The reason for using decibels is the compress the immense range of watts we can hear, 0.000000000001 watts up to 100 watts or more, into manageable numbers. Similarly the threshold of hearing is 0.00002 Pascals up to 200 Pascals. Converting these values to dBs results in a range of numbers 0 dB up 140 dB, much easier to 'handle'.
However manipulating these figures mathematically can be confusing for two reasons the first is Watts and Pascals are different quantities, but both linked to our hearing and then there is the 'little understood' logarithmic base to the dB scale.
The object of this web page is to demystify the use of dBs in acoustics using a couple of simple tables
Sound Power Level (Lw) = 10 log (w/wo) dB where w is the sound power in watts and wo is the reference sound power level of 10-12 watts = 0 dB
3 dB = a factor of 2 in sound energy (double or half the sound power)To see a table of typical sound power levels click here
Sound Pressure Level (SPL) = 20 log (p/po) dB, where p is the sound pressure in pascals and po is the reference sound pressure of 0.00002 pascals = 0 dB in air ≡ to the threshold of hearing at 1KHz
6 dB = a factor of 2 in sound pressure (double or half the sound pressure)To see a table of typical sound pressure levels click here
To understand the 10 log sound power vs the 20 log sound pressure formulae see our power quantity and the root power quantity entries.
If you have access to an Excel Spreadsheet or similar then it's relatively simple to add, subtract and average acoustic levels, as the following table shows, or just use Google
|SWL dB||antilogs/10||antilog totals||dB totals|
|60 dB||1,000,000||2,000,000||63.01 dB|
|70 dB||10,000,000||12,000,000||70.79 dB|
|61 dB||1,258,925||13,258,925||71.23 dB|
To check the above table using Google, type 10^(61/10) into the search box to get the antilog of 61 dB = 1,258.925
Then to check to the addition of 60 db + 60 dB + 70 dB + 61 dB = 71.23 dB enter Log(13258925)*10 into Google
Note 1: the /10 converts the decibels to Bels and the *10 converts the Bels back to decibels
Note 2: the 3 dB increase when you add 2 x 60 dB confirms the 3 dB rule of thumb stated above
You can also get the dB average of the 4 sound power levels by entering Log(13258925/4)*10 into Google the answer should be 65.2 dB and not the arithmetic average of 251/4
|SWL dB||antilogs/20||antilog totals||dB totals|
|60 dB||1,000||2,000||66.02 dB|
|70 dB||3,162||5,162||70.79 dB|
|61 dB||1,122||6,284||75.97 dB|
The sound pressure dB calculations are the same as the sound power levels except for the 20 log factor, instead of the 10 log power factor, which results in the 6 dB 'rule of thumb' concerning doubling or halving sound field quantities, not the 3 dB rule that applies to sound energy levels.The power quantity and the root power quantity entries explain the reason for the 10 and 20 dB factors.
** The above calculations do not hold true if the original dB measurements were not consistent, i.e. similar situation, identical measurement periods and the same frequency or time weightings if used.