Sound Intensity and Decibels — How Loud Is Loud?

Intensity: power per area

A sound wave carries energy, and intensity $I$ measures how much power flows through each square meter: $I = P/A$, in watts per square meter. Shout with the same vocal power into a small room versus an open field — the field spreads the power over a huge area, so the intensity (and loudness) at any spot drops.

For a point source, intensity falls as $1/r^2$: double your distance, quarter the intensity. That inverse-square spreading is why the back of a concert hall hears so much less than the front row.

The trillion-fold problem

Human hearing handles intensities from $10^{-12}\ \mathrm{W/m^2}$ (the faintest audible 1 kHz tone) to about $1\ \mathrm{W/m^2}$ (where pain begins) — a range of a trillion. No linear scale survives that: on a linear plot, everything below a jet engine rounds to zero.

The fix is a logarithm. The decibel compresses the whole range of hearing into 0–120 dB, where each 10 dB step is one factor of ten in intensity.

The decibel formula

Sound level $\beta$ in decibels compares an intensity to the reference $I_0 = 10^{-12}\ \mathrm{W/m^2}$ via the formula below.

The arithmetic is friendly: $\times 10$ in intensity is $+10$ dB; $\times 2$ is $10\log_{10} 2 \approx +3$ dB; $\times 100$ is $+20$ dB. A whisper at 30 dB carries a thousand times the intensity of a 0 dB threshold tone — the log scale keeps the numbers small while the physics spans orders of magnitude.

Reading the scale: reference levels

Textbook reference points anchor the scale: 0 dB is the hearing threshold, ~30 dB a whisper, ~60 dB normal conversation, ~70–80 dB a busy street, ~100 dB a subway train, ~110 dB a rock concert, ~120 dB the threshold of pain.

Notice the compression at work: conversation feels 'a bit louder' than a whisper, but 60 dB vs 30 dB is a thousand-fold intensity gap. Your ears are logarithmic instruments — the decibel just speaks their language.

Intensity vs loudness

Intensity is physics; loudness is perception. They track each other but not linearly: intensity grows as the square of wave amplitude ($I \propto A^2$), and perceived loudness roughly doubles every $+10$ dB — a tenfold intensity jump.

That is why audio engineers think in decibels: a volume knob that 'feels' even is really multiplying power exponentially. The math of perception is logarithmic all the way down.

Adding decibels: two speakers

Two identical 70 dB speakers don't make 140 dB — decibels add as logs, not numbers. Doubling the sources doubles the intensity: $10\log_{10}(2I/I_0) = 10\log_{10}(I/I_0) + 10\log_{10} 2$, which is $70 + 3 = 73$ dB.

General rule: doubling the number of equal sources adds 3 dB; ten times the sources adds 10 dB. To sound 'twice as loud' you need roughly +10 dB — ten times the power, not twice.

The 20-log variant

You will also meet $\beta = 20\log_{10}(p/p_0)$ for sound pressure. No contradiction: intensity is proportional to pressure squared ($I \propto p^2$), and $10\log(p^2) = 20\log(p)$. Same scale, different starting quantity.

The reference pressure $p_0 = 20\ \mu\mathrm{Pa}$ is chosen to match $I_0 = 10^{-12}\ \mathrm{W/m^2}$ in air — the two formulas agree. Whenever a decibel formula surprises you, check whether it's tracking power (10 log) or amplitude (20 log).

Variables and units

Common mistakes

Real-world applications

• Noise regulations set legal exposure limits in dB for workplaces and neighborhoods.

• Audio engineers mix tracks using dB faders matched to logarithmic hearing.

• Hearing protection is rated by how many dB it subtracts (NRR ratings).

• Speaker and headphone specs quote sensitivity as dB per watt at 1 meter.