 
10. Manipulation with
Decibels
In the study of acoustics,
calculations are mainly performed in decibels, e.g. addition, averaging, and subtraction.
(a) Adding Decibels
For n numbers of sound power
levels acting together, the ith sound power, Wi, is given by :
Total sound power,
So total sound power level,
(17)
Similarly, for sound pressure levels,
(18)
Example 1
For the addition of 3 sound
pressure levels : 90dB, 95dB and 88dB, the total sound pressure level :
Lpt
= 10 log10 [log1(90/10)
+ log1(95/10) + log1(88/10)]
= 10 log10
(109 + 109.5 + 108.8)
= 96.8dB
It can be seen that addition of
decibels in this way is quite tedious. In practice, graphs are designed to give the sum of
two decibel values (see Fig. 6). Values obtained by using these graphs are less accurate,
but are generally sufficient.
(b) Averaging Decibels
It follows from Equation (18) that
the time average decibel level, Lp, is given by :
(19)
and similarly,
(20)
Figure 6 Graph for Adding
Decibels
Example 2
Suppose that the 4 different
measurements of the sound pressure level at a particular location are 96dB, l00dB, 90dB,
and 97dB, then the average sound pressure level is given by :
Lp =
10 log10 [(1/4) (109.6 +
1010 + 109 + 109.7)]
= 97dB
When the fluctuation in sound level is
small, Equations (19) and (20) can be approximated by simplified forms :
(i) If
(21)
(22)
(ii) If
(23)
(24)
(c) Subtracting Decibels
In certain instances, it is
desirable to subtract an ambient or background sound pressure level from a total measured
level. This allows one to determine the sound pressure level produced by a particular
source. In general, it is not possible to make meaningful measurements of sound pressure
level unless the background sound pressure level is at least 3 dB below the level of the
source under consideration.
Let the subscripts t, b, and s
represent the total, background, and the source, respectively. Then,
and,
(25)
Rearrange Equation (25),
or,
(26)
Example 3
The sound pressure level at a
point is 85dB and 94dB with a particular machine 'off' and 'on' respectively, i.e.
Lp,b
= 85dB
Lp,t
= 94dB
The sound pressure level due to the
machine is :
Lp,s
= 10 log10 [log1(94/10)
 log1(85/10)]
= 10 log10
(2.196 x 10)
= 93.4dB
Once again, graphs are devised to help
the subtraction of decibel values (see Fig. 7).
In case the background noise level is
close to or even greater than the sound level of the source, more advanced methods which
measure sound intensity have to be used for analysis of sound power.
Figure 7 Graph for Subtracting
Decibels
