Sensitivity, power and distance: how loud a system actually plays

How loud a system plays is decided by four numbers: the sensitivity of the speaker, the power of the amplifier, the distance to the listener, and the crest factor of the material. Three of those are usually stated, one is usually forgotten, and the one that is forgotten is often the largest term in the budget.

The calculator runs the chain in both directions: from a power to the level at a distance, and from a target level back to the power required.

Sensitivity, and the convention that hides in it

Sensitivity is the level a speaker produces for a stated input, at a stated distance. Two conventions are in common use and they are not equivalent:

• dB SPL at 1 W at 1 m — the electrical convention, used by most professional loudspeaker datasheets.
• dB SPL at 2.83 V at 1 m — the voltage convention, common in hi-fi, because 2.83 V is the voltage that dissipates 1 W into 8 ohm.

The second is only a power specification when the impedance is 8 ohm. Into 4 ohm, 2.83 V is 2 W, so a 4 ohm speaker specified at 90 dB/2.83 V is really 87 dB/W — and a 16 ohm speaker specified the same way is really 93 dB/W. That 6 dB spread between two "90 dB" speakers is entirely a convention, and it is the most common way to mis-compare loudspeakers:

At 8 ohm the correction is 0.0 dB, at 4 ohm it is −3.0 dB, and at 16 ohm it is +3.0 dB. The calculator reports both conventions whichever one you enter, so the comparison is at least on one footing.

The inverse square law, and where it stops being true

A point source radiating into free space spreads its power over a sphere, so the intensity falls as the square of the distance and the level falls by 6.02 dB for every doubling. That is the whole of the distance calculation:

Two caveats matter in practice. At distances comparable to the size of the box the source is not a point and the law over-predicts the loss — measuring half a metre from a 15-inch driver will not give you the 6 dB the formula promises. And in a room the reverberant field eventually dominates, so the level flattens out instead of falling forever. The free-field figure is therefore a best case, and the honest way to use it is as a lower bound on what a room will deliver.

A worked example with the defaults: a 90 dB/1 W speaker driven with 100 W produces 110 dB at 1 m, then 104.0 dB at 2 m, 98.0 dB at 4 m and 91.9 dB at 8 m. Most of the level in any system is spent in the first few metres.

Power, and the compression of returns

Power enters the calculation logarithmically, so it buys much less than the numbers suggest:

Amplifier power	Gain over 1 W	Level at 1 m from 90 dB/W	At 4 m
1 W	0 dB	90.0 dB	78.0 dB
10 W	+10 dB	100.0 dB	88.0 dB
100 W	+20 dB	110.0 dB	98.0 dB
400 W	+26 dB	116.0 dB	104.0 dB
1000 W	+30 dB	120.0 dB	108.0 dB

Every doubling is 3.01 dB and every decade is 10 dB. Going from 100 W to 1000 W is 10 dB — real, but four times the amplifier for a change that a listener would call "louder" rather than "twice as loud". The corollary is more useful in the other direction: if you are 3 dB short, you need to double the power, and if you are 10 dB short there is no amplifier that will fix it — you need a more sensitive speaker, more of them, or a shorter distance.

The crest factor, which is usually the biggest term

Continuous and peak levels differ by the crest factor of the material, and that difference is the difference between two wildly different amplifiers. Consider 105 dB at 4 m from a 90 dB/1 W speaker:

• 105 dB continuous (a sine test tone) requires 117.0 dB at 1 m, which is 506 W — 63.6 V into 8 ohm.
• 105 dB peak with a typical 12 dB programme crest factor means 93 dB continuous, which requires only 31.9 W.

Sixteen times the power, for the same peak number, depending entirely on which question is being asked. Sizing an amplifier on continuous SPL for music is the most expensive mistake in this article; sizing it on peak SPL without allowing for the crest factor is the one that clips.

The practical rule is to set the target from the peaks, then check the continuous requirement against the driver's thermal rating. Music with a 12 to 15 dB crest factor needs an amplifier rated for the peaks and speakers rated for the continuous level, which is why tweeters in small systems fail before woofers do and why the same tweeter that survives a sine test can die on a clipped signal.

Air absorption, humidity and why it is usually ignored

Air absorbs sound, and the absorption rises steeply with frequency. At 20 °C and 50% relative humidity the figures are roughly 0.004 dB/m at 1 kHz, 0.025 dB/m at 4 kHz and 0.075 dB/m at 8 kHz. Over 20 m that is 0.08 dB at 1 kHz, 0.5 dB at 4 kHz and 1.5 dB at 8 kHz — negligible for the midband and worth knowing for the top end of a long throw.

Two details are worth carrying: humidity matters more than temperature, with dry air absorbing two to three times as much high frequency as humid air, and the effect is strongly frequency-dependent, so it tilts the response rather than attenuating it uniformly. A 40 m throw in a dry hall can lose 3 dB at 8 kHz while losing nothing at 500 Hz.

Stacking, and what +6 dB really means

Two identical cabinets fed the same signal do not simply add their levels. Where they are within a quarter wavelength of each other the outputs add coherently, which is +6.02 dB per doubling of count; where they are far apart, or above the frequency at which they stop being acoustically close, the powers add instead and the gain is only 3.01 dB per doubling. The calculator uses the coherent figure, which is the right one for stacked or flown boxes at low and mid frequencies and optimistic above them.

A realistic example: two 98 dB cabinets each driven with 500 W, at 20 m in the 2 kHz region. The level at 1 m is 131.0 dB, the distance loss takes 26.0 dB, the air takes another 0.5 dB, and the listener hears 104.8 dB continuous or 114.8 dB peak with 10 dB of crest factor. That is a comfortable concert-level figure, and it is exactly the kind of estimate worth running before specifying amplifiers rather than after.

Five mistakes worth avoiding

1. Comparing sensitivities across conventions. A 2.83 V specification means a different power at every impedance; convert to 1 W before comparing.
2. Using the 1 m figure at the listening position. Four metres is 12 dB, which is a large fraction of the entire system budget.
3. Sizing the amplifier on continuous level. Programme material has a crest factor of 10 to 15 dB, so the peaks need the power and the continuous level needs the thermal rating.
4. Expecting 6 dB per doubling close to the box. The source is not a point at a fraction of a metre.
5. Quoting an SPL without saying where it was measured. "105 dB" is half a specification; the distance, the bandwidth and whether it is continuous or peak are the other half.

Summary

Convert the sensitivity to one convention, add 10 log₁₀ of the power, subtract 20 log₁₀ of the distance, then subtract the air and add the stacking gain. Then decide whether the number you were aiming at was continuous or peak, because that single choice moves the amplifier requirement by more than an order of magnitude — 506 W against 31.9 W in the example above.

Values shown are engineering aids rather than measurements; see the disclaimer and the tool index.