 Local wavelength

For a locally oscillatory wave written $u(\mathbf{x},t)=A(\mathbf{x},t)\cos \varphi (\mathbf{x},t)$, the local wavevector is ${\mathbf{k}}_{\mathrm{loc}}=\nabla \varphi$, where $\nabla$ is the spatial gradient. The local wavelength $2\pi /|{\mathbf{k}}_{\mathrm{loc}}|$ measures local wave phase spacing normal to the wavefront. Interpreting it as a crest separation requires the amplitude and local wavevector to vary little over one wavelength, as in the WKB approximation. Along a unit direction $\mathbf{e}$ the corresponding local repeat distance is $2\pi /|{\mathbf{k}}_{\mathrm{loc}}\cdot \mathbf{e}|$, if that dot product is nonzero; oblique sampling measures this projected spacing.

A real snapshot alone does not uniquely specify $A$ and $\varphi$: their meaning requires a chosen oscillatory component or physical approximation. Strong interference, zeros of amplitude, and rapidly changing media can defeat a single-carrier description. If $\nabla \varphi =0$, the local reciprocal formula gives no finite wavelength; it does not establish an exact global spatial period.