SOUND INTENSI

TY

SOUND PRESSURE AND INTENSITY

Sound pressure (p) = force per square centimeter(dynes/cm2)

Intensity (I) = power per square centimeter(Watts/cm2)

I = kp2

Smallest audible sound = 2 x 10-4 dynes/cm2

= 10-16 Watts/cm2

A problem: Between a just audible sound and a sound at the pain threshold, sound pressures vary by a ratio of 1:10,000,000, and intensities vary by a ratio of 1: 100,000,000,000,000! More convenient to use scales based on logarithms.

Decibels (dBSPL,IL) = 20 log (p1/p0)= 10 log (I1/I0)

where p1 is the sound pressure and I1 is the intensity of the sound of interest, and p0 and I0 are the sound pressure and intensity of a just audible sound.

DECIBEL SCALE

INTENSITY OF SOUND WAVES

SI units are W/m2

Intensity is proportional to squareof amplitude (pressure modulation)

Power

AreaI EA t

PA

INTENSITY RANGE FOR HUMAN HEARINGThreshold of Hearing• 10-12 W/m2

• P ~ 10-10 atm !

Threshold of Pain• 1.0 W/m2

DECIBEL SCALESensation is logarithmic

• I0 is threshold of hearing (0 dB)• Threshold of Pain is therefore 120 dB

10 log10IIo

I I010 /10

INTENSITY VS. INTENSITY LEVEL

• INTENSITY is P/A, W/m2 • INTENSITY LEVEL is in decibels (dimensionless)

SOUND LEVEL DEMO

EXAMPLE 14.2A noisy machine in a factory produces a sound

with a level of 80 dB. How many machines can the factory house without exceeding the 100-dB limit?

a) 12.5 machinesb) 20 machinesc) 100 machines

SPHERICAL WAVES

Energy propagates equally in all directions

ÞI P

4r2I1I2

r2

2

r12

EXAMPLE 14.3 (SKIP)A train sounds its horn as it approaches an intersection. The horn can just be heard at a level of 50 dB by an observer 10 km away. Treating the horn as a point source and neglect any absorption of sound by the air or ground,

a) What is the average power generated by the horn?

b) What intensity level of the horn’s sound is observed by someone waiting at an intersection 50 m from the train?

a) 126 W

b) 96 dB

EXAMPLE 14.4

Bozo Bob buys a 20-W train whistle and figures out that he won’t have any trouble standing 2 meters from the whistle since his stereo speakers are rated at 100 W and he has little trouble with the speakers turned all the way up. What is the intensity level of the whistle?

116 dB

INTESITYAll waves carry energy. In a typical sound wave the pressure doesn’t vary much from the normal pressure of the medium. Consequently, sound waves don’t transmit a whole lot of energy. The more energy a

1 m 2

wavefronts

sound wave transmits through a given area in a given amount of time, the more intensity it has, and the louder it will sound. That is, intensity is power per unit area:

I = PASuppose that in one second the green wavefronts carry one joule of sound energy through the one square meter opening. Then the intensity at the red rectangle is 1 W / m 2. (1 Watt = 1 J / s.)

INTENSITY EXAMPLEIf you place your alarm clock 3 times closer to your bed, how many times greater will the intensity be the next morning?

answer:Since the wavefronts are approximately spherical, and the area of a sphere is proportional to the square of its radius (A = 4 r 2), the intensity is inversely propotional to the square of the distance (since I = P / A). So, cutting the distance by a factor of 3 will make the intensity of its ring about nine times greater. However, our ears do not work on a linear scale. The clock will sound less than twice as loud.

THRESHOLD INTENSITYThe more intense a sound is, the louder it will be. Normal sounds carry small amounts of energy, but our ears are very sensitive. In fact, we can hear sounds with intensities as low as 10-12 W / m 2 ! This is called the threshold intensity, I 0. I 0 = 10

-12 W / m 2

This means that if we had enormous ears like Dumbo’s, say a full square meter in area, we could hear a sound delivering to this area an energy of only one trillionth of a joule each second! Since our ears are

thousands of times smaller, the energy our ears receive in a second is thousands of times less.

SOUND LEVEL IN DECIBELSThe greater the intensity of a sound at a certain place, the louder it will sound. But doubling the intensity will not make it seem twice as loud. Experiments show that the intensity must increase by about a factor of 10 before the sound will seem twice as loud to us. A sound with a 100 times greater intensity will sound about 4 times louder. Therefore, we measure sound level (loudness) based on a logarithmic scale. The sound level in decibels (dB) is given by:

Ex: At a certain distance from a siren, the intensity of the sound waves might be 10 –5 W / m 2 . The sound level at this location would be:Note: According to this definition, a sound at the intensity level registers zero decibels:

= 10 log I I 0

(in decibels)

10 log (10 –5 / 10

–12) = 10 log (10 7 )

= 70 dB10 log (10

–12 / 10 –12) = 10 log (1 ) =

0 dB

THE DECIBEL SCALE

Source DecibelsAnything on the verge

of being audible 0

Whisper 30Normal Conversation 60

Busy Traffic 70Niagara Falls 90

Train 100Construction Noise 110

Rock Concert 120Machine Gun 130

Jet Takeoff 150Rocket Takeoff 180

PainDamage

}Constant exposure leads to permanent hearing loss.

The chart below lists the approximate sound levels of various sounds. The loudness of a given sound depends, of course, on the power of the source of the sound as well as the distance from the source. Note: Listening to loud music will gradually damage your hearing!

INTENSITY & SOUND LEVEL = 10 log I I 0Every time the intensity of a sound is increased by a factor of 10, the

sound level goes up by 10 dB (and the sound seems to us to be about twice as loud). Let’s compare a 90 dB shout to a 30 dB whisper. The shout is 60 dB louder, which means its intensity is 10 to the 6th power (a million) times greater. Proof:

60 = 1 - 2 = 10 log (I 1 / I 0 ) - 10 log(I 2 / I 0 ) = 10 log I 1 / I 0

I 2 / I 0

60 = 10 log (I 1 / I 2 ) 6 = log (I 1 / I 2 )

10 6 = I 1 / I 2

Compare intensities: 80 dB vs. 60 dBCompare intensities: 100 dB vs. 75 dBCompare sound levels: 4.2 · 10 –4 W / m 2 vs. 4.2 · 10 –7 W / m 2

answers:

factor of 100

factor of 316 (10 2.5 = 316)

differ by 30 dB ( I’s differ by 3 powers of 10 )

DECIBEL EXAMPLESuppose a 75 g egg is dropped from 50 m up onto the sidewalk. The splat takes 0.05 s. Nearly all of the gravitational potential energy the egg had originally is converted into thermal energy, but a very small fraction goes into sound energy. Let’s say this fraction is only 6.7582 · 10 –11. How loud is the splat heard from the point at which the egg was dropped? Hints: Answers:

1. How much energy does the egg originally have?

2. How much of that energy goes into sound?

3. Calculate sound power output of the egg.

4. Figure intensity at 50 m up. (Assume the hemispherical wavefronts.)

5. Compute sound level in decibels.

36.75 J

2.4836 · 10 –9 J

4.9673 · 10 –8 W

3.1623 · 10 –12 W / m 2

5 dB, very faint