Dispersion relation

Remarks

The equivalence holds because $\partial_t$ acts on the plane wave as multiplication by $-i\omega$ and $\nabla$ as multiplication by $i k$. For systems such as Maxwell's equations one takes a vector amplitude, and the condition becomes that a matrix built from $\omega$ and $k$ is singular. The equation is called non-dispersive when every branch is linear in $\abs{k}$, and dispersive otherwise. In a linear medium with refractive index $n(\omega)$, the dispersion relation of light is $\abs{k} = n(\omega) \, \omega / c$.

Example: for the wave equation $u_{tt} - c^2 \Delta u = 0$, $P(\tau, \xi) = \tau^2 - c^2 \xi \cdot \xi$, and $P(-i\omega, ik) = -\omega^2 + c^2 \abs{k}^2$, so the relation is $\omega = \pm c \abs{k}$. For the Klein–Gordon equation $u_{tt} - c^2 \Delta u + m^2 u = 0$, which also models light in a cold plasma, the relation is $\omega^2 = c^2 \abs{k}^2 + m^2$, which is dispersive.

Non-example: the pair $\omega = 2c$, $k = (1, 0, 0)$ does not satisfy the dispersion relation of the wave equation, since $-4c^2 + c^2 \neq 0$; so $e^{i(x - 2ct)}$ is not a solution.