= Local wavelength
{title2=$\lambda_{\mathrm{loc}}=2\pi/|\nabla\phi|$}
For a locally oscillatory <wave> written $u(\mathbf x,t)=A(\mathbf x,t)\cos\phi(\mathbf x,t)$, the local <wavevector> is $\mathbf k_{\mathrm{loc}}=\nabla\phi$, 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 $\phi$: 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 $\nabla\phi=0$, the local reciprocal formula gives no finite <wavelength>; it does not establish an exact global spatial <period>.
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