The base unit is the bel, which is the logarithm base 10 of the ratio of the sound being measured to the threshold of hearing. Thus, if the sound energy is 10 times louder than our threshold of hearing it is log(10/1)=1. If the sound is 100 times louder, it is 2 bels, 1000 times louder is 3 bels, and so on. Just count the number of zeroes after the 1.
You can hear over a range of greater than 12 bels (120 decibels), which, when written out, is 1,000,000,000,000 to 1. Writing 120 dB is much easier. The decibel becomes confusing when you have, for instance, only twice the energy. The math works out to 10 (to convert from bels to decibels) � logarithm (2/1). Use a calculator to do the logarithm, which works out to a two-fold increase, or 3.010299957... decibels, which you can round off to 3. With 10 times as much energy (1 bel), you get 10 decibels. If you double the sounds' energy again, it cannot be 20 decibels because that is 100 times as loud.
You now know the secret of logarithms: add the 3 from the doubling to the 10 from the 10-fold increase to determine that the energy increases by 13 dB. Thus, 40 times as much energy would be 3 + 3 + 10, or 16dB. Eighty times, of course, is 3 + 3 + 3 + 10, or 19 dB.
If you know that 20 dB is 100 times and 19 dB is 80 times, then what is 1 dB? Well, 100 divided by 80 is 1.25. The log of 1.25 is .0969100. Ten times that is close enough to 1 that you should see how this works. The precise answer is 10 to the exponent (1 dB/10 decibels to bel), which equals 1.258925412..., or 1.
Another consideration is the accuracy to which you can measure. It is fine measuring a voltage on a digital meter that is accurate to four digits. A change in barometric pressure of 1˝ will change some sound levels by about 0.2 dB. Wind and air movement also affects the readings. That is why sound pressure levels are rarely used with a resolution of greater than 1 dB.
This text was featured in audioXpress May 2007, as a side note to the author's "Speech Intelligibility" article series. We thought this would be a good time to share it again.
And check the great decibel meter apps. Click on the two pictures.
