Plane wave

Remarks

The planes $x \cdot n = \text{const}$ are the wavefronts. A sinusoidal plane wave is travelling with speed $\omega / \abs{k}$ when $\omega > 0$; its wavelength is $2\pi / \abs{k}$. Complex notation $u = \mathrm{Re}\big(a \, e^{i(k \cdot x - \omega t)}\big)$, with complex amplitude $a$, is standard. Every twice differentiable travelling plane wave $F(x \cdot n - ct)$ solves the wave equation $u_{tt} = c^2 \Delta u$, since both sides equal $c^2 F''$.

Example: $u(x, y, z, t) = \sin(2x - 3t)$ is a sinusoidal plane wave with direction $n = (1, 0, 0)$, wave vector $k = (2, 0, 0)$ and $\omega = 3$; it travels with speed $3/2$.

Non-example: the spherical wave $u(x, t) = \cos(\abs{x} - t) / \abs{x}$ on $x \neq 0$ is not a plane wave. At $t = 0$ its values at $(1, 0, 0)$ and $(1, 1, 0)$ differ, though both points lie on the plane $x \cdot e_1 = 1$. Similar pairs exist for every direction $n$.