Sound intensity — decibels (logarithmic scale): Visual walkthrough

Step 1 — What "loudness" really is: energy spreading over an area

WHAT. Before any formula, picture sound leaving a source. Sound is a travelling ripple of air pressure — see Sound waves — pressure & displacement — and it carries energy (that energy is what pushes your eardrum). Energy delivered per second is power, measured in watts (W).

WHY. "Loud" isn't about total power alone. A loudspeaker putting out $10$ watts sounds deafening up close and faint far away, even though the watts never change. What matters is how concentrated that power is when it reaches you — how many watts land on each square metre of your eardrum.

PICTURE. In the figure, the same amber energy-arrows leave the source. Near the source they crowd through a small patch; far away the same arrows are spread over a big patch, so each square metre catches fewer.

Step 2 — How intensity shrinks with distance (the sphere argument)

WHAT. A tiny source radiates equally in all directions. At distance $r$ , all its power $P$ has spread over the surface of a sphere of radius $r$ . A sphere's surface area is $A = 4 π r^{2}$ , so

$I (r) = \frac{P}{sphere surface 4 π r ^{2}} .$

WHY. No energy is destroyed as it travels (in free air with no echoes). It just gets painted over an ever-bigger balloon. Bigger balloon → thinner paint → smaller $I$ . This is the Inverse-square law for radiation.

PICTURE. Two cyan spheres, radius $r$ and $2 r$ . The outer one has 4× the surface area, so the same power is diluted to one-quarter the intensity. Watch how the amber "paint" thins.

Step 3 — The problem: the numbers are absurd

WHAT. Let's list the honest intensities the ear handles.

Sound	$I$ (W/m²)
Faintest you can hear	$0.000000000001$
A conversation	$0.000001$
Pain / damage	$1$

WHY show this? So you feel the disaster. The faintest sound and the loudest sound differ by a factor of a trillion ( $1 0^{12}$ ). Plot these on an ordinary ruler and the whisper sits invisibly against zero while the painful sound is a mile away.

PICTURE. A linear axis from $0$ to $1$ W/m². Almost every everyday sound is crushed into a sliver near the left edge — useless.

Step 4 — The tool that fixes it: the logarithm

WHAT. We bring in $lo g_{10}$ . Read $lo g_{10} (x)$ out loud as: "how many times do I multiply 10 by itself to get $x$ ?" So $lo g_{10} (1000) = 3$ because $10 \times 10 \times 10 = 1000$ .

WHY this tool and not another? Our numbers are all powers of ten multiplied together, and multiplying gets ugly fast. The one function on Earth whose whole job is to turn multiplication into addition is the logarithm: $lo g_{10} (a \times b) = lo g_{10} a + lo g_{10} b .$ That is exactly the "×10 becomes +constant" behaviour Step 3 demanded. No other elementary tool does this — that's why it's the log and nothing else. (Full toolkit: Logarithms and exponentials.)

PICTURE. The same intensities from Step 3, now placed by their $lo g_{10}$ . The trillion-fold crush becomes evenly spaced tick marks. The log has unbent the scale.

Step 5 — Choosing the zero point: the reference intensity $I_{0}$

WHAT. We need a starting line — the value that gives "0". We pick the quietest sound a healthy ear can detect: $I_{0} = 1 0^{- 12} W/m^{2} (the ==threshold of hearing== at 1 kHz) .$

WHY. With this choice, a sound sitting exactly at the threshold gives $I / I_{0} = 1$ , and $lo g_{10} (1) = 0$ . So silence-threshold reads 0 — a natural, tidy floor. Everything audible is a positive count of "factors of 10 above the quietest whisper".

PICTURE. A number line whose origin is nailed to $I_{0}$ . Each step to the right = one more ×10. The amber marker sits at 0 for $I = I_{0}$ .

Step 6 — Finer resolution: the decibel

WHAT. One bel is a whole factor of ten — a huge jump. Between "conversation" and "shouting" you'd only move a fraction of a bel, awkward to speak. So we chop each bel into ten equal parts (deci = one-tenth) by multiplying by 10:

$β = deci: tenths of a bel 10 count factors of ten lo g_{10} (\frac{I your sound}{the zero-line I _{0}}) dB$

WHY the extra 10? Purely for a comfortable scale: it stretches the useful audible range from "0 to 12 bels" into "0 to 120 dB", so ordinary conversation isn't buried in decimals. The physics is untouched — a 10× cosmetic rescaling.

PICTURE. Two aligned rulers: the top in bels ( $0 \dots 12$ ), the bottom in decibels ( $0 \dots 120$ ). Same landmarks, just ten times more tick marks below.

Step 7 — The payoff: every ×10 in intensity adds exactly 10 dB

WHAT. Multiply intensity by 10 and watch the level: $β^{'} = 10 lo g_{10} \frac{10 I}{I _{0}} = 10 [= 1 lo g_{10} 10 + lo g_{10} \frac{I}{I _{0}}] = 10 + β .$ The split in the middle is only possible because the log turns the "×10" into "+1" (Step 4). Multiply by two levels' worth of intensities and subtract: $Δ β = β_{2} - β_{1} = 10 lo g_{10} \frac{I _{2}}{I _{1}} .$ The $I_{0}$ cancels — for changes you never need the reference at all.

WHY it matters. This is the rule you actually use: you rarely know absolute intensities, but you often know a ratio (twice as many singers, four times the distance). Feed the ratio straight in.

PICTURE. A staircase: each tread is one ×10 in intensity, each riser is a fixed +10 dB. Amber landmarks: ×2 ≈ +3 dB, ×100 = +20 dB.

Step 8 — Edge & degenerate cases (never leave a gap)

WHAT. Every scale has boundaries. Let's walk each one so no scenario surprises you.

$I = I_{0}$ (exactly threshold): $β = 10 lo g_{10} (1) = 0 dB$ . The floor, by design.
$I < I_{0}$ (below threshold): $I / I_{0} < 1$ , and $lo g_{10}$ of a number below 1 is negative. So $β$ goes negative — e.g. $I = 1 0^{- 13}$ gives $β = - 10$ dB. Perfectly legal; it just means "quieter than the faintest audible sound". Anechoic chambers reach about $- 9$ dB.
$I \to 0$ (true silence): $lo g_{10} (0) \to - \infty$ . The formula sends $β$ to minus infinity. That's the maths honestly telling you "no sound at all has no place on a ratio scale" — you can never reach it, only approach it.
Adding two equal sources: intensities add, so $I \to 2 I$ , giving $+ 3$ dB — not doubling the dB number. (Step 7's ×2 rule.)
Combining with distance: from Step 2, $r \times 4 \Rightarrow I \div 16 \Rightarrow Δ β = 10 lo g_{10} \frac{1}{16} \approx - 12$ dB.

WHY show these. The negative and the $- \infty$ cases are where beginners panic ("can dB be negative?!"). Yes — the scale is a comparison, and comparisons can go either way.

PICTURE. The full number line: amber $I_{0}$ at 0 dB, cyan region to the right (audible, positive), a hatched region to the left (below threshold, negative) trailing off toward $- \infty$ at true silence.

The one-picture summary

Below, the whole derivation lives in a single frame: raw intensities crushed on a linear ruler (top), the log unbending them into the bel ladder (middle), and the ×10 = +10 dB staircase with all landmarks (bottom). Trace it left→right, top→bottom and you have re-walked Steps 1–8.

Recall Feynman retelling — the whole walkthrough in plain words

Sound is energy flying off a source. Spread that energy over a big area and each patch gets little (quiet); squeeze it small and each patch gets lots (loud) — that's intensity, watts per square metre. As you back away, the energy coats a bigger and bigger balloon, so it thins as one-over-distance-squared.

The trouble: the quietest sound you can hear and a painful sound differ by a trillion times. Written normally, the quiet one is invisible next to the loud one. So we count zeros instead of raw numbers — that's what a logarithm does: it turns "times ten" into "plus one". We measure each sound against the faintest audible one, $I_{0} = 1 0^{- 12}$ , so the whisper-threshold scores a tidy 0. Then, because whole factors of ten are too coarse to speak, we slice each into ten and call the slice a decibel — that's the lone "10" out front.

The magic that pops out: every ×10 in intensity is a fixed +10 dB step, so loudness becomes a friendly staircase from 0 to 120. Two equal sources? Just +3 dB. Four times farther? −12 dB. Quieter than the threshold? Negative dB — totally fine. Absolute silence? The formula runs off to minus infinity, its polite way of saying "that point isn't on a comparison scale." That's the decibel, built from a single idea: count the zeros.

Recall

Why is $lo g_{10}$ , and not some other function, the right tool here? ::: Our numbers span factors of ten multiplied together; only the logarithm turns multiplication into addition, making "×10 louder" a fixed step. Why does the reference $I_{0}$ vanish in the difference rule $Δ β = 10 lo g_{10} (I_{2} / I_{1})$ ? ::: Subtracting the two level formulas, $10 lo g_{10} (I_{2} / I_{0}) - 10 lo g_{10} (I_{1} / I_{0})$ , the $lo g_{10} I_{0}$ terms cancel. What does a negative dB value mean? ::: The sound is quieter than the threshold reference $I_{0}$ , since $lo g_{10}$ of a ratio below 1 is negative. Where does the standalone factor of 10 in the formula come from? ::: From "deci" — we split each coarse bel into ten decibels for a comfortable 0–120 scale.

Parent: Sound intensity — decibels (logarithmic scale) · See also perceived loudness · Doppler effect