Sound Intensity — Concert Speaker (Worked Example)

Given

• P = 100 W

• r = 10 m

Find

The intensity I.

Principle

Power spreads over a sphere of area 4πr², so intensity I = P/(4πr²) — the inverse-square law for sound.

Formula

Solution

1. Write the formula: $I = P/(4\pi r^2)$

2. Substitute: $I = 100/(4\pi \times 10^2)$

3. Denominator: $4\pi \times 100 = 1256.6$

4. Divide: $I = 100/1256.6 = 0.0796\ \mathrm{W/m^2}$

Unit check

W ÷ m² = W/m² — correct.

What this means physically

0.0796 W/m² — about 109 dB, genuinely loud. Double the distance to 20 m and intensity quarters to 0.02 W/m² — the inverse-square law in action.