3.3 Decibel scale and sound measurement

Decibel Scale Fundamentals

Decibels provide a way to compress an enormous range of sound intensities into manageable numbers. Because human hearing spans roughly 12 orders of magnitude in intensity, a linear scale would be impractical. The decibel scale solves this by using logarithms, which also happen to align well with how we actually perceive loudness.

Logarithmic Nature of Decibels

A decibel (dB) expresses the ratio between a measured value and a reference value on a logarithmic scale. Rather than saying one sound is 1,000,000 times more intense than another, you can say it's 60 dB higher.

The key relationship to internalize:

• A 10 dB increase = 10× the sound intensity
• A 20 dB increase = 100× the sound intensity
• A 30 dB increase = 1,000× the sound intensity

Each 10 dB step multiplies intensity by another factor of 10. This is why a rock concert at 110 dB isn't just "a bit louder" than a 70 dB conversation; it's 10,000 times more intense.

Conversions in the Decibel Scale

There are two main quantities you'll convert to decibels: sound pressure and sound intensity. Each has its own formula and reference value.

Sound Pressure Level (SPL):

$SPL = 20 \log_{10}\left(\frac{P}{P_{ref}}\right) \text{ dB}$

The reference pressure $P_{ref}$ is 20 μPa in air, which corresponds to the threshold of human hearing at 1 kHz.

Sound Intensity Level (SIL):

$SIL = 10 \log_{10}\left(\frac{I}{I_{ref}}\right) \text{ dB}$

The reference intensity $I_{ref}$ is $10^{-12}$ W/m², again corresponding to the hearing threshold.

Notice SPL uses a factor of 20 while SIL uses 10. This is because intensity is proportional to pressure squared ($I \propto P^2$), and the logarithm rule $\log(x^2) = 2\log(x)$ pulls that factor of 2 out front: $10 \times 2 = 20$.

In free-field conditions (open outdoor spaces with no reflections), SPL and SIL are numerically equal. In enclosed or reverberant spaces, they can diverge.

Common Sound Levels in Decibels

These benchmarks are worth memorizing for quick reference:

The threshold of pain sits around 130–140 dB. Prolonged exposure above 85 dB can cause hearing damage, which is why occupational noise regulations typically set limits in that range.

Sound Level Meters for Measurement

A sound level meter converts acoustic pressure fluctuations into a dB reading. Its signal chain has four main stages:

1. Microphone captures sound waves and converts them to an electrical signal
2. Amplifier boosts the signal to a usable level
3. Weighting network shapes the frequency response according to the selected weighting curve
4. Display shows the resulting dB value

Frequency weighting scales adjust which frequencies the meter emphasizes:

• A-weighting (dBA) rolls off low and very high frequencies to approximate how the human ear responds at moderate levels. This is the most commonly used weighting for environmental and occupational noise.
• C-weighting (dBC) is much flatter, with only slight rolloff at the extremes. It's used for louder industrial noise and peak measurements.
• Z-weighting (dBZ) applies no filtering at all, giving a flat response across the full frequency range. This is used for scientific and engineering analysis where you need the raw data.

Time weighting controls how quickly the meter responds to changes:

• Fast (125 ms time constant) tracks fluctuating sounds and captures short peaks
• Slow (1 s time constant) smooths out variations, giving a more stable reading for steady noise

Common measurement types:

• $L_{eq}$ (equivalent continuous level) averages the sound energy over a measurement period. This is the single most useful metric for characterizing a noise environment.
• $L_{max}$ records the highest level reached during the measurement.
• Statistical levels like $L_{10}$, $L_{50}$, and $L_{90}$ indicate the dB level exceeded for 10%, 50%, or 90% of the measurement time, respectively. $L_{90}$ often represents the background noise floor, while $L_{10}$ captures the louder events.

Calibration should be performed before and after every measurement session using a known reference source (typically a pistonphone or acoustic calibrator). This confirms the meter is reading accurately and that no drift occurred during the session.