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Choose $k,m>0$, $K=(k^2+m^2{)}^{1/2}$ and the positive angular frequency $\omega =Nk/K$. Then the phase velocity is $Nk(k,m)/K^3$ and the group velocity is $N(m^2,-km)/K^3$. Their dot product vanishes and their sum is horizontal with magnitude $N/K$. Draw the group velocity from the tip of the phase velocity: the two form the perpendicular sides of a right triangle. The horizontal hypotenuse is their sum, not the horizontal phase velocity $\omega /k$.

A free top and rigid base impose transverse wavenumbers $(n+1/2)\pi /h$ on guided SH modes between rigid and free planes. Their phase velocity and group velocity are $c={\beta }^{\prime }/\sqrt{1-{\omega }_n^2/{\omega }^2}$ and $U={\beta }^{\prime }\sqrt{1-{\omega }_n^2/{\omega }^2}$ for $\omega >{\omega }_n$. A real Love wave replaces the rigid base by an evanescent elastic load; the approximation becomes reliable sufficiently above the corresponding strong-dispersion transition, not at the actual mode cutoff.

The Yanai wave is the $n=0$ branch of the equatorial shallow-water dispersion relation with positive frequency:Its group velocity is $d\omega /dk=c{\omega }^2/({\omega }^2+\beta c)>0$. At low frequency it has westward phase velocity but transports energy eastward; at high frequency its phase and group velocities approach $c$.

For a resting midlatitude layer, the slow Rossby wave equation is $({\nabla }_h^2-a^2){\eta }_t=-\beta {\eta }_x$, where $a=f_0/\sqrt{gH}$. With the plane wave convention $e^{i(kx+ly-\omega t)}$,The zonal phase velocity is westward for $\beta >0$, but the zonal group velocity changes sign at $k^2=l^2+a^2$.

For a semi-infinite stratified fluid with thermal wind $U=\Lambda z$ above a gently sloping boundary $z=\alpha y$, a zero-interior-potential vorticity perturbation decays as $e^{-\mu z}$, where $\mu =N|k|/|f_0|$. The boundary buoyancy perturbation obeys a single oscillatory equation with phase velocity $c=(\Lambda -\alpha N^2/f_0)/\mu$. The topographic and thermal-wind contributions cancel when the boundary follows a background isopycnal.

Slowness is the reciprocal of phase velocity, measured in time per distance. A plane wave with wavevector $\mathbf{k}$ and angular frequency $\omega$ has vector slowness $\mathbf{p}=\mathbf{k}/\omega$. Its direction is the wavefront normal, which need not be the group velocity direction in an anisotropic medium.

The Taylor–Goldstein equation governs linear two-dimensional disturbances of an inviscid, vertically sheared, stably stratified parallel flow. For horizontal phase velocity $c$, base velocity $U(z)$, buoyancy frequency $N(z)$, and horizontal wavenumber $k$,

For the three-layer stratified piecewise-linear shear flow, set $\alpha =kh/2$. The dispersion relation is quadratic in squared nondimensional phase velocity. Its constant term is $[2\alpha -(1+J)(1+e^{-2\alpha })][2\alpha -(1+J)(1-e^{-2\alpha })]/(4{\alpha }^2)$. Throughout the displayed band this is negative, so one squared speed is negative and gives an exponentially growing normal mode.

A monochromatic wave in vacuum has frequency $\nu$ and wavelength $\lambda$ related by $\nu \lambda =c$, where $c$ is the speed of light. During one oscillation period $T=1/\nu$, a constant-wave phase surface travels one wavelength, so $\lambda =cT$. Equivalently, the plane electromagnetic wave relation $\omega =ck$ with angular frequency $\omega =2\pi \nu$ and wavenumber $k=2\pi /\lambda$ gives the same result.

As a calculated example, a vacuum radio wave with $\nu =300\mathrm{MHz}=3.00\times 10^8\mathrm{Hz}$ has $\lambda =0.999308\mathrm{m}$ using $c=299792458m/s$. Doubling frequency halves wavelength, independently of wave amplitude and wave intensity. The frequency spectrum of a pulse contains multiple frequencies; the relation applies to each vacuum component, rather than assigning one exact wavelength to the whole pulse. In a homogeneous medium replace $c$ by the appropriate phase velocity; the vacuum equation does not assert that all media have the same wavelength at a given frequency.

A wave crest is a local maximum of the wave's displacement. In a monochromatic travelling wave it lies at a fixed phase and propagates at the phase velocity.

Wave dispersion means that phase velocity depends on wavenumber or, equivalently, wavelength. Wave packets generally travel with group velocity, which need not equal phase velocity. The linear rotating shallow-water dispersion relation ${\omega }^2=f^2+c^2{k}^2$ has faster long-wave phase motion but slower long-wave packet motion when $f\ne 0$.

A wavefront connects points with the same wave phase at one specified time. It is a surface, rather than the trajectory of a particular piece of light. For a scalar monochromatic wave written as $u(\mathbf{r},t)=A(\mathbf{r})\cos \Phi (\mathbf{r},t)$, a wavefront is a connected level set of the phase function $\Phi$; different surfaces can share the same wave phase modulo $2\pi$. A plane wave has planar wavefronts. An outward wave from an ideal point source in a uniform medium has spheres as wavefronts, while an optical lens can turn nearly planar wavefronts into converging ones. The wavefront describes wave phase, not wave amplitude: drawing it does not say how bright the beam is. ^[ref]

The relation to light rays becomes precise in a stationary medium with isotropy, negligible loss and a real positive scalar refractive index $n(\mathbf{r},\omega )$. In the geometrical optics limit, spatial changes in the refractive index and wave amplitude are slow on the local wavelength scale. For a single smooth propagation branch, writeHere $\omega >0$ is the angular frequency, $c$ is the vacuum speed of light, $S$ has units of length, and the last relation is the optical eikonal equation. The local wavevector is $\mathbf{k}=\nabla \Phi =k_0\nabla S$. Its direction is normal to a smooth wavefront, since a gradient is normal to its level sets. A forward light ray has unit tangent vector $d\mathbf{r}/ds=\nabla S/n$, so it follows this normal under the stated assumptions. Along that light ray, $dS/ds=\nabla S\cdot (d\mathbf{r}/ds)=n$: the change in $S$ is the optical path length. This is the short geometric argument linking wavefronts, light rays and optical path length. ^[ref]

The diagram shows incident and transmitted branches separately, with the reflected branch omitted. Its stipulated refractive indices are $n_1=1$ and $n_2=3/2$, and its incident angle is $30^{\circ }$ to the interface normal. At a stationary interface, the frequency and tangential wavevector agree on the two sides: Snell's law gives $\sin {\theta }_2=1/3$, hence ${\theta }_2\approx 19.4712^{\circ }$. The wavefronts turn while their normal spacing decreases from ${\lambda }_0$ to $2{\lambda }_0/3$, where ${\lambda }_0$ is the vacuum wavelength. More explicitly, with $x$ normal to the interface and $y$ tangential, take $S_1=\sqrt{3}x/2+y/2$ for $x<0$ and $S_2=\sqrt{2}x+y/2$ for $x>0$. These functions agree at $x=0$; the drawn wavefronts are $S_j=m{\lambda }_0$ at $t=0$ with ${\varphi }_0=0$. This is an exact piecewise plane wave phase construction for the stipulated transparent media, not measured material data. ^[ref]

The normal-to-wavefront rule requires its assumptions. In a medium with anisotropy, the wavevector remains normal to a smooth wavefront, but the group velocity and energy flux need not follow it. ^[ref] Near a wave caustic, where multiple light rays meet, one smooth phase function may no longer describe the field; diffraction and the full wave amplitude matter. A wavefront is also not generally a surface of equal pulse arrival time. In a medium with optical dispersion, a monochromatic wave phase moves with the phase velocity, while a sufficiently narrow wave packet follows the group velocity. Even identical optical path lengths at one frequency can give different group delays. The usual equal-travel-time picture needs a common launch reference and a nondispersive ray model; an arbitrary broadband disturbance has no single monochromatic wavefront.^[ref]

Above a mode's waveguide cutoff frequency ${\omega }_c$, its phase velocity is $c/[1-({\omega }_c/\omega {)}^2{]}^{1/2}$ and its group velocity is $c[1-({\omega }_c/\omega {)}^2{]}^{1/2}$. Differentiate ${\omega }^2=c^2{k}^2+{\omega }_c^2$ to obtain the latter. Phase travels faster than $c$ while the wave packet travels slower, so the phase speed does not imply superluminal information transport.

A wavenumber measures how rapidly a wave phase changes with distance. Two conventions coexist: angular wavenumber counts radians per unit length, while ordinary spatial frequency counts cycles per unit length. For a monochromatic wave with wavelength $\lambda >0$,Here $K\ge 0$ is the magnitude of the angular wavenumber. The symbol $k$ can instead denote a signed one-dimensional angular wavenumber: a factor $e^{ikx}$ has wavelength $2\pi /|k|$ for real $k\ne 0$. Both conventions have dimensions of inverse length; writing $\mathrm{rad}{\mathrm{m}}^{-1}$ or “cycles per metre” exposes the convention. A quoted value in ${\mathrm{m}}^{-1}$ alone does not resolve the factor of $2\pi$. Spectroscopic wavenumber usually uses the cycles-per-length convention.^[ref]

For a real monochromatic plane wave, fix the conventionThe wavevector $\mathbf{k}$ supplies both the spatial wave phase slope and the normal to the wavefronts; its dot product with position makes the wave phase constant on parallel planes. The angular frequency is $\omega =2\pi f$, with $f>0$ the ordinary frequency. Moving a distance $\lambda$ along $\mathbf{k}$ changes the wave phase by $2\pi$. Holding a wave crest's wave phase fixed gives its phase velocity $\omega \mathbf{k}/K^2$ and phase speedIn one dimension the signed phase velocity is $\omega /k$. This identifies a moving pattern; it does not identify the motion of the medium's particles or a wave packet's group velocity.

For the synthetic plane wave below, choose $\lambda =0.40\mathrm{m}$ and $f=5\mathrm{Hz}$. Then $K=5\pi \mathrm{rad}{\mathrm{m}}^{-1}$, ${\nu }_x=2.5$ cycles per metre, $\omega =10\pi \mathrm{rad}{\mathrm{s}}^{-1}$, and $v_p=2\mathrm{m}{\mathrm{s}}^{-1}$. The wave crests are $0.40\mathrm{m}$ apart at one instant; after a quarter oscillation period, each has advanced $0.10\mathrm{m}$. The lower panel retains whole turns of wave phase, making its slope visible instead of reducing it modulo $2\pi$.

A wave packet is a localized superposition of Fourier modes. In a narrow packet, nearby wavenumbers interfere to produce a slowly varying wave envelope around a rapid carrier wave. Its leading translation is governed by group velocity; the carrier's wave phase follows phase velocity. Localized does not necessarily mean compact support: a Gaussian function decays rapidly but extends over the whole line. The Gaussian envelope in a Klein-Gordon medium gives an original example with both translation and spreading.

A waveplate uses birefringence to change polarization of an electromagnetic wave through phase retardation. Its fast axis has the smaller refractive index, hence the larger phase velocity; its slow axis has the larger index. Consider normal incidence with the optic axis in the plate face, so both components follow the same straight path. A linear polarization at $45^{\circ }$ to these axes excites equal wave amplitudes.

A complementary experiment uses two sheet polarizers with a cellophane sheet between them. First cross the polarizers and remove the sheet: the field is dark. Insert and rotate the sheet about the beam: light returns at intermediate orientations, but extinction recurs at two perpendicular orientations. The sheet has changed the polarization of an electromagnetic wave, without requiring a resolved double image. With white light, different wavelengths can produce different transmitted colors.^[ref]. Brightness alone does not establish circular polarization: an ideal rotating linear polarizer also gives orientation-independent light intensity for unpolarized light.

For the homogeneous one-dimensional wave equation $u_{tt}=c^2{u}_{xx}$ with constant $c>0$, wave speed means the speed at which a travelling wave profile translates. In $F(x-ct)$, a feature initially at $x=s$ follows $x=s+ct$, giving signed velocity $+c$; in $G(x+ct)$ it follows $x=s-ct$, giving signed velocity $-c$. The characteristic lines for the one-dimensional wave equation therefore have slopes $dx/dt=\pm c$. The speed of either profile is $c$, even though the displacement at a fixed point can change sign or pass through zero.^[ref]

The same $c$ bounds the causal cone for the constant-speed wave equation. For a wave equation on a string, $c=\sqrt{T/\mu }$ depends on tension and linear mass density; for small acoustic waves in stationary uniform air it is the speed of sound. This interpretation assumes the stated ideal wave equation. In media with wave dispersion, phase velocity, group velocity and the earliest signal arrival have different meanings: a measured wave crest speed need not be the speed of a wave envelope, and neither alone establishes a causal cone for the constant-speed wave equation.