The waves that are most familiar in daily life are sound waves, light waves, electrical waves, water waves, and mechanical waves (say a standing wave on a piano string). These familiar waves have the property that their magnitude decreases as they propagate away from their source or, for standing waves, once their source is turned off. The decrease in magnitude is a result of dissipative effects in the medium such as fluid viscosity, electrical resistance, or friction that drain energy from the wave and that restore the medium to thermal equilibrium.

These familiar waves have the additional property of often being accurately described by a linear evolution equation such as the wave equation. Because the evolution equation is linear, one can superimpose sinusoidal waves to get localized pulses of arbitrary shape, and these pulses can also propagate. (For example, clapping your hands once loudly creates a localized sound pulse that propagates away.) Because each Fourier component in the superposition is itself damped in typical media, the propagating pulses also damp out and disappear over time. Even in the absence of damping, dispersive effects can cause the different Fourier components to travel at different speeds so, again, waves and pulses change their shape and decrease in magnitude during propagation.

We have seen in earlier chapters that sustained nonequilibrium systems allow many dynamical states that can propagate or exist in a local spatial region but are such that these states do not damp out over time or they preserve their shape and speed as they propagate away from a source.