A wave is a time-dependent function that describes an oscillation or a spatial profile propagating through a domain.
In one dimension, right- and left-traveling profiles with constant speed 
have the forms 
and 
,
respectively. Their superposition is d'Alembert's
solution,
 |
and every sufficiently differentiable function of this form satisfies the one-dimensional wave equation 
.
Conversely, every sufficiently differentiable solution on the real line has this
form (Whitham 1974).
A sinusoidal traveling wave can be written
 |
where 
is the amplitude,
is the phase,
is the spatial period,
and 
is the temporal period.
A relation between 
and 
is called a dispersion
relation. For a linear wave equation, waves can be combined using the superposition
principle, and periodic waves can be decomposed into Fourier
series.
