Decibel

The decibel (dB) expresses a level: the base-10 logarithm of the ratio between a measured quantity and a reference quantity of the same kind. It is not an absolute unit but a comparison, which always requires stating the chosen reference.

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How to measure it in class

FizziQ’s sound level meter uses the smartphone’s microphone and displays the sound level in decibels in real time, with a graphical recording that can be analyzed.

Steps:

Open FizziQ’s sound level meter and record the background noise of the room, with silence: typically 35 to 45 dB
Play a continuous sound with a speaker and note the level L₁ obtained at a fixed spot
Add a second identical speaker, set to the same volume, placed next to the first one, and record L₂
Compare the measured difference L₂ − L₁ with the expected value of +3 dB for two independent sources
Repeat while moving the microphone away: doubling the distance to the source should lose about 6 dB in free field
Plot L as a function of log(r): you should get a straight line

Scientific activities on this topic

Sound level and distance: Chloe at the concert - establish experimentally how sound level decreases with distance from the source
Adding sound levels: do two sources make twice as much noise? - verify the +3 dB rule and discover why decibels do not add up

Learn more

Why a logarithmic scale, and why “deci”?

The human ear covers twelve orders of magnitude in intensity: from the threshold of hearing (10⁻¹² W·m⁻²) to the threshold of pain (1 W·m⁻²), the ratio is one trillion. A linear scale would be unmanageable. The logarithm compresses this immense interval into a scale from 0 to 120, far more practical.

The primary unit is the bel, named after Graham Bell, defined by log(P/P₀): it comes from the engineers of the Bell laboratories who, in the 1920s, were looking for a way to quantify the attenuation of a telephone line. The bel proved too large for everyday use, hence the decibel, which is one tenth of it - this is where the factor 10 in the formula comes from.

Power or amplitude: 10 log or 20 log?

This is a common trap. The decibel is always defined on a ratio of powers (or intensities):

L = 10 × log(I / I₀)

But we often measure an amplitude - an acoustic pressure p, an electrical voltage U - and not a power. Since power is proportional to the square of the amplitude, the 10 log becomes 20 log:

L = 20 × log(p / p₀), with p₀ = 20 μPa

The two formulas give the same result in decibels: they are not two competing conventions, but two ways of writing the same definition. Using the wrong factor amounts to doubling or halving all your levels. Remember: coefficient 10 for an energy quantity, 20 for an amplitude quantity.

Adding sound levels

Decibels never add up directly. Two sources at 60 dB do not give 120 dB. You must go back to the intensities, add them, then convert back:

L_total = 10 × log(10^(L₁/10) + 10^(L₂/10))

Two cases must be distinguished, and this is the heart of the matter:

Incoherent sources (two independent noises, two voices, two distinct instruments). The intensities add: I_tot = 2I, hence a gain of 10 × log(2) ≈ +3 dB. Two sources at 60 dB give 63 dB. This is by far the most frequent case in practice.
Coherent sources in phase (the same signal sent to two loudspeakers, at a point where the two waves arrive in phase). It is the amplitudes that add: p_tot = 2p, so I_tot = 4I, that is a gain of 10 × log(4) ≈ +6 dB.
Still in the coherent case, but in phase opposition, the amplitudes subtract and the level drops - down to complete cancellation if the amplitudes are equal. This is the principle of anti-noise.

Another useful consequence: when two sources have very different levels, the weaker one is negligible. Adding a 50 dB noise to a 70 dB noise gives 70.04 dB, an undetectable variation. To lower the noise of a workshop, you therefore have to treat the noisiest machine - treating the others is pointless.

Weighted decibels: dB, dB(A), dB(C)

The ear is not equally sensitive to all frequencies: it is poorly sensitive to low frequencies and most sensitive between 3 and 4 kHz. A raw physical level, in dB, therefore does not reflect the perceived annoyance. Sound level meters apply an A-weighting, a filter that attenuates low frequencies to get closer to the sensation. The levels are then written dB(A), and these are the ones used as the regulatory reference in occupational health. The C-weighting, which is flatter, is used for very high peak levels. The same sound can give quite different values in dB and in dB(A), especially if it is rich in low frequencies: comparing two measurements requires the same weighting.

The decibel does not exist without a reference

Since it expresses a ratio, a level in decibels only makes sense if you specify what you are comparing to. In airborne acoustics, the reference is p₀ = 20 μPa (that is I₀ = 10⁻¹² W·m⁻²), chosen because it corresponds to the threshold of hearing of a young ear at 1 kHz. In underwater acoustics, the reference is 1 μPa: the levels quoted there look enormous, without being comparable to airborne levels. In electronics, the dBm refers to 1 milliwatt. Reading “130 dB” without knowing which reference is meant is strictly meaningless. Nor is the decibel reserved for acoustics: it is used in electronics and telecommunications for gains and attenuations (dBm, dBW), in signal processing for the signal-to-noise ratio, and in radio for antenna gain (dBi).

A consequence worth remembering: a level in decibels can be negative. It simply means that the intensity is below the reference - a sound of −5 dB exists, it is just below the conventional threshold of hearing.

Does adding two sound sources really increase the sound level by 3 decibels? (FizziQ blog) covers the distinction between coherent and incoherent sources in detail.

Formula

Sound intensity level, from the intensity:

L = 10 × log(I / I₀)

Sound pressure level, from the pressure amplitude:

L = 20 × log(p / p₀)

Level difference between two situations:

L₂ − L₁ = 10 × log(I₂ / I₁)

Addition of two levels (incoherent sources):

L_total = 10 × log(10^(L₁/10) + 10^(L₂/10))

where:

L: sound level (dB)
I: sound intensity (W·m⁻²)
I₀: reference intensity, 10⁻¹² W·m⁻²
p: acoustic pressure amplitude (Pa)
p₀: reference pressure, 20 μPa = 2 × 10⁻⁵ Pa

Benchmarks to remember:

+3 dB → intensity × 2 (two identical independent sources)
+6 dB → intensity × 4 (amplitude × 2, sources in phase, or distance divided by 2)
+10 dB → intensity × 10, perceived as “twice as loud”
+20 dB → intensity × 100
−6 dB → distance to the source doubled, in free field

Application examples

Reference scale: 0 dB threshold of hearing; 20 dB rustling leaves; 40 dB library; 60 dB conversation; 80 dB busy street; 100 dB jackhammer at 5 m; 110 dB concert; 120 dB threshold of pain; 140 dB aircraft engine at 25 m.
Two identical motorcycles. A single motorcycle produces 90 dB as it passes. Two motorcycles side by side give 93 dB, not 180. It would take ten to reach 100 dB, and one hundred for 110 dB: each +10 dB step requires multiplying the number of sources by ten.
Occupational regulations. The regulatory action threshold is set at 85 dB(A) over 8 hours. Since the noise dose received is proportional to intensity multiplied by duration, +3 dB halves the admissible exposure time: 88 dB(A) for 4 h is equivalent to 85 dB(A) for 8 h.
Sound insulation. Acoustic double glazing provides an attenuation of 30 to 40 dB, that is a transmitted intensity divided by 1,000 to 10,000. A highway noise barrier typically gains 5 to 10 dB for nearby residents.
Active noise-cancelling headphones. They generate a wave in phase opposition with the ambient noise and gain 20 to 30 dB, but mainly at low frequencies, where the wavelength is large compared with the size of the headset.
Signal-to-noise ratio. A digital link with a signal 1,000 times more powerful than the noise shows an SNR of 30 dB. Below about ten decibels, the transmission becomes unreliable.

FAQ

Q: Does the decibel measure sound intensity? A: No, and this is the most widespread confusion. Sound intensity I is a power per unit area, in W·m⁻². The decibel measures a level L, that is, the logarithm of the ratio I/I₀. The two quantities are related but distinct: I varies over twelve orders of magnitude, L varies from 0 to 120.

Q: Do two speakers at 80 dB make 160 dB? A: No, about 83 dB. It is the intensities that add, not the levels: doubling the intensity adds 10 × log(2) ≈ 3 dB. If the two speakers play exactly the same signal and the waves arrive in phase at the measurement point, the gain rises to 6 dB - but that is a special case, difficult to achieve over a whole room.

Q: Why do we sometimes see 10 log and sometimes 20 log? A: Because the starting quantity is not the same. You use 10 log for a ratio of powers or intensities, and 20 log for a ratio of amplitudes (pressure, voltage). Since power is proportional to the square of the amplitude, the two expressions give the same level: log(p²) = 2 log(p).

Q: Can a sound level be negative? A: Yes. A negative level means the intensity is lower than the reference intensity I₀. This is the case in an anechoic chamber, where you can go below 0 dB. Nothing abnormal: 0 dB is not silence, it is simply the conventional threshold of hearing.

Q: Why does my smartphone not display the same value as the teacher’s sound level meter? A: Because a smartphone’s microphone is not calibrated in absolute terms and its response depends on the model, the position and the weighting applied. Differences of a few decibels are normal. On the other hand, the variations measured with the same device are reliable, and that is what matters for classroom activities.

Sound Intensity - Sound Level - Sound level meter - Microphone - Inverse Square Law - Active Noise Reduction - Pink Noise