The intensity ( I ) of a sound is quantitatively different than how sound is perceived by the ear. The perception of sound is referred to as ” loudness “. When a sound’s intensity increases by 10 decibels ( B ), the ear perceives this increase as a ” doubling ” in loudness. Logarithmic mathematics is used to relate these two phenomena to one another.
Intensity ( I ) of sound is defined as power per unit area ( P / A ). Power ( P ) is expressed in Joules ( J ) per second ( s ) in units referred to as Watts ( W ), and area ( A ) is expressed in square meters ( m^2 ). The threshold of human hearing is quantified as an intensity ( I ) having a magnitude of 1.0 * 10^-12 W / m^2. If this threshold intensity ( I ) increased tenfold, it would have a value of 1.0 * 10^-11 W / m^2.
It is easier to express this tenfold increase by using the threshold of sound as a reference to which higher values are compared. This is accomplished by setting the intensities of sound ( I ) into ratios with the threshold of sound in the denominator and relative values in the numerator. If the logarithm of such a ratio is determined, the result is a number that lets us know how many factors of ten the threshold of sound has increased. For example, the log ( I / Io ), where Io = threshold of sound in 1.0 * 10^-12 W / m^2, and I = 1.0 * 10^-11 W / m^2 gives us log ( ( 1.0 * 10^-11 W / m^2 ) / ( 10 * 10^-12 W / m^2 ) ) = log ( 1.0 * 10^-11 W / m^2 ) – log ( 1.0 * 10^-12 W / m^2 ) = -11 – ( – 12 ) = 1.
The number above is referred to as 1 Bel. Converting Bels to decibels ( B ), we use a conversion fraction which states that 1 Bel = 10 dB. Therefore, ( 1 Bel )( 10 dB / 1 Bel ) = 10 dB. THE SAME RESULT IS OBTAINED BY SIMPLY MULTIPLYING log ( I / Io ) by 10. Thus, the previous derivation of log ( I / Io ), where I = 1.0 * 10^-11 W / m^2, and the threshold of sound ( Io ) = 1.0 * 10^-12 W / m^2 in decibels ( B ) = 10 log ( I / Io ) = 10 dB. This notation clearly denotes that an intensity ( I ) of 1.0 * 10^-11 W / m^2 is one factor of 10 larger than the threshold of hearing ( Io ) = 1.0 * 10^-12. The value for the threshold of hearing need not be the reference point, however. The important thing to know is that EVERY 10 DECIBEL CHANGE FROM SOME REFERENCE POINT IS RELATED TO A SOUND THAT IS INTERPRETED BY THE EAR AS HAVING ” DOUBLED “.
When a quantity has doubled, we must know what quantity was present when doubling began. For example, in cell biology, a single cell may divide into two cells, then two cells divide into four cells, then four cells divide into eight cells, etc. If we have two cells, and one doubling occurs, we will end up with four cells. If we begin with eight cells, and two doublings occur, we will end up with thirty-two cells. WHEN USING LOGARITHMS TO DETERMINE HOW MANY FACTORS OF 10 SOME ORIGINAL VALUE OF INTENSITY HAS CHANGED, THE DECIBEL VALUE WE OBTAIN MUST BE DIVIDED BY 10 TO DETERMINE HOW MANY DOUBLINGS HAVE OCCURRED. This value will be our input ( x ) value in 2^x = y.
If a source of sound increases in intensity by 50 dB, regardless of what the original intensity ( Io ) is, it will have gone through five ( 10 dB ) increases. This means that the loudness of the source has doubled by a factor of 5. Thus, 2^5 = 32. REGARDLESS OF WHAT THE ORIGINAL INTENSITY OF THE SOURCE WAS, IT WILL NOW BE INTERPRETED AS BEING THIRTY-TWO TIMES AS LOUD.
Example: If B = 10 log ( ( 1.0 * 10^-10 W / m^2 ) / ( 1.0 * 10^-12 W / m^2 ) ) = 20 dB, the intensity has gone through two ( 10 dB ) increases. This represents two doublings, and the sound will be interpreted as having gotten 2^2 = 4 times louder.
Finally, if we know the intensity level ( B ) of a sound in decibels, we can also determine its intensity ( I ) relative to some threshold. Recall that B = 10 log ( I / Io ). Let’s say that B = 75 dB. Dividing both sides by 10 yields ( B / 10 ) = log ( I / Io ). Now, ( 75 dB / 10 ) = log ( I / Io ). The Rule of Logs states that the output ( 7.5 dB ) is an exponent used to raise a base of 10 to obtain ( I / Io ). Therefore, 10^7.5 = ( I / Io ). Multiplying both sides by the threshold of sound ( Io ) yields ( Io )( 10^7.5 ) = I. Finally, ( 1.0 * 10^-12 W / m^2 )( 10^7.5 ) = I = 3.16 * 10^-5 W / m^2
RENAISSANCE PHYSICS 