe two points:
Just before b
Just after b
Both points change direction before and after b, so at b all points must have zero velocity.
Every point in in SHM, with the amplitude fixed for each position

47. Standing Waves II
A string is clamped at both ends and plucked so it vibrates in a standing mode between two extreme positions a and b. Let upward motion correspond to positive velocities. When the string is in position c, the instantaneous velocity of points on the string:
a) is zero everywhere
b) is positive everywhere
c) is negative everywhere
d) depends on the position along the string a b c

48. Standing Waves II
A string is clamped at both ends and plucked so it vibrates in a standing mode between two extreme positions a and b. Let upward motion correspond to positive velocities. When the string is in position c, the instantaneous velocity of points on the string:
a) is zero everywhere
b) is positive everywhere
c) is negative everywhere
d) depends on the position along the string
When the string is flat, all points are moving through the equilibrium position and are therefore at their maximum velocity. However, the direction depends on the location of the point. Some points are moving upward rapidly, and some points are moving downward rapidly. a b c

49. Beats
Two waves with close (but not precisely the same) frequencies will create a time-dependent interference

50. Beats Beats are an interference pattern in time, rather than in space.
If two sounds are very close in frequency, their sum also has a periodic time dependence: f beat = f1 - f2

51. In tuning a string, a 262-Hz tuning fork is sounded at the same time as the string is plucked. Beats are heard withy a frequency of 6 Hz. What is the frequency emitted by the string?

52. Beats
The traces below show beats that occur when two different pairs of waves interfere. For which case is the difference in frequency of the original waves greater?
a) pair 1
b) pair 2
c) same for both pairs
d) impossible to tell by just looking
Pair 1
Pair 2

53. Beats
The traces below show beats that occur when two different pairs of waves interfere. For which case is the difference in frequency of the original waves greater?
a) pair 1
b) pair 2
c) same for both pairs
d) impossible to tell by just looking
The beat frequency is the difference in frequency between the two waves: fbeat = f2 – f1.
Pair 1 has the greater beat frequency (more oscillations in same time period), so pair 1 has the greater frequency difference.
Pair 1
Pair 2

54. Open and Closed Pipes
You blow into an open pipe and produce a tone. What happens to the frequency of the tone if you close the end of the pipe and blow into it again?
a) depends on the speed of sound in the pipe
b) you hear the same frequency
c) you hear a higher frequency
d) you hear a lower frequency

55. Open and Closed Pipes
You blow into an open pipe and produce a tone. What happens to the frequency of the tone if you close the end of the pipe and blow into it again?
a) depends on the speed of sound in the pipe
b) you hear the same frequency
c) you hear a higher frequency
d) you hear a lower frequency
In the open pipe, of a wave “fits” into the pipe, and in the closed pipe, only of a wave fits. Because the wavelength is larger in the closed pipe, the frequency will be lower.
Follow-up: What would you have to do to the pipe to increase the frequency?

56. Decibel Level
When Mary talks, she creates an intensity level of 60 dB at your location. Alice talks with the same volume, also giving 60 dB at your location. If both Mary and Alice talk simultaneously from the same spot, what would be the new intensity level that you hear?
a) more than 70 dB
b) 70 dB
c) 66 dB
d) 63 dB
e) 61 dB
Answer: c

57. Decibel Level
When Mary talks, she creates an intensity level of 60 dB at your location. Alice talks with the same volume, also giving 60 dB at your location. If both Mary and Alice talk simultaneously from the same spot, what would be the new intensity level that you hear?
a) more than 70 dB
b) 70 dB
c) 66 dB
d) 63 dB
e) 61 dB
With two voices adding up, the intensity increases by a factor of 2, meaning that the intensity level is higher by an amount equal to Δβ = 10 log(2) = 3 dB. The new intensity level is β = 63 dB.

58. Workplace Noise
A factory floor operates 120 machines of approximately equal loudness. A plant safety inspection shows that the sound intensity level of 101 dB is too high, and must be lowered to 91 dB. How many of the machines, at least, would need to be turned off to bring the sound level into compliance?
a) about 10
b) about 12
c) about 60
d) about 108

59. Workplace Noise
A factory floor operates 120 machines of approximately equal loudness. A plant safety inspection shows that the sound intensity level of 101 dB is too high, and must be lowered to 91 dB. How many of the machines, at least, would need to be turned off to bring the sound level into compliance?
a) about 10
b) about 12
c) about 60
d) about 108
increase level by 10 dB ⇒⇒ increase intensity by factor of (10)
If the level is 10 dB too high, the loudness is too high by a factor of 2, but the sound intensity is too high by a factor of 10! Only 10% of the machines can remain on, so 108 machines need to be turned off.