Foundations — Sound intensity — decibels (logarithmic scale)

This page assumes you know nothing. Every letter, every squiggle used in the decibel note is unpacked below, in the order that lets each one lean on the one before it.

1. Energy, power, and the watt (the very bottom)

The picture: imagine filling a bucket with water. The total water is energy. How fast the tap runs — litres per second — is power. A trickle over an hour and a gush over a minute can deliver the same water; they differ in power.

Why the topic needs it: a loud sound isn't about total energy, it's about how much energy arrives per second. That "per second" is exactly power. Every $P$ you see in the parent note is measured in watts. See Wave energy and power for where a wave's power comes from.

2. Area, and "per unit area" (the $/A$ idea)

The picture — why we divide power by area: hold your hand up in the rain. The rate of water hitting your whole hand is like power. But whether it feels heavy depends on how that rate is shared across the skin — that's rate per square metre. A big umbrella catches the same rain spread thin; a thimble catches a fierce trickle concentrated.

Why the topic needs it: intensity is power per unit area, so you must be comfortable with dividing by an area before the formula $I=P/A$ means anything.

3. Intensity $I=P/A$ (assembling the first two)

The picture: the same $3\text{W}$ of sound landing on $1{\text{m}}^2$ gives $I=3{\text{W/m}}^2$; spread over $6{\text{m}}^2$ it gives only $0.5{\text{W/m}}^2$. Same power, different intensity, because the area changed.

Why the topic needs it: $I$ is the quantity the decibel scale measures. Everything downstream is a comparison of one $I$ against another. The energy itself comes from vibrating air; see Sound waves — pressure & displacement for what is actually moving.

4. The sphere and the inverse-square law ($4\pi r^2$)

Now we need to know how $I$ changes as you walk away from a source. This is where the geometry of a sphere enters.

The picture — why a sphere at all: a tiny "point" source pumps sound out equally in every direction. After a time, that energy sits on the skin of an expanding ball centred on the source. All the source's power $P$ is smeared over that skin.

So the intensity at distance $r$ is $I=\frac{P}{4\pi r^2}.$

Why the topic needs it: worked example (d) in the parent note walks a speaker from $2\text{m}$ to $8\text{m}$. You must know $I\propto 1/r^2$ to find the intensity ratio before touching decibels.

5. Powers of ten and exponents ($10^{-12}$, $10^{10}$)

The parent note is drowning in numbers like $10^{-12}$. Here is what they mean.

The two rules you actually use:

• Multiplying adds exponents: $10^a\times 10^b=10^{a+b}$.
• Dividing subtracts them: $\frac{10^a}{10^b}=10^{a-b}$.

Why the topic needs it: the threshold intensity is $I_0=10^{-12}{\text{W/m}}^2$, and every intensity ratio is a clash of powers of ten. If you can't subtract exponents in your head, the decibel formula stays opaque.

6. Logarithms — the star of the show (${\log }_{10}$)

Powers of ten make big numbers; logarithms count them back.

The picture — a ruler where each tick is ×10:

On an ordinary ruler, steps are added (1, 2, 3). On a log ruler, steps are multiplied (1, 10, 100, 1000) yet they land at evenly spaced positions 0, 1, 2, 3. The log reads off that position. This is why a trillion-fold range collapses into a tidy 0-to-12 spread.

7. Ratios, the reference $I_0$, and "dimensionless"

The picture: $I_0=10^{-12}{\text{W/m}}^2$ is the faintest sound a healthy ear detects — the "zero line" of loudness. Every sound is quoted as a multiple of that baseline, so 0 dB means "equal to the quietest audible sound", not "no sound at all".

Why the topic needs it: the decibel formula $\beta =10{\log }_{10}\left(\frac{I}{I_0}\right)$ feeds the ratio $I/I_0$ into the log. Skip the reference and you'd be taking the log of something with units — meaningless.

8. The decibel unit and the Greek letters ($\beta$, $\Delta$)

Why the topic needs it: the difference rule $\Delta \beta =10{\log }_{10}(I_2/I_1)$ uses $\Delta$ to compare two situations (two distances, two numbers of violins), and every answer is stamped in dB. Recognising $\Delta$ as "the gap between" and dB as "the unit of the level" is essential.

Prerequisite map

Read it top-down: joules build watts, watts and area build intensity; separately, powers of ten build the logarithm; the two streams meet at the decibel.

Equipment checklist

Test yourself — you should be able to answer each before reading the parent note.