THE DECI……..BEL (dB)
INTRODUCTION TO DECIBEL CALCULATIONS:
A bel (symbolized B) was named by Bell Laboratories in honor of Alexander Graham
Bell. The bel was invented to deal with amplifier gains and attenuator reductions in
electronic circuits. (This concept may be observed via Windows Media Player with a
song in progress. Go to Windows Media Player Tool Bar
=>>View=>>Visualizations=>>Bars and Waves =>> Scope or Ocean Mist or Bars.)
A bel is an indicator (without dimensional units) which converts ratios between reference
values and actual values into small, convenient, dimension-free values.
Mathematically: If one circumstance is measured and found to be ten times larger than
the reference value, it is one power of 10 larger - or "1 bel greater".
If another circumstance is 100 times larger than the reference, it is two powers of 10
larger - or "2 bels greater".
If a third is 1,000,000 times larger than the reference value, it is six powers of 10 larger or "6 bels greater"!er of ten is
because
is --- bels
100
1
by mathematical definition 0
101
10
by mathematical definition 1
102 100 10 X 10
2
103 1,000 10 X 10 X 10 3
104 10,000 10 X 10 X 10 X 10 4
105 100,000
10 X 10 X 10 X 10 X 10
5
106 1,000,000
10 X 10 X 10 X 10 X 10 X 10 6
For decibels of sound-pressure level (symbolized dB Lp), the reference value is usually
0.00002 Newton/M squared.
For a bel or decibel "to mean anything under practical application", the reference value
must be known.
In the metric system of measurements, "deci" means one tenth. This means that there
are 10 decibels (symbolized dB) for each bel. Mathematically:
dB = 0.1 bel
bel = 10 dB
The basic equation is ten times the log of a ratio (of the squared actual measure
divided by the squared reference measure): It is this form of the equation that is
useful when adding and subtracting decibels as meter readings (but not when
dealing directly with pressures).
The Trick is Back-calculating the Value for the Ratio
Divide the dB value by 10 to get bels:
Then, raise 10 to the power of the bels to get the value of the ratio:
Example: Back-calculating the Ratio
A machine produces a sound-pressure level of 95 decibels at a given position of a soundlevel meter. What is the value associated with the ratio?
divide the dB Lp by 10: result... 9.5 bels (Excel Input: =95/10)
raise 10 to the 9.5th power: result.. 3162277660 [Excel Input: =POWER(10,9.5)]
Work it backwards to check our work!
Determine the number of decibels if the square of the actual pressure divided by the
reference pressure is 3162277660 (the pressure units cancel)?
log 3162277660 = 9.5 (note: bels!) [Excel Input: =LOG10(3162277660)]
decibels = 9.5 X 10 = 95 (Excel Input: =9.5*10)
(Because we didn't round,
the number back-calculates exactly!) DO NOT ROUND RATIOS NUMBERS!!!
ADDITION OF DECIBELS
RULE: DEIBELS CANNOT BE ADDED TOGETHER. RATHER EACH
DEIBEL’S RATIO MUST BE CALCULATED, THEN ADD THE RATIOS
TOGETHER, AND CONVERT THE COMBINED RATIOS BACK TO A
DECIBEL.
EXAMPLE: Suppose, at a given position of a sound-level meter, two aircraft fly over
your home.
Aircraft A flys over, the meter reads a maximum sound level of 88 decibels.
Aircraft B flys over and the meter reads a maximum sound level of 90 decibels.
Problem: What is the combined maximum sound level as both aircraft fly over your home
at the same time.
With only aircraft A flying over, the sound meter indicates 88 decibels.
Divide 88 decibels by 10 to get bels: 88/10 = 8.8 bels (Excel Input: =88/10)
Aircraft A's ratio = 10 to the power 8.8 = 630957344.5 [Excel Input: POWER(10,8.8)]
( A back-check gives 88 dB, so the ratio value is correct. )
[Excel Input: =LOG10(630957344.5 )*10]
With the aircraft B fly-over, the meter indicates 90 decibels.
Divide 90 decibels by 10 to get bels: 90/10 = 9.0 bels (Excel Input: =90/10)
Aircraft A's ratio = 10 to the power 9.0 = 1,000,000,000 (Excel Input: POWER(10,9)
( A back-check gives 90 dB, so the ratio value is correct. )
[Excel Input: =LOG10(1,000,000,000 )*10]
The two ratios may now be added:
630957344.5 + 1,000,000,000 = 1630957345
(Excel Input: = 630957344.5 + 1,000,000,000)
Take the log of 1630957345 to get 9.212442603
Multiply by 10 to get decibels: 92 (answer) (Excel Input: =LOG10(1630957345)*10)
( Note: since the original measures have two significant figures, the combined
expectation of 92 dB was rounded appropriately.)
Yes, 88 dB + 90 dB = 92 dB!
RULE: Whenever two equal decibel vales are combined, the result is always 3 dB
higher than either of the individual dB values.
Note: if the dB readings for two sound sources (both running) are equal and one is turned
off, the reading is also reduced by 3 dB.
The table below indicates the dB value that is added to the higher dB reading.
Example: 55dB + 55dB. The difference between the two values is zero. Locate 0 on the
x axis of the chart and move your pencil vertically until the curve is met. Read the
corresponding value on the y axis = 3. Solution: 55dB + 55dB = 58dB.
Example: 78dB + 83dB. . The difference between the two values is 5. Locate 5 on the x
axis of the chart and move your pencil vertically until the curve is met. Read the
corresponding value on the y axis = 1.2 dB. Solution: 78dB + 83dB = 84.2dB.
Let's Subtract!
Two aircraft flying over your house records 94 dB in the sound meter . When aircraft A
passes beyond the meter range, the sound drops to 89 dB. With aircraft A and B gone, the
background dB level is 61 dB.
How many dB's are associated with aircraft A?
How many dB's are associated with aircraft B?
(Hint: the total is A + B + background)
The dB level from aircraft B plus the 61 dB background produces 89 dB at the given
measurement point.
The ratio associated with the background is:
10 to the power 6.1 = 1258925.412 [Excel Input: =POWER(10,6.1)]
The ratio associated with aircraft B plus the background is:
10 to the power 8.9 = 794328234.7 [Excel Input: =POWER(10,8.9)]
The "B + background" ratio minus the background ratio is aircraft B's ratio:
794328234.7 - 1258925.412 = 793069309.3 (Excel Input: = 794328234.7 - 1258925.412)
Converting this back to decibels:
10 log 793069309.3 = 88.99311144 ( Answer: 89 dB for aircraft B )
[Excel Input: =LOG10(793069309.3)*10]
How I find aircraft A's dB...
As was seen, the background dB being (89dB –61dB= 28 dB) lower than aircraft B's
reading contributes virtually nothing to the overall noise field. The dB level from aircraft
A plus the 89 dB from aircraft B produces 94 dB at the given measurement point.
The ratio associated with the total (background + A + B) is:
10 to the power 9.4 = 2511886432 [Excel Input: =POWER(10,9.4)]
The ratio associated with aircraft B plus the (negligible) background is:
10 to the power 8.9 = 794328234.7 [Excel Input: =POWER(10,8.9)]
The ratio from the total dB minus aircraft B's ratio is aircraft A's ratio:
2511886432 - 794328234.7 = 1717558197 for aircraft A.
(Excel Input: = 2511886432 - 794328234.7)
Converting this back to decibels:
10 log 1717558197 = 92.34911461 ( Answer: 92 dB for Aircraft A )
[Excel Input: =LOG10(1717558197)*10]
Let's back check ...
Let's add up the three dB levels and see if we get 94 dB:
The ratio associated with the background is:
1258925.412
The ratio associated with aircraft B is:
793069309.3
The ratio from aircraft A is:
1717558197
Add them up:
1258925.412 + 793069309.3 + 1717558197 = 2511886432
(Excel Input: =1258925.412 + 793069309.3 + 1717558197)
Convert this to decibels:
10 log 2511886432 = 94 ( Answer: 94 dB EXACTLY! )
[Excel Input: =Log10(2511886432)*10]
Let's review this problem ...The background of 61 dB is insignificant when compared to
the 89 decibels produced by aircraft B.
Aircraft A contributes 92 dB to the overall level of 94 dB.
Because aircraft B is 3 dB less than aircraft A, aircraft B contributes slightly less than 2
dB to aircraft A's dB. The background adds slightly more, totaling up to 94 dB.