Sound Level Converter

Decibels use a logarithmic scale, not a linear one.

Formulas:

• dB = 10 × log₁₀(power ratio)
• dB = 20 × log₁₀(amplitude ratio)
• Power ratio = 10^(dB/10)
• Amplitude ratio = 10^(dB/20)

Which one applies, and why it is not a style choice

The two formulas are not alternatives. Which you use depends on what kind of quantity you are measuring, and picking wrong gives an answer that is off by a factor of two in dB.

• 10 log is for power quantities: acoustic power, sound intensity, energy, electrical power.
• 20 log is for amplitude quantities: sound pressure, voltage, current, particle velocity.

Sound intensity is a power quantity, so it takes the 10 log rule. Sound pressure is an amplitude quantity, so it takes 20 log. The two land on the same decibel number for the same physical sound, which is the whole point of the convention: intensity goes as pressure squared, and squaring inside a logarithm is what turns the 10 into a 20.

So 3 dB is twice the power and 1.41 times the pressure, which are two descriptions of one event, not two different events.

Examples:

Common sound levels:

• Whisper: 30 dB
• Normal conversation: 60 dB
• Rock concert: 110 dB

The reason sound uses a logarithmic scale is the sheer range involved: the loudest sound the ear tolerates carries about a trillion times the power of the faintest one it can detect. A linear scale would be unusable, so the decibel compresses that range into manageable numbers.

The catch is that decibels don’t add the way intuition expects. Every 10 dB is a tenfold jump in power, yet the ear hears it as only a rough doubling of loudness. So 70 dB isn’t “a little louder” than 60, it carries ten times the acoustic power. Two identical 60 dB sources together make 63 dB, not 120, because you add power, not the decibel figures.

That is what the second-source box above does. Add the power ratios, take the logarithm of the sum, and you have the combined level. The consequence is counter-intuitive and useful: a 60 dB source next to an 80 dB one makes essentially no difference, because 60 dB carries a hundredth of the power. Silencing the quiet machine in a noisy room buys you 0.04 dB.

Safe exposure time

The other thing the calculator now works out is how long a level is safe. NIOSH puts the limit at 85 dBA for an eight-hour day, and applies a 3 dB exchange rate, meaning every 3 dB above that halves the allowable time.

Hours = 8 × 2^((85 − level) / 3)

The numbers this produces surprise people. 94 dBA gives one hour. 100 dBA gives fifteen minutes. 110 dBA, which is a normal rock concert, gives about ninety seconds. That is not a warning about a whole evening at a concert; it is the whole budget for the day.

Note that OSHA’s legal standard in the US is more permissive, using 90 dBA and a 5 dB exchange rate. The NIOSH figure is the health-based one, and it is the one used here.