Mr. Titus Tulay recently purchased a home four blocks from a busy tollway. On a typical evening, the decibel level resulting from tollway traffic is 62 dB at the location of his house. Determine the decibel level of a house 1 block from the tollway (four times closer).
Essentially, a decibel reading tells us how many factors of 10 larger ( or smaller ) an intensity-level ( I ) is in regard to some reference point ( Io ). Intensity is expressed as ( I = W/m2 ), where power in Watts ( W ) is a measure of energy per second ( J/s ), and area is expressed in square meters. The intensity-level ( 𝜷 ) in decibels is defined as 𝜷 = 10 log10 ( I / Io ).
Ex. 𝜷 = 10 log10 ( 10-6 W/m2 / 10-12 W/m2 ) = 10 log10 106 = ( 10 )( 6 ) = 60 dB.
Thus, in the scenario above, the intensity-level of sound was determined to be 6 factors of 10 higher ( 106 ) than the threshold of sound.
The threshold of hearing ( 10-12 W/m2 ) is oftentimes the vantage point intensity ( Io ) that other intensity measurements are compared to, but that need not be the case. A 60 dB difference just as well exists between an intensity ratio of ( 10-4 W/m2 / 10-10 W/m2 ).
Of equal importance is the psychological perception of sound in relation to tenfold increases in sound intensity. For every increase in 10 decibels, the human brain interprets a doubling of a sound’s loudness to occur. For example, a sound that is 50 dB above some other reference point is ( 50 dB / 10 ) = 5 factors of 10 higher than some other intensity-level. Therefore, the sound that is heard is 25 = 32 times louder than another sound it is being compared to. To give further clarity to this point, another example is due.
Q: Dr. Chris Dennis, Quadratic Al-kwarisma Young, Varney Kromah, Peter C Dorkpoh Jr, Dr. Saki Golafale, Theophilus Mambu, Anderson Keah, Cephus Acolatse, Fallah Nyandemor, and Desmond Worji have joined an orchestra. They each have learned to play a violin, and during a concert opening, they each join in one-at-a-time until all 10 violins are playing in sequence. If each violinist tunes their instrument to play at 70 dB, what will the sound-level be as each one has begun to play, and what will the sound-level be when they all play together?
A: When one violin plays, its sound-level is 70 dB = 10 log10 ( I1 / Io ). When the second violinist begins to play, we can use the rules of logarithms as follows:
10 log10 ( 2I1 / Io ) = 10 log10 ( 2 ) + 10 log10 ( I1 / Io ).
3 dB + 70 dB = 73 dB.
NOTE: NOTICE THAT A DOUBLING IN INTENSITY DID NOT CAUSE THE OVERALL dB VALUE TO INCREASE BY 10 dB! DOUBLING THE INTENSITY INCREASED OUR NUMERATOR VALUE BY A FACTOR OF 2, NOT 10. Let’s see what happens if we continue to incrementally increase the intensity-level by adding a third violin.
10 log10 ( 3I1 / Io ) = 10 log10 ( 3 ) + 10 log10 ( I / Io )
5 dB + 70 dB = 75 dB.
Subsequent increases in dB measurements are as follows: With 4 violins, 76 dB; 5 violins, 77 dB; 6 violins, 78 dB; 7 violins, 78 dB; 8 violins, 79 dB; 9 violins, 80 dB; 10 violins, 80 dB.
Ten violins increased the first violin’s intensity by one factor of 10 ( 70 dB to 80 dB ).
10 log10 ( 10I1 / Io ) = 10 log10 10 = ( 10 )( 1 ) = 𝚫10 dB.
Furthermore, log10 10 = 1, and 21 = 2, which corresponds with one doubling of loudness.
Any time we divide a given dB value by the number 10, we have an exponent that tells us how many factors of 10 an intensity is ( relative to some other ), and we also have an exponent that can be used to see how much louder a sound has become by some factor of 2y .
Addressing Mr. Tulay’s circumstance will necessitate usage of the Law of Conservation of Energy. Let’s assume that an outwardly expanding spherical sound wave carries with it some measure of energy in Joules ( J ). Within this sphere, there is a defined intensity I = P/A = W/m2 . When the sphere gets larger, the same amount of energy exists within it; however, the energy is now spread out ( less concentrated ). The fact that the flow of energy ( W = J/s ) in a smaller sphere continues to exist in a larger sphere is quantified as follows:
Pi = Pf
The energy per second that exists within the first frame of reference now exists within the second, and since I = ( P/A ), and the area of a sphere is A = 4𝝅r2 , ( I )( 4𝝅r2 ) = P. Thus, Pi = Pf is expressed as follows:
( Ii )( 4𝝅r12 ) = ( If )( 4𝝅r22 )
Recall that the sound-intensity measured at Mr. Tulay’s home is 62 dB. The decibel level of a house 1 block from the tollway (four times closer) can be determined in the following manner:
( 62 dB / 10 ) = log10 ( I / Io ) = 6.2
106.2 = ( I / Io )
( 10-12 W/m2 )( 106.2 ) = I = 1.5849 x 10-5 W/m2
If we assign an arbitrary value of “ 1 “ to r1 ( the distance from Mr. Tulay’s home to the tollway ), we can assign an arbitrary value of “ ¼ “ to r2 , which describes the location of a house only ¼ the distance away ( regardless of what the actual distance happens to be. ¼ of any total distance = ¼ the total distance. )
Pi = Pf
( Ii )( 4𝝅r12 ) = ( If )( 4𝝅r22 )
( 1.5849 x 10-5 W/m2 )( 4𝝅12 ) = ( If )( 4𝝅( ¼ )2 )
The 4𝝅 terms cancel, and we’re left with…
( Ii )( r12 ) = ( If )( r22 )
Solving for I2 will give us the intensity that will be measured ¼ the distance that Mr. Tulay’s home is located from the tollway.
If = ( 1.5849 x 10-5 W/m2 )( 16 ) = 2.53584 x 10-4 W/m2
𝜷 = 10 log10 ( I / Io ) = 10 log10 ( 2.53584 x 10-4 W/m2 / 10-12 W/m2 )
𝜷 = 10 log10 ( 2.5358400 x 107 )
𝜷 = ( 10 )( 7.40412 )
𝜷 = 74 dB
Since the sound-intensity at Mr. Tulay’s home is 62 dB, the sound intensity at ¼ the distance from the tollway to his home is 12 dB higher. 12 dB corresponds to 1.2 doublings in loudness.
21.2 = a perceived increase in loudness of 2.3 times!
Needless to say, Mr. Tulay gets more restful sleep at night than the folks located ¼ the distance he is located from the tollway.
RENAISSANCE PHYSICS 