Sound Intensity and Intensity Level

Homework #3Friday’s Standing Wave Lab

will be due Friday.

Alexander Graham BellPhoto taken 1914 to 1919

Bell's father, grandfather, and brother had all been associated with work on elocution and speech, and both his mother and wife were deaf, profoundly influencing Bell's life's work.[

Bell considered his most famous invention an intrusion on his real work as a scientist and refused to have a telephone in his study.

Bell became one of the founding members of the National Geographic Society

Sound: Intensity and Intensity Level

I. Sound Intensity

Intensity of sound is the ____________________ of the sound. Sound propagates through the air via fluctuations in pressure of the gas carrying the sound. The greater the amplitude of the fluctuations in the pressure, the louder the sound appears.

loudness or “volume”

Intensity is defined equal to the amount of energy transferred per second through the sound wave divided by the area through which the energy passes.

Imagined surface surrounding the sound source.

All sound from the source must pass through the area as the sound energy expands out. Intensity is the rate of flow of sound energy per unit of area.

The size of the area depends on the distance from the source.

Image shows sound expanding our from a source. The shaded region is the (imagined) area through which the sound energy passes.

This shows an ideal source of sound energy. The energy spreads uniformly in all directions (spherical symmetry).

This is called an isotropic source of sound.

Intensity is calculated by dividing the power, in watts, by the cross-sectional area, in m2, through which the sound travels. The formula is:

PIA

Here, I is the intensity in W/m2, P is the power in watts, and A is the area, in m2, through which the sound travels.

Some useful values for reference are related to human hearing. The least amount of sound intensity that the average human ear can perceive, averaged over frequency, is 10-12 W/m2. This value is usually known as the threshold of hearing, and has the symbol Io. On the other end of the spectrum is the greatest intensity the average human ear can withstand without pain. This value is on the order of 1 W/m2. This is called the threshold of pain. (Values on the table are given in decibels.)

12210o

WIm

21painWIm

Typical response of human ear to sound levels. 0 dB represents the threshold of hearing.

Example #1: A sound wave has a power of 14.0 W as its passes through an area that measures 49.0 cm by 20.0 cm.(a) What is the intensity of this sound wave?

PIA

14.00.490 0.200

Wm m

2142 WIm

(b) If this sound wave entered your ear canal at this same intensity, what would be the rate energy is transferred from the sound wave to your ear drum? Assume the diameter of your ear canal is 5.00 mm.

PI P IAA

23

2

5.00 101432

W mPm

32.81 10P W

II. Spherical Distributions of Sound Waves: IntensityWhen sound is emitted by a source, the energy of the sound

wave spreads in all direction. By the conservation of energy, the total amount of sound energy is fixed. As the sound spreads farther from the source, it passes through successively greater areas. The intensity drops as the sound travels away from the source. The formula for the intensity of sound, I, as a function of distance from the source, r, is:

24PI

r

A source of sound that spreads the sound outwards equally in all directions is known as an isotropic source. The above equation is valid for isotropic sources.

Example #2: At a distance of 36.0 meters from an isotropic source, sound is measured to carry energy at a rate of 90.0 W. (a) What is the intensity of the sound at this location?

24PI

r

2

90.04 36.0

Wm

325.52 10 WI

m

(b) At what distance from the source would the sound be at the threshold of pain, 1 W/m2?

224 4

P PI rr I

2

90.0

4 1.00

WrWm

2.68r m

III. Sound Intensity LevelAs can be seen in the examples

above, the intensity of sounds varies over a great range of values. Even for the human ear, this ranges form 10-12 W/m2 to 1 W/m2. Dealing with these differences in order of magnitude is often frustrating, making direct comparisons difficult. A new scale, similar to the pH scale, is needed to put these values in a range that easier to form comparisons. The intensity level scale makes this possible:

1010logo

II

is the symbol for intensity level, measured in decibels or dB.I is the intensity of the wave to be studied, and Io is the threshold of hearing, 10-12 W/m2.

Example #3: What is the decibel level or intensity level of (a) the threshold of hearing and (b) the threshold of pain?

1010logo

II

1010log o

o

II

0

1010logo

II

2

1012

2

110log

10

Wm

Wm

120

Example #4: (a) If the intensity level of a jet aircraft engine is 150 dB, what is the intensity of the engine?

1010logo

II

10log10o

II

1010 / oI I

15012 10210 10WI

m

321.0 10 WI

m 10log

1010 10 o

II

(b) If this value is measured at a distance of 50.0 m from the aircraft, what is the rate at which sound energy is produced by the jet engine?

22 4

4PI P r I

r

2 324 50.0 1.0 10 WP m

m

73.14 10P W

Example #5: In a business office, the typewriters are rated at 72 dB intensity level each. How many typewriters can run at any given instant and not exceed the maximum of 85 dB as established by OSHA?

Only intensities, I, are additive. Convert to intensity.

1010total oI I

8512 10210 10total

WIm

423.16 10total

WIm

1010one oI I

7212 10210 10one

WIm

521.58 10total

WIm

The total number of machines that may run at any given time is the ratio of the two intensities:

42

52

3.16 10

1.58 10

total

one

WI mN

WIm

20N machines

Example #6: A local ordinance in the city of Torrance limits sounds produced by any motorized vehicle to be under 83 dB when measured at a distance of 15.0 m from the source. A policeman measures the sound output of a car’s stereo system to be 77.0 dB at 90.0 meters distance. Is this car in violation of the sound ordinance?

1010oI I

7712 10210 10W

m

525.01 10 WI

m

This is the intensity of the sound at a distance of 90.0 m. Now calculate to the intensity at 15.0 m.

22 1

2 21 24

I rPIr I r

21

2 1 22

rI Ir

25

22

90.05.01 10

15.0

mWm m

32 21.80 10 WI

m

This is the intensity of the sound at a distance of 15.0 m. Now find the intensity level for this sound level.

1010logo

II

32

1012

2

1.80 1010log

10

Wm

Wm

92.6 dB This exceeds the intensity level set by the local ordinance.

Noise Cancelling Technology