<< Chapter < Page Chapter >> Page >

The sound intensity level     β of a sound, measured in decibels , having an intensity I in watts per meter squared, is defined as

β ( dB ) = 10 log 10 ( I I 0 ) ,

where I 0 = 10 −12 W/m 2 is a reference intensity, corresponding to the threshold intensity of sound that a person with normal hearing can perceive at a frequency of 1.00 kHz. It is more common to consider sound intensity levels in dB than in W/m 2 . How human ears perceive sound can be more accurately described by the logarithm of the intensity rather than directly by the intensity. Because β is defined in terms of a ratio, it is a unitless quantity, telling you the level of the sound relative to a fixed standard ( 10 −12 W/m 2 ). The units of decibels (dB) are used to indicate this ratio is multiplied by 10 in its definition. The bel, upon which the decibel is based, is named for Alexander Graham Bell , the inventor of the telephone.

The decibel level of a sound having the threshold intensity of 10 −12 W/m 2 is β = 0 dB, because log 10 1 = 0 . [link] gives levels in decibels and intensities in watts per meter squared for some familiar sounds. The ear is sensitive to as little as a trillionth of a watt per meter squared—even more impressive when you realize that the area of the eardrum is only about 1 cm 2 , so that only 10 −16 W falls on it at the threshold of hearing. Air molecules in a sound wave of this intensity vibrate over a distance of less than one molecular diameter, and the gauge pressures involved are less than 10 −9 atm .

[1] Several government agencies and health-related professional associations recommend that 85 dB not be exceeded for 8-hour daily exposures in the absence of hearing protection.
Sound intensity levels and intensities
Sound intensity level β (dB) Intensity I ( W/m 2 ) Example/effect
0 1 × 10 12 Threshold of hearing at 1000 Hz
10 1 × 10 11 Rustle of leaves
20 1 × 10 10 Whisper at 1-m distance
30 1 × 10 9 Quiet home
40 1 × 10 8 Average home
50 1 × 10 7 Average office, soft music
60 1 × 10 6 Normal conversation
70 1 × 10 5 Noisy office, busy traffic
80 1 × 10 4 Loud radio, classroom lecture
90 1 × 10 3 Inside a heavy truck; damage from prolonged exposure [1]
100 1 × 10 2 Noisy factory, siren at 30 m; damage from 8 h per day exposure
110 1 × 10 1 Damage from 30 min per day exposure
120 1 Loud rock concert; pneumatic chipper at 2 m; threshold of pain
140 1 × 10 2 Jet airplane at 30 m; severe pain, damage in seconds
160 1 × 10 4 Bursting of eardrums

An observation readily verified by examining [link] or by using [link] is that each factor of 10 in intensity corresponds to 10 dB. For example, a 90-dB sound compared with a 60-dB sound is 30 dB greater, or three factors of 10 (that is, 10 3 times) as intense. Another example is that if one sound is 10 7 as intense as another, it is 70 dB higher ( [link] ).

Ratios of intensities and corresponding differences in sound intensity levels
I 2 / I 1 β 2 β 1
2.0 3.0 dB
5.0 7.0 dB
10.0 10.0 dB
100.0 20.0 dB
1000.0 30.0 dB

Calculating sound intensity levels

Calculate the sound intensity level in decibels for a sound wave traveling in air at 0 °C and having a pressure amplitude of 0.656 Pa.


We are given Δ p , so we can calculate I using the equation I = ( Δ p ) 2 2 ρ v w . Using I , we can calculate β straight from its definition in β ( d B ) = 10 log 10 ( I I 0 ) .


  1. Identify knowns:
    Sound travels at 331 m/s in air at 0 °C .
    Air has a density of 1.29 kg/m 3 at atmospheric pressure and 0 °C .
  2. Enter these values and the pressure amplitude into I = ( Δ p ) 2 2 ρ v .
    I = ( Δ p ) 2 2 ρ v = ( 0.656 Pa ) 2 2 ( 1.29 kg/m 3 ) ( 331 m/s ) = 5.04 × 10 −4 W/m 2 .
  3. Enter the value for I and the known value for I 0 into β ( dB ) = 10 log 10 ( I / I 0 ) . Calculate to find the sound intensity level in decibels:
    10 log 10 ( 5.04 × 10 8 ) = 10 ( 8.70 ) dB = 87 dB .


This 87-dB sound has an intensity five times as great as an 80-dB sound. So a factor of five in intensity corresponds to a difference of 7 dB in sound intensity level. This value is true for any intensities differing by a factor of five.

Got questions? Get instant answers now!

Questions & Answers

Suppose the master cylinder in a hydraulic system is at a greater height than the cylinder it is controlling. Explain how this will affect the force produced at the cylinder that is being controlled.
Louise Reply
Why is popo less than atmospheric? Why is popo greater than pipi?
The old rubber boot shown below has two leaks. To what maximum height can the water squirt from Leak 1? How does the velocity of water emerging from Leak 2 differ from that of Leak 1? Explain your responses in terms of energy.
David rolled down the window on his car while driving on the freeway. An empty plastic bag on the floor promptly flew out the window. Explain why.
Louise Reply
the pressure differential exerted a force on the bag greater than the gravitational force holding it on the floor.
what is angular velocity
Sthandazile Reply
The rate of change in angular displacement is defined as angular velocity.
a length of copper wire was measured to be 50m with an uncertainty of 1cm, the thickness of the wire was measured to be 1mm with an uncertainty of 0.01mm, using a micrometer screw gauge, calculate the of copper wire used
Nicole Reply
What is the answer please
If centripetal force is directed towards the center,why do you feel that you're thrown away from the center as a car goes around a curve? Explain
Maira Reply
if there is a centripetal force it means that there's also a centripetal acceleration, getting back to your question, just imagine what happens if you pull out of a car when it's quickly moving or when you try to stop when you are running fast, anyway, we notice that there's always a certain force..
... that tends to fight for its previous direction when you try to attribute to it an opposite one ou try to stop it.The same thing also happens whe a car goes around a curve, the car it self is designed to a"straight line"(look at the position of its tyres, mainly the back side ones), so...
... whenever it goes around a curve, it tends to throw away its the occupiers, it's given to the fact that it must interrupt its initial direction and take a new one.
Which kind of wave does wind form
Matthias Reply
calculate the distance you will travel if you mantain an average speed of 10N m/s for 40 second
Abdulai Reply
hw to calculate the momentum of the 2000.0 elephant change hunter at a speed of 7.50 m/s
Kingsley Reply
how many cm makes 1 inches
Hassan Reply
how do we convert from m/s to km/hr
Toni Reply
When paddling a canoe upstream, it is wisest to travel as near to the shore as possible. When canoeing downstream, it may be best to stay near the middle. Explain why?
SANA Reply
Explain why polarization does not occur in sound
one ship sailing east with a speed of 7.5m/s passes a certain point at 8am and a second ship sailing north at the same speed passed the same point at 9.30am at what distance are they closet together and what is the distance between them then
Kuber Reply
density of a subtance is given as 360g/cm,put it in it s.i unit form
Linda Reply
if m2 is twice of m1. find the ration of kinetic energy in COM system to lab system of elastic collision
Raman Reply
What is a volt equal to?
Clifton Reply
Practice Key Terms 8

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now

Source:  OpenStax, University physics volume 1. OpenStax CNX. Sep 19, 2016 Download for free at http://cnx.org/content/col12031/1.5
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'University physics volume 1' conversation and receive update notifications?