= Wavelength
{title2=$\lambda$}
{wiki}
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 $2\pi$:
$$
\lambda=\frac{2\pi}{|\mathbf k|}=\frac{v_p}{f}.
$$
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>, so
$$
f_1=f_2,\qquad \frac{\lambda_2}{\lambda_1}=\frac{v_{p,2}}{v_{p,1}}.
$$
For <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>.<College Physics by Paul Peter Urone and others/27.1 The Wave Aspect of Light: Interference>{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.
Back to article page
