Two speakers are three decibels, not twice as loud

Two 80 dB speakers together produce 83 dB, not 160. Ten of them produce 90. Decibels are a logarithmic ratio rather than a quantity, so they combine by adding powers and taking the logarithm again — and that is why the intuition fails so completely.

Three numbers cover almost all of it: 3 dB doubles power, 6 dB doubles voltage, and about 10 dB doubles perceived loudness. Those three are different doublings of different things, which is the source of most of the confusion.

Why are there two formulas?

Because decibels measure a ratio and the ratio can be of power or of amplitude. Power uses 10·log₁₀ and voltage or pressure uses 20·log₁₀, and the factor of two exists because power goes with the square of amplitude.

A voltage gain from 0.1 V to 2 V is a factor of 20, which is 26 dB. The same figure read as a power ratio would be 13 dB — same numbers, different question, and stating which one you mean is not optional.

Why does adding sources give so little?

Because you are adding powers, not levels. Two equal sources double the power, which is 3 dB. Four sources quadruple it, which is 6 dB. Ten sources give 10 dB — audibly about twice as loud for a tenfold increase in equipment.

That is the arithmetic behind a great deal of disappointment in live sound. Doubling the number of speakers is a 3 dB gain, and 3 dB is close to the smallest difference most listeners reliably notice.

It also works in reverse and more usefully: removing one noise source from a room full of them changes almost nothing, which is why noise control targets the loudest source rather than the count.

What does distance do?

Costs 6 dB per doubling from a point source in free space. Two metres away is 6 dB quieter than one metre; four metres is 12 dB down.

That figure is optimistic indoors. Reflections keep energy in the room, so beyond a certain distance the level stops falling and settles at whatever the reverberant field is — which is why a large room has a distance past which moving further away barely helps.

Outdoors and for a line source rather than a point, the fall is gentler still: about 3 dB per doubling, which is one reason a motorway is audible so much further away than a single vehicle.

How long is a sound wave?

Speed divided by frequency, and the range is enormous. Concert A at 440 Hz is 0.78 metres in air; a 40 Hz bass note is 8.6 metres and a 10 kHz treble note is 34 millimetres.

That thousandfold span between 20 Hz and 20 kHz explains most of what is difficult about room acoustics. A treatment panel that absorbs treble is a few centimetres thick; one that absorbs the same fraction of a 40 Hz note has to be metres deep or use a resonant trick instead.

The speed itself moves with temperature — about 0.6 m/s per degree Celsius, from 331 m/s at 0 °C to about 349 at 30 °C. A five per cent shift across an ordinary temperature range is enough to detune a wind instrument, which is why they warm up before playing.

Where else does a logarithm show up?

In pitch. Each semitone multiplies frequency by the twelfth root of two, about 1.05946, so twelve semitones double it exactly — an octave.

That is the same structure as decibels: equal musical intervals are equal ratios rather than equal differences, which is why the frequency gap between two adjacent notes grows as you go up the keyboard. Three semitones above A440 is 523.25 Hz, middle C.

The reference pitch is itself a convention that has moved. A440 was only standardised internationally in 1955, and baroque ensembles commonly play at 415 Hz — a semitone lower — because that is closer to what the music was written for.

Which reference does a decibel use?

It has to be stated, and the suffix is what states it. dB SPL references the threshold of hearing in air. dBu and dBV reference specific voltages in audio equipment. dBFS references digital full scale and is therefore always negative. A bare "dB" with no suffix is a ratio between two things you named, and nothing else.

That is why a level of −18 dBFS and a level of +4 dBu can describe the same signal at the same moment. They are two measurements against two different zeros, and the relationship between them is a property of the converter rather than of the sound.

Questions people ask

Is 0 dB silence? No. Decibels are a ratio against a reference, so 0 dB SPL is the nominal threshold of hearing rather than nothing. Negative values are perfectly valid.

Why do amplifier power ratings feel misleading? Because doubling the watts buys 3 dB. Going from a 50 W amplifier to a 100 W one is a barely noticeable change in level, which is not what the number suggests.

Does the 6 dB distance rule work in a car? Not usefully. A car interior is a small reverberant space where reflections dominate almost everywhere, so the free-field rule does not describe it.

Why 20·log for microphones? Because a microphone produces a voltage proportional to pressure, and both are amplitude quantities. Anything measured as a pressure or a voltage uses 20.

Three, six and ten decibels are three different doublings, and knowing which is which is most of the subject. The dB calculator handles both ratio forms, the dB gain calculator covers amplification and distance loss, and frequency to wavelength and wavelength to frequency convert between the two ways of describing the same wave.