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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.

The wavelength $\lambda$ is the spatial period of a monochromatic wave, measured at a fixed time along the wavevector. For a plane wave with a sinusoid as its profile, it is the distance between consecutive wave crests, or between any consecutive points with the same wave phase modulo $2\pi$:Here $\mathbf{k}$ uses angular wavenumber, $f$ is ordinary frequency, and $v_p$ is phase speed. A distance between arbitrary zero crossings need not be a wavelength: consecutive zeros of a sinusoid are only $\lambda /2$ apart, with opposite slopes. In a standing wave, consecutive nodes of a standing wave are also $\lambda /2$ apart; $\lambda$ refers to the constituent travelling waves, even though the nodes of a standing wave do not travel.

Across a stationary interface between stationary linear media, the incident, reflected and transmitted monochromatic waves have the same frequency. The boundary conditions must hold at every time: terms oscillating with different angular frequencies cannot cancel throughout the entire time record. Each medium's dispersion relation then determines its own angular wavenumber and wavelength, soFor electromagnetic radiation in a transparent medium with refractive index $n$, $v_p=c_0/n$ and $\lambda ={\lambda }_0/n$, where $c_0$ is the vacuum speed of light. A synthetic example with ${\lambda }_0=600\mathrm{nm}$ and $n=1.5$ gives $\lambda =400\mathrm{nm}$ at the same frequency. This conservation argument assumes time-independent linear material response and a stationary interface in the chosen frame. A moving interface can produce a Doppler effect; time-dependent or nonlinear response can change or generate frequencies.^[ref]

A broad frequency spectrum generally supplies many component wavelengths, rather than one wavelength for an entire pulse. Even two tones have both their individual wavelengths and a different overall repetition distance: the $2$- and $8$-cycle-per-metre sum in spatial sampling of a wave repeats after $0.50\mathrm{m}$, but contains the finer $0.125\mathrm{m}$ structure. A narrow wave packet can have an approximate carrier wavelength; the wave envelope propagates approximately at group velocity, determined by the slope of the dispersion relation. Substituting that group velocity into $\lambda =v_p/f$ generally gives the wrong carrier wavelength. Where the medium varies slowly, local wavelength gives a more useful description.