Waves transport energy from one region to another. The local medium may only oscillate around equilibrium, but energy can propagate with the wave.

Amplitude and energy

For many linear waves, the transported energy grows with the square of amplitude. Doubling the amplitude therefore typically multiplies the energy transfer by four.

The exact proportionality depends on the physical wave system.

Power and intensity

If a source transfers wave energy at rate $P$, the intensity is power per unit area perpendicular to the propagation direction:

$$I=\frac{P}{A}.$$

Its SI unit is

$$\mathrm{W/m^2}.$$

Geometrical spreading

If a point-like source radiates uniformly in three dimensions, the same power crosses spheres of increasing area

$$A=4\pi r^2.$$

The intensity therefore falls approximately as

$$I\propto\frac1{r^2}$$

when absorption can be neglected.

Absorption

Real media can convert part of the wave energy into internal energy. The wave amplitude and intensity then decrease with propagation distance in addition to any geometrical spreading.

Intensity describes energy flow, not the speed at which the wave pattern travels.