Loud and soft — the size of a wave

A big wave is a loud sound. Grow the wave and shrink it, then meet the decibel — a sneaky ruler for loudness that's really a logarithm in disguise.

works for:Ages 6–9Ages 10–13Ages 14+

Turn any sound up and the wave gets taller. That height has a name — amplitude — and it is exactly what your ear calls loudness. Grow the wave below, hear it get louder, then meet the sneaky ruler we use for loudness: the decibel, which is really a logarithm wearing a disguise.

amplitude — the height of the wave is the loudness

drag to make the wave big and smallamplitude 0.50 (of a full swing)

taller than the quietest

× 10.0

amplitude ÷ 0.05 = 0.50 ÷ 0.05

relative loudness

+20.0 dB

dB = 20 · log₁₀(× 10.0)

power carried

× 100

power ∝ amplitude² = (× 10.0)²

◆ ×10 taller → +20 dB exactly. Ten times the height is just twenty decibels.

a decibel ruler for everyday sounds

0 dB … 120 dB

0 dB
Softest sound you can hear (threshold)
20 dB
Rustling leaves
30 dB
A whisper
60 dB
Normal talking
85 dB
Busy traffic — damage risk
110 dB
Rock concert
120 dB
Jet engine / pain

From the faintest sound you can hear to the threshold of pain, the power in the air grows by a factor of a trillion (10¹²). Nobody wants to say “a trillion times louder,” so we fold that whole range into a friendly 0–120 with a logarithm. Every +10 dB is ten times the power — and sounds to us roughly twice as loud.

Bigger wave, louder sound

Pitch is how fast the wave wiggles. Loudness is how tall it is. A gentle tap makes a short wave; a hard whack makes a tall one — same note, more push.

Loudness grows very fast

Make the wave twice as tall and it carries four times the power — power is amplitude squared. Sounds pile up energy far faster than they look like they should.

The decibel is a log

Our ears feel multiplying, not adding — just like octaves. So we measure loudness with a log ruler: +6 dB doubles the wave, +20 dB makes it ten times taller.

For the classroom

Learning goals

Amplitude is the height of the wave and is what we hear as loudness; power grows as the square of amplitude, so doubling the wave quadruples the power.
The decibel is a ratio on a logarithmic scale: dB = 20·log₁₀(amplitude ratio). +6 dB doubles the amplitude and +20 dB makes it ten times taller — big size changes become small dB numbers.
Hearing spans a trillion-to-one (10¹²) range in power from 0 to ~120 dB; the logarithm is what folds that enormous range into a readable scale, exactly as cents do for pitch.

Try this

1.Set the slider so it reads × 2 (amplitude 0.10) and note it lands on +6 dB; then × 10 (amplitude 0.50) for +20 dB. Ask: why does making the wave ten times taller add only twenty to the number? (Because the ruler is a logarithm.)
2.Walk the everyday-sounds table together. Have students place a whisper, a chat and a rock concert on the 0–120 scale, then work out that jumping from 60 dB to 90 dB is +30 dB — a thousand times the power, but only 'about eight times as loud' to our ears.