Result

Average power of a wave

P = \frac{1}{2} μ ω^{2} A^{2} v

Statement

The average rate at which a sinusoidal wave transports energy along a string.

Symbols and units

$P$	average power transmitted	W
$μ$	mass per unit length	kg/m
$A$	amplitude	m
$ω$	angular frequency	rad/s
$v$	wave speed	m/s

When it applies

A sinusoidal travelling wave on a uniform string.

Don't use when

Specific to a sinusoidal traveling wave on a uniform string — a standing wave transports no net power, and other wave types (sound, EM) use different power formulas.

Derivation

Kinetic and potential energy per unit length, averaged over a cycle, both scale as ω²A²; multiplying the energy per unit length by the wave speed gives the rate of transport:

P = \frac{1}{2} μ ω^{2} A^{2} v

Worked examples

How power scales with amplitude and frequency

Since power depends on the square of both amplitude and frequency:

P \propto A^{2} ω^{2}

Doubling either quadruples the power; doubling both increases it sixteen-fold.

Watch out for

Power scales with the square of amplitude and the square of frequency — doubling either quadruples the power carried, and doubling both increases it sixteen-fold.