Energy and intensity of waves

Wave intensity tracks energy flux

A wave can have zero average displacement while transporting energy. Intensity measures average power crossing unit area and reveals how amplitude and geometry affect what a detector receives.

Relate amplitude to power

For a harmonic oscillator, average energy scales as A² because both kinetic and potential terms contain a squared amplitude. If waves spread without loss over a sphere, the same power crosses area 4πr², so intensity I = P/(4πr²) and falls as 1/r². In a plane wave, area may stay constant and intensity need not spread geometrically.

A 100 W isotropic source at 10 m gives I = 100/[4π(100)] = 0.0796 W/m². At 20 m intensity is one quarter, 0.0199 W/m², while amplitude is halved if I ∝ A².

Near-field geometry matters

The inverse-square rule assumes spherical spreading and no absorption or reflection. A waveguide, wall, or focused nozzle can keep intensity from following that law.

Work it: A speaker produces 0.20 W/m² at 4 m in an open field. Predict intensity at 12 m and the amplitude ratio if the spherical model holds.