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 [log-1(90/10) + log-1(95/10) + log-1(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)

Re-arrange 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 [log-1(94/10) - log-1(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