The reason for using decibels is to compress the immense range of watts we can hear, 0.000000000001 watts up to 100 watts or more, into manageable numbers. Similarly the range of sound pressure levels we can hear is 0.00002 pascals up to 200 pascals. Converting these levels into dBs results in a range of values 0 dB up 140 dB, much easier to 'handle'.
However sound level calculations involve logarithms which many people are not comfortable with, so we describe a simpler modern way.
First the basics
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 **Sound Power Level is a sound energy quantity and uses the 10·log factor, so as a rule of thumb: 3 dB = a factor of 2 in sound energy (+3dB is double the sound power and 3dB less is half the power)
To see a table of typical sound power levels and other 'rules of thumb', click here
** When you divide watts by the reference watts, for example, the result is a simple ratio (no units).
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 = the threshold of hearing at 1KHz
To understand the 10·log sound power vs the 20·log sound pressure formula see our root power quantity entry. See also the IEC decibel definition
Sound Level Calculations
Acoustic engineers often use spreadsheets to add, subtract and average acoustic levels, like the following tables. If you don't have access to this software or the experience to programme spreadsheet columns you can simply ask Google to convert the dB levels back into sound level ratios, add them up and then ask Google to convert the 'new' total sound level ratio into decibels again.
|10·log (w/wo)||w/wo||w/wo totals||10·log (w/wo) 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|
Column 1 lists the dB levels we want to add up, note the column headings
In column 2 we removed the 10·log decibel conversion by simply typing 10^(60/10) into Google, to convert both the 60 dB levels and up comes the answer 1,000,000
Then enter 10^(70/10) into the search engine to convert the 70dB level and finally 10^(61/10) to convert the 61dB level
The column 3 levels are the arithmetic running totals of the w/wo (sound power quantities).
Finally in Column 4 we have re-converted the individual totals back to decibels using Google again, this time entering 10*log (2,000,000) for row 2, the addition of (60 dB + 60 dB). Next enter 10*log (12,000,000) into the search engine for the row 3 total and finally enter 10*log (13,258,925) = 71.23 dB, the sum of the four sound power dB levels.
It also follows that the average sound power level, of the four dB levels is 10*log (13,258,925/4) = 65.20 dB
The above example also confirms the sound energy rule of thumb that 3 dB is a factor of 2, double or half and 10 dB = a factor of 10.
|20·log (p/po)||p/po||p/po totals||20·log (p/po) totals|
|60 dB||1,000||2,000||66.02 dB|
|70 dB||3,162||5,162||74.26 dB|
|61 dB||1,122||6,284||75.96 dB|
Sound pressure level calculations are the same as sound power level calculations detailed above, except the 20 log(p/po) factor applies, instead of the 10 log(w/wo) ratios.
For example to convert the 60 dB sound pressures to the p/po levels type 10^(60/20) into Google and the answer is 1,000. Repeat this procedure for the 70 and 61 dB levels.
Then to convert the total p/po levels to decibels type 20*log (6,284) into Google and the sum of the four sound pressure dB levels is 75.96 dB.
Similarly the average sound pressure level = 20*log (6,284/4) = 63.92 dB
The above example also confirms the sound pressure rule of thumb that 6 dB is a factor of 2, double or half and 10 dB = a factor of 3.
We use Google in our examples, only because it is currently the most popular search engine.