Intensity and Intensity Level
Loudness feels like a simple, single-scale sensation, but the physical quantity behind it — intensity — swings across such an enormous range that physicists don't measure it directly at all. Instead they compress it onto a logarithmic scale, and that compression is the whole trick behind the decibel.
Intensity: power spread over area
The intensity of a sound wave is defined as the sound power passing through a given area, per unit area:
with units of watts per square metre, $\text{W m}^{-2}$. Intensity is the physical quantity that underlies what we perceive as loudness: broadly, a more intense sound wave delivers more energy per second to your ear, and is perceived as louder. But "broadly" is doing a lot of work in that sentence — the relationship between the physical intensity of a sound and how loud it's perceived to be is not a simple linear one, which is exactly why a different scale is needed to describe it usefully.
Why a logarithmic scale?
The human ear is sensitive across an extraordinary range of intensities. The quietest sound most people can just detect (the threshold of hearing) and the loudest sounds that can be tolerated without immediate pain differ by a factor of roughly $10^{12}$ or more in intensity. Trying to describe this range using the linear intensity value directly is impractical for two separate reasons:
- Sheer scale. Quoting intensities directly would mean working with numbers from around $10^{-12}\text{ W m}^{-2}$ up to $1\text{ W m}^{-2}$ or beyond — an unwieldy spread of magnitudes to work with, compare, or plot on any sensible scale.
- Perceptual match. Human hearing doesn't respond to loudness in a linear way — perceived loudness increases roughly in step with equal ratios of intensity, not equal linear differences. Doubling the intensity of a sound doesn't double how loud it sounds; a much bigger jump in intensity is needed at high intensities to produce the same perceived increase in loudness as a much smaller jump produces at low intensities.
A logarithmic scale solves both problems at once: it compresses the enormous range of intensities into a much more manageable range of numbers, and because a logarithm turns ratios into differences, equal steps on a logarithmic scale correspond reasonably well to equal steps in perceived loudness — which is exactly the behaviour a linear intensity scale fails to capture.
Intensity level, dB and dBA
The quantity actually used to describe how intense a sound is, on this logarithmic scale, is the intensity level, $IL$, measured in decibels (dB):
where $I$ is the intensity of the sound being measured and $I_0$ is a fixed reference intensity, $I_0 = 1.0\times10^{-12}\text{ W m}^{-2}$. This reference value is conventionally taken as approximately the quietest sound intensity a healthy human ear can just detect, so setting $I = I_0$ gives $IL = 0\text{ dB}$ — the threshold of hearing sits at (approximately) zero on the decibel scale, with everything audible above it corresponding to a positive number of decibels.
Combining sources: intensities add, intensity levels don't
A classic source of error: if two independent sound sources are combined, it's the intensities that add directly (since intensity is a linear physical quantity, power per unit area), not the intensity levels. To find the combined intensity level, you must first add the intensities, then convert the total back into dB using the formula — you cannot simply add two dB values together.
For example, two identical sources each producing an intensity level of 70 dB on their own do not combine to give 140 dB. Converting 70 dB back to an intensity gives $I = I_0\times10^{7}$ for each source; adding the two intensities doubles the total intensity, which (from the worked example above) raises the intensity level by only about 3 dB — so two 70 dB sources together produce approximately 73 dB, not 140 dB.
Plain dB measures intensity level exactly as defined above: a purely physical quantity, calculated directly from the ratio of intensities, that makes no reference to how a human ear actually perceives different frequencies. dBA is a related but distinct unit: it applies a frequency-dependent weighting to the physical dB measurement, correcting it to better reflect how loud a sound is actually perceived by a typical human ear, since (as the next lesson explains in detail) the ear is not equally sensitive at every frequency. dB is what a meter measures from the raw physics; dBA is an attempt to make that number track subjective loudness.
Key takeaways
- Intensity is defined as power per unit area, $I = P/A$, measured in $\text{W m}^{-2}$.
- The range of intensities detectable by the human ear spans roughly twelve orders of magnitude, from the threshold of hearing to the threshold of pain.
- A logarithmic scale is used because it compresses this huge range into manageable numbers, and because perceived loudness tracks intensity ratios more closely than raw intensity differences.
- Intensity level is defined as $IL = 10\log_{10}(I/I_0)$ dB, with reference intensity $I_0 = 1.0\times10^{-12}\text{ W m}^{-2}$, chosen so that 0 dB is approximately the threshold of hearing.
- Doubling the intensity always adds about 3 dB to the intensity level, regardless of the starting value.
- When independent sound sources combine, intensities add directly, but intensity levels do not — the intensities must be added first, then converted back to dB.
- Plain dB is a purely physical, objective measurement of intensity level; dBA applies a frequency-dependent correction intended to approximate perceived loudness.