Decibel, sound pressure and SPL

The decibel appears everywhere in audio, yet it is one of the most frequently misunderstood concepts. It is not a physical quantity by itself; it is a logarithmic way of expressing a ratio. Therefore the meaning of +3 dB depends on whether we are discussing power, voltage, sound pressure or perceived loudness.

1. dB is a dimensionless ratio

For a power ratio:

Symbols: L is a power ratio in decibels, P₁ reference power and P₂ measured power. The powers must be positive and expressed in the same unit; doubling power gives approximately +3 dB.

For amplitude-like quantities where power is proportional to the square of amplitude - for example voltage at the same impedance or acoustic pressure in the same medium - the form becomes:

Symbols: A₁ and A₂ are like-for-like amplitude quantities, such as voltage across unchanged impedance or sound pressure in the same medium. The factor 20 is valid only when power is proportional to amplitude squared.

The factor of 20 follows directly from P=U²/Z:

Symbols: U₁ and U₂ are RMS voltages and Z is the same purely resistive load in both cases. The derivation shows why a voltage ratio uses 20·log₁₀ when impedance is unchanged.

If Z₁ ≠ Z₂, voltage-level difference and power-level difference are no longer equivalent. The reference quantity and loading condition must then be stated explicitly.

2. What is SPL?

SPL means Sound Pressure Level. In air the reference pressure is 20 µPa:

Symbols: L_p is sound-pressure level in decibels, p_rms RMS sound pressure in pascals and 20 µPa the reference pressure in air. Their ratio is dimensionless.

1 Pa RMS corresponds to approximately 93.98 dB SPL, which is why 94 dB SPL at 1 kHz is a common acoustic-calibrator level. 10 Pa is about 114 dB SPL. A 20 dB increase means ten times the pressure amplitude.

3. +3, +6 and +10 dB are three different stories

+3 dB corresponds to approximately twice the power or intensity. +6 dB corresponds to twice an amplitude quantity such as voltage or sound pressure. +10 dB is ten times the power. The popular statement that “+10 dB sounds twice as loud” is only a rough psychoacoustic rule under certain conditions; it is not a physical identity and depends on frequency, level, signal and listener.

4. Adding two sound sources

Two independent, equal-level, uncorrelated noise sources add energetically to about +3 dB. Two perfectly coherent equal-amplitude signals in phase at one point can theoretically produce +6 dB SPL. Real loudspeaker systems sit between these simplified cases because phase and path length vary with frequency and position.

Symbols: L₁, L₂, … are levels of energetically additive independent sources and L_sum is the total level. Convert each decibel value to a linear energy ratio, add the ratios, then convert the result back to decibels.

5. Distance and the inverse-square law

For an ideal point source in free field, intensity falls with the square of distance and pressure approximately with 1/r:

Symbols: ΔL_p is change in sound-pressure level and r₁, r₂ are initial and new source distances. The equation assumes far-field free-space spreading from an ideal point source; the minus sign denotes level loss with distance.

Doubling distance therefore gives approximately -6 dB. This is a geometric-spreading result, not a universal “speaker law”: line-like sources, near-field conditions, room reflections and atmospheric absorption modify it.

Boundary conditions also matter at low frequencies. An ideal source radiating into free space (4π steradians), half space (2π), quarter space (π) and eighth space (π/2) can gain roughly +3, +6 and +9 dB respectively relative to the preceding larger radiation space, provided the boundaries are acoustically rigid and dimensions are small relative to wavelength.

6. dBA, dBC and dBZ

Frequency weighting filters approximate selected aspects of human sensitivity. A-weighting strongly attenuates low frequencies and is widely used for occupational and environmental noise legislation. C-weighting is substantially flatter through the audible band and retains much more low-frequency energy. Z-weighting is approximately flat (“zero” weighting) within its specified band.

7. Peak, Max, RMS and Leq are not interchangeable

Peak describes instantaneous extrema; Max is the maximum of whatever detector or time weighting is being used; RMS-type quantities represent energy-related effective values; and L_eq is an equivalent continuous energy-average level over a defined time interval. The labels require detector bandwidth, weighting and integration time to have engineering meaning.

Crest factor - peak-to-RMS ratio - is especially relevant for music. Two signals can share the same Leq but have very different transient peaks and therefore different headroom requirements.

8. Dynamic range and headroom

Dynamic range is the span between a defined noise floor or minimum usable level and a defined maximum. Headroom is the margin between normal operating level and a limiting point such as clipping, limiter action or mechanical excursion. In gain staging, a few dB of numerical headroom can be the difference between transparent transients and hard clipping.

9. dBu, dBV and dBFS - same notation, different references

The suffix identifies the reference. 0 dBu = 0.775 V RMS; 0 dBV = 1 V RMS; 0 dBFS is the maximum representable digital full-scale level in a fixed-point system.

The 0.775 V dBu reference comes from legacy 600 Ω systems: 1 mW into 600 Ω requires U=√(0.001·600)=0.7746 V RMS, historically 0 dBm on a 600 Ω termination. Modern dBu is a voltage reference and does not require an actual 600 Ω load.

10. Leq and SEL - integrating time

Equivalent continuous level is logarithmic energy averaging, not an arithmetic average of dB readings:

Symbols: L_eq,T is equivalent continuous level over duration T and L(t) is the instantaneous or short-time level in decibels. The integral averages linear acoustic energy over time, not the numerical dB readings themselves.

Sound Exposure Level (SEL) normalises the total energy of an event to a reference duration, typically one second. This makes events of different length easier to compare energetically.

11. Frequency weighting and time weighting must be stated together

A reading such as “96 dB” is incomplete. Was it LAeq, LCpeak, LAFmax, Z-weighted RMS, or something else? A/C/Z frequency weighting and Fast/Slow/Impulse or explicit integration time define different measurement operators. Reliable reporting includes the complete descriptor.

12. Measurement microphone and calibration

A measurement microphone is useful only as part of a traceable chain. Capsule response, preamplifier, ADC, calibration file, acoustic calibrator, environmental conditions and placement all contribute uncertainty. A 94 dB/1 kHz calibrator provides a known pressure reference so the software can map digital level to physical SPL.

13. Why RTA smoothing can be deceptive

Octave and fractional-octave smoothing are useful for readability but can hide narrow resonances, cancellations or high-Q problems. Conversely, an unsmoothed single-point response can exaggerate spatially local comb filtering. The correct amount of smoothing depends on the question being asked.

14. Quick rules for decibel arithmetic

• Double power: about +3 dB.
• Half power: about -3 dB.
• Double pressure or voltage at unchanged impedance/reference: about +6 dB.
• Ten times power: +10 dB.
• Ten times pressure amplitude: +20 dB.
• Double free-field distance from a point source: about -6 dB.

These are excellent mental checks, but every practical application must still state what physical quantity, reference and field condition are involved.