Sound Intensity & the Decibel Scale

A sound wave transports energy, and the quantity a listener meets is the intensity I: the time-averaged power per unit area crossing a surface that faces the source, in W m⁻². Instantaneously the power flux is the excess pressure p multiplied by the particle velocity u, and for a travelling plane wave in a non-dispersive fluid those two are in phase with p = ρv u, so the flux is p²/(ρv). Averaging a squared sinusoid over one period, (1/T)∫₀ᵀ sin²(ωt) dt = ½, leaves I = pₘₐₓ²/(2ρv). The product z = ρv is the medium's characteristic impedance; for air at 20 °C, z = 1.20 × 343 = 412 Pa s m⁻¹. Feed the reference intensity into that relation and the pressure amplitude at the threshold of hearing is √(2 × 412 × 1.00 × 10⁻¹²) = 2.87 × 10⁻⁵ Pa, about 20 μPa rms — two parts in ten thousand million of atmospheric pressure, and a displacement amplitude at 1 kHz of only 1.1 × 10⁻¹¹ m.

Surround a small source with a sphere of radius r. If the medium neither absorbs nor reflects, every watt radiated must cross every such sphere, so P = I × 4πr² and I = P/(4πr²). Nothing about sound entered that argument: it is conservation of energy and the area of a sphere. Since I ∝ pₘₐₓ², the pressure amplitude falls only as 1/r. A 0.25 W source gives I = 0.25 ÷ (4π × 16.0) = 1.24 × 10⁻³ W m⁻² at 4.0 m, and exactly a quarter of that at 8.0 m. Three assumptions are buried here: the source is small compared with r, so the field is spherical; the field is free, so no reflected energy returns; and the air does not absorb, which fails above a few kilohertz over tens of metres. A line source such as a motorway spreads over a cylinder instead, so I ∝ 1/r and the fall per doubling is halved.

From the faintest audible 1 kHz tone to the onset of pain, intensity spans twelve decades, 1.00 × 10⁻¹² to about 1 W m⁻². No linear axis can show that, and the ear's response to a change is closer to a ratio than to a difference. So levels are defined as β = 10 log₁₀(I/I₀), with the reference fixed by definition at I₀ = 1.00 × 10⁻¹² W m⁻² rather than measured. A decibel is therefore a compressed ratio, and a level quoted without its reference means nothing. Two consequences are worth committing to memory. Doubling the intensity adds 10 log₁₀ 2 = 3.01 dB, and a factor of ten adds exactly 10 dB. Doubling the distance from a point source quarters I, so it subtracts 10 log₁₀ 4 = 6.02 dB — the six decibels per doubling an acoustician quotes without calculating.

Microphones are calibrated in pascals, so field data are usually reported as a sound pressure level, Lₚ = 20 log₁₀(pᵣₘₛ/p₀) with p₀ = 20 μPa. The 20 is not a rival convention. Because I ∝ p², taking the logarithm brings the square down as a factor of two: 10 log₁₀(p²/p₀²) = 20 log₁₀(p/p₀). The two references were chosen to agree, since p₀²/(ρv) = I₀ requires ρv = 400 Pa s m⁻¹ exactly; air at 20 °C offers 412, so the two levels differ by 10 log₁₀(412/400) = 0.13 dB and the difference is ignored in practice. The factor 20 also fixes the pressure arithmetic: doubling the pressure amplitude adds 6.02 dB, and halving the distance from a point source, which doubles p, does the same.

Two uncorrelated sources — machines whose vibrations hold no fixed phase relation — deliver intensities that add, because the cross term in (p₁ + p₂)² averages to zero over a cycle. Levels do not add, because the logarithm of a sum is not the sum of the logarithms. The procedure is therefore fixed: convert each level with I = I₀ × 10^(β/10), add the intensities, convert back with β = 10 log₁₀(I/I₀). Take 86.0 dB and 80.0 dB at one point. The intensities are 3.98 × 10⁻⁴ and 1.00 × 10⁻⁴ W m⁻², their sum is 4.98 × 10⁻⁴ W m⁻², and the level is 87.0 dB. Silencing the quieter machine completely would save 1.0 dB; silencing the louder would save 7.0 dB. That asymmetry is the whole of noise control: the loudest source sets the answer, and anything more than about 10 dB below it is already irrelevant.

Indoors only part of the energy reaches you directly; the rest arrives after reflections, and the reverberant field is roughly uniform through the room. Near the source the direct field wins and the level falls at 6.02 dB per doubling, but beyond a critical distance the reverberant field dominates and the level stops falling, so walking further away buys nothing. Real sources also radiate unevenly: a directivity factor Q multiplies the free-field intensity, and a source sitting on a hard floor radiates into a hemisphere, giving Q = 2 and a level 3.01 dB above the free-field value. Finally, intensities add only for uncorrelated sources. Two loudspeakers driven by one signal are coherent, so where their pressures arrive in phase the amplitudes add, the intensity quadruples and the level rises 6.02 dB, while in antiphase it collapses.