LOUDNESS OF A SOUND
A LOGARITHMIC SCALE
We use a logarithmic scale when there is a wide range of values, and when a change in a value depends not on the absolute size of the change but on proportion to the value itself. Adding 1 to a value is absolutely the same whether the original value is 1 or 1000, a linear scale makes sense. If doubling a value is just the same change whether is it from 1 to 2 or from 1000 to 2000, a logarithmic scale is appropriate.
We are going to investigate the decibel scale for loudness. There are two reasons why a logarithmic scale is useful:
Quantities of interest exhibit such ranges of variation that a dB scale is more convenient than a linear scale. For example, sound pressure radiated by a submarine may vary by eight orders of magnitude depending on direction.
The human ear interprets changes in loudness within a logarithmic scale.
WHAT IS A DECIBEL (dB)?
The sensation of loudness of sound is not proportional to
the energy intensity, but rather is a logarithmic function. Loudness, in Bels (after Alexsander
Bell), of a sound of intensity I is
defined to be 
, where 
is the minimum
intensity detectable by the human ear (such as the tick of a watch at 6 m under
quiet conditions), 
. When a sound is 10 times as intense as another, its
loudness is 1 Bel greater. If a sound is 100 times as intense as another, it is
louder by 2 Bels, and so on. The bell is a large unit, so a subunit, a decibel, is usually used. For L in decibels, the formula is as
follows:
HOW IS SOUND MEASURED
A sound level meter is the principal instrument for general noise measurement. The indication on a sound level meter indicates the sound pressure, p, as a level referenced to 0,00002 Pa, calibrated on a decibel scale.
Sound Pressure Level = 
EXERCISES:
Find the loudness, in decibels, of the sound in a radio
studio, for which the intensity I is
199 times 
.
Equation: .
Answer: ...
Find the loudness of a heavy truck, for which the intensity
is 
times 
.
Equation: .
Answer: ...
Find the loudness of a supersonic plane, for which the
intensity ( 30 m apart) is 13
.
Equation: .
Answer: ...
What is the intensity of a talk, which loudness is between 40 and 65 dB?
Equation: .
Answer: ...
What is the intensity of a car on the city drive, which loudness is between 80 and 90dB?
Equation: .
Answer: ...
There was a rock concert in town. It was very noisy and the musicians had to reduce the intensity to a half. What was the loudness if the initial one was 130 dB?
Equation: .
Answer: ...
Aunt Zofi is complaining about noisy neighbours. They have got seven children. The first one is crying with loudness of 60 dB, the second one lucky sleeps. The third one and the fourth one are quarrelling with each other at the loudness of 65 dB, the fifth one is listening to the radio of 80 dB. The sixth one is learning permanently (fortunately very quietly) and the seventh one is playing truant. Aunt Zofi has added the loudness to be 270 dB. This is what she cant stand and she has decided to sue the neighbours. Is she right? Is it possible, that the neighbours are so loud?
Your opinion: ..
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Equations: .
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Answer: ...