# Wave vector

A **wave vector** is a vector that describes a traveling wave by its direction of propagation and its spatial frequency: its magnitude is the wavenumber \( k = 2\pi/\lambda \), where \( \lambda \) is the wavelength, and it points perpendicular to the wavefronts, in the direction in which surfaces of constant phase advance.<sup>[1](https://www.rp-photonics.com/wave_vector.html)</sup><sup> • </sup><sup>[2](https://www.physics.utoronto.ca/~phy485/ModOpt/dloads/DispPrimer.pdf)</sup> It is the central quantity connecting a wave's geometry, its speed through the dispersion relation, and, in quantum mechanics, its particle momentum.

| Key fact | Detail |
|---|---|
| Definition | Vector perpendicular to the wavefronts, pointing along propagation, with magnitude \( k = 2\pi/\lambda \) for the wavelength in the medium<sup>[1](https://www.rp-photonics.com/wave_vector.html)</sup> |
| Phase meaning | A point of constant phase (\( \mathbf{k} \cdot \mathbf{x} - \omega t = \text{const} \)) moves along \( \mathbf{k} \) at speed \( v = \omega/k \)<sup>[3](https://www.phys.lsu.edu/~jarrell/COURSES/ELECTRODYNAMICS/Chap7/chap7.pdf)</sup> |
| Dispersion | \( \omega = v k \) for sound and light in non-dispersive media; phase velocity \( \omega/k \), group velocity \( \nabla_{\mathbf{k}} \omega \)<sup>[4](https://uclnatsci.github.io/2021/Electromagnetism-Fluids-and-Waves/waves/dispersionrelations.html)</sup> |
| Conventions | Physics uses \( 2\pi/\lambda \); spectroscopy and some crystallography use \( 1/\lambda \), differing by the factor \( 2\pi \)<sup>[5](https://goldbook.iupac.org/terms/view/09441)</sup><sup> • </sup><sup>[6](https://goldbook.iupac.org/terms/view/W06664)</sup> |
| Quantum link | \( \mathbf{k} = \mathbf{p}/\hbar \), the quotient of momentum and the reduced Planck constant (ISO 80000-12)<sup>[7](https://cdn.standards.iteh.ai/samples/102847/4936259ea5e3457788627e7389c18634/ISO-80000-12-2019.pdf)</sup> |
| Typical magnitudes | About 18.3 rad/m for 1 kHz sound in air; about \( 9 \times 10^{6} \) to \( 1.6 \times 10^{7} \) rad/m for visible light<sup>[8](https://roboculator.com/calculator/wave-number-calculator)</sup> |
| Anisotropy | In non-isotropic media the Poynting vector and group velocity can deviate from \( \mathbf{k} \) (spatial walk-off)<sup>[1](https://www.rp-photonics.com/wave_vector.html)</sup> |

## Definition and mathematical form

For a plane wave in an isotropic medium, the wave vector \( \mathbf{k} \) points in the direction of propagation and is always perpendicular to the wavefronts.<sup>[1](https://www.rp-photonics.com/wave_vector.html)</sup> Its magnitude is the wavenumber,

\[ k = \frac{2\pi}{\lambda}, \]

where \( \lambda \) is the wavelength in the medium, not the vacuum wavelength; the unit is m⁻¹.<sup>[1](https://www.rp-photonics.com/wave_vector.html)</sup> Equivalently, \( \mathbf{k} \) is the gradient of the spatial phase \( \theta(\mathbf{x}) = \mathbf{k} \cdot \mathbf{x} \), so it points in the direction of maximum spatial-phase advance, and it decomposes as \( \mathbf{k} = k(\cos\alpha, \cos\beta, \cos\gamma) \) along its direction unit vector.<sup>[9](https://ccrma.stanford.edu/~jos/waveguide/Vector_Wavenumber.html)</sup> The wavenumber \( k \) is the spatial frequency in radians per meter along the direction of travel.<sup>[9](https://ccrma.stanford.edu/~jos/waveguide/Vector_Wavenumber.html)</sup>

**Two conventions.** The factor \( 2\pi \) is a recurring source of confusion. The angular wavenumber \( k = 2\pi/\lambda \) measures phase in radians per meter, while the ordinary wavenumber \( 1/\lambda \) counts full cycles per unit length; wavelength measures crest-to-crest distance, so the \( 2\pi \) factor converts between the two descriptions.<sup>[6](https://goldbook.iupac.org/terms/view/W06664)</sup><sup> • </sup><sup>[10](https://web.ics.purdue.edu/~dchavas/download/teaching/EAPS53400_S24/Chavas_EAPS53400_M05.2cPRIMER_WavesBasics_PUBLIC.pdf)</sup> IUPAC notes that different fields take the wave vector as \( 2\pi/\lambda \) or as \( 1/\lambda \); in [X-ray reflectometry](https://www.edgepedia.org/x-ray-reflectometry) the former is usually used.<sup>[5](https://goldbook.iupac.org/terms/view/09441)</sup> The spectroscopic wavenumber \( \tilde{\nu} = 1/\lambda \) is quoted in m⁻¹ or, traditionally, cm⁻¹, and the two are related by \( k = 2\pi\tilde{\nu} \).<sup>[6](https://goldbook.iupac.org/terms/view/W06664)</sup><sup> • </sup><sup>[8](https://roboculator.com/calculator/wave-number-calculator)</sup> Early spectroscopy workers used the reciprocal of the wavelength in air; [Heinrich Kayser](https://www.edgepedia.org/heinrich-kayser) recognized that the wavelength in vacuum is essential for complex spectra, and the name Kayser was advocated for the cm⁻¹ unit.<sup>[11](https://preview-www.nature.com/articles/170043b0)</sup>

## Role in the plane-wave solution

The plane wave is the basic traveling wave in two and three dimensions, and the wave vector enters through the phase of its complex representation,

\[ \psi(\mathbf{r}, t) = A\, e^{i(\mathbf{k} \cdot \mathbf{r} - \omega t)}. \]

A point of constant phase satisfies \( \mathbf{k} \cdot \mathbf{x} - \omega t = \text{constant} \) and moves along the direction of \( \mathbf{k} \) with speed \( v = \omega/k \).<sup>[3](https://www.phys.lsu.edu/~jarrell/COURSES/ELECTRODYNAMICS/Chap7/chap7.pdf)</sup> The surfaces of constant phase are planes perpendicular to \( \mathbf{k} \), which is what makes the wave "plane"; successive wavefronts are separated by one wavelength.<sup>[3](https://www.phys.lsu.edu/~jarrell/COURSES/ELECTRODYNAMICS/Chap7/chap7.pdf)</sup><sup> • </sup><sup>[12](https://phys.libretexts.org/Bookshelves/University_Physics/Radically_Modern_Introductory_Physics_Text_I_(Raymond)/02%3A_Waves_in_Two_and_Three_Dimensions/2.02%3A_Plane_Waves)</sup> The vector thus does double duty: its orientation fixes the propagation direction, and its length fixes the wavenumber.<sup>[12](https://phys.libretexts.org/Bookshelves/University_Physics/Radically_Modern_Introductory_Physics_Text_I_(Raymond)/02%3A_Waves_in_Two_and_Three_Dimensions/2.02%3A_Plane_Waves)</sup>

## Dispersion relation and wave speed

Substituting the plane-wave form into the wave equation yields the **dispersion relation** \( \omega = \omega(k) \), which connects angular frequency to wave vector.<sup>[13](https://physics.uwo.ca/~mhoude2/courses/PDF%20files/astro9620/Ch5-Wave_propagation.pdf)</sup> For sound in air (about 340 m/s at normal temperatures) and light in vacuum (\( 3 \times 10^{8} \) m/s) the relation is linear, \( \omega = ck \); in a transparent medium of refractive index \( n \), \( \omega = k\, c_{\text{vac}}/n \).<sup>[14](https://phys.libretexts.org/Bookshelves/University_Physics/Radically_Modern_Introductory_Physics_Text_I_(Raymond)/01%3A_Waves_in_One_Dimension/1.03%3A_Types_of_Waves)</sup> A linear relation between \( \omega \) and \( k \) is characteristic of non-dispersive media, where the phase velocity is independent of frequency.<sup>[13](https://physics.uwo.ca/~mhoude2/courses/PDF%20files/astro9620/Ch5-Wave_propagation.pdf)</sup><sup> • </sup><sup>[15](https://scholar.harvard.edu/files/david-morin/files/waves_dispersion.pdf)</sup>

The relation need not be linear. Light in vacuum obeys \( \omega = c(k_x^2 + k_y^2)^{1/2} \), so the same symbol \( \mathbf{k} \) governs qualitatively different wave systems.<sup>[12](https://phys.libretexts.org/Bookshelves/University_Physics/Radically_Modern_Introductory_Physics_Text_I_(Raymond)/02%3A_Waves_in_Two_and_Three_Dimensions/2.02%3A_Plane_Waves)</sup> When the refractive index depends on frequency, \( n(\omega) \), the wave velocity depends on frequency and one must distinguish the phase velocity \( v_p = \omega/k \) from the group velocity \( \mathbf{v}_g = \nabla_{\mathbf{k}} \omega \).<sup>[16](https://web2.ph.utexas.edu/~vadim/Classes/2024f-emt/dispersion.pdf)</sup><sup> • </sup><sup>[4](https://uclnatsci.github.io/2021/Electromagnetism-Fluids-and-Waves/waves/dispersionrelations.html)</sup>

## By the numbers

Two anchors give the scale of wave vectors in everyday physics. In air, a sound wave of frequency 1 kHz measures 34.3 cm from crest to crest, giving a wave number near 18.3 rad/m—equivalent to approximately three complete cycles within each meter.<sup>[8](https://roboculator.com/calculator/wave-number-calculator)</sup> Visible light spans \( k \approx 9 \times 10^{6} \) rad/m for red light near 700 nm up to \( 1.6 \times 10^{7} \) rad/m for violet near 400 nm, equivalent to about 14,000 to 25,000 cm⁻¹ spectroscopically.<sup>[8](https://roboculator.com/calculator/wave-number-calculator)</sup>

## Wave vectors at boundaries and in media

**Snell's law from \( \mathbf{k} \) conservation.** When a plane wave strikes a boundary, the tangential component of the wave vector must be continuous. For the transmitted wave this conservation yields [Snell's law](https://www.edgepedia.org/snells-law), \( n \sin i = n' \sin r \); the reflected wave is a solution of the same wave equation as the incident wave and therefore has the same wave number, which gives equality of the angles of incidence and reflection.<sup>[3](https://www.phys.lsu.edu/~jarrell/COURSES/ELECTRODYNAMICS/Chap7/chap7.pdf)</sup>

Inside a medium, the wave vector's magnitude is constrained to be proportional to the refractive index \( n \).<sup>[17](https://ocw.mit.edu/courses/2-71-optics-spring-2009/339199275a4ea3997df533690f65270b_MIT2_71S09_lec13.pdf)</sup> In media with absorption or gain the wave vector can have complex components, and for an evanescent wave it can have a purely imaginary component, describing decay rather than propagation.<sup>[1](https://www.rp-photonics.com/wave_vector.html)</sup>

**Metasurfaces extend the rule.** A metasurface with a designed phase profile adds its own transverse momentum, generalizing Snell's law to a wave-vector balance of the form \( \Delta\mathbf{k}_{//}^{\text{total}} + \mathbf{k}_i + \boldsymbol{\mathscr{K}}(\mathbf{k}_i) = \mathbf{k}_r \); the periodic metasurface architecture supplies transverse momentum compensation while conserving the total wave-vector magnitude \( |\mathbf{k}| = k_0 \) in free space.<sup>[18](https://www.nature.com/articles/s41377-025-02009-3)</sup>

## Anisotropic media: wave vector, Poynting vector, and group velocity

In anisotropic crystals the wave vector loses its monopoly on describing where the wave goes. The direction of energy flow, given by the [Poynting vector](https://www.edgepedia.org/poynting-vector), can deviate from the wave vector; this phenomenon is called *spatial walk-off*.<sup>[1](https://www.rp-photonics.com/wave_vector.html)</sup> Equivalently, the energy moves at the group velocity \( \mathbf{v}_g = \partial\omega/\partial\mathbf{k} \), which in an anisotropic medium is generally not parallel to \( \mathbf{k} \).<sup>[4](https://uclnatsci.github.io/2021/Electromagnetism-Fluids-and-Waves/waves/dispersionrelations.html)</sup>

The geometry is captured by Fresnel's equation. In the space spanned by \( n_x, n_y, n_z \) it describes a fourth-order surface, the wave-vector surface, giving the allowed wave-normal vectors; the ray direction \( \mathbf{v}_g \) is the normal to this surface at the given \( \mathbf{k} = (\omega/c)\,\mathbf{n} \).<sup>[19](https://galileoandeinstein.phys.virginia.edu/Elec_Mag/2022_Lectures/EM_52_Anisotropic_Medium.html)</sup>

## Reciprocal space, crystals, and quantum mechanics

In condensed matter physics the wave vector lives in **reciprocal space**, the Fourier-dual of the crystal lattice. For phonons, the magnitude is \( k = 2\pi/\lambda \) for the wavelength of the lattice vibrations, and the angular wavenumber is often simply called the wavenumber in this field.<sup>[7](https://cdn.standards.iteh.ai/samples/102847/4936259ea5e3457788627e7389c18634/ISO-80000-12-2019.pdf)</sup> The reciprocal lattice concept traces to the "polar lattice" of Gibbs and Ewald, described as the great ancestor of the reciprocal lattice.<sup>[20](https://www.cambridge.org/core/journals/powder-diffraction/article/abs/history-of-the-reciprocal-lattice/005E3423D62834CF941C070A3EABFC36)</sup>

The theory of the **Brillouin zone** was formalized in a 1936 [Physical Review](https://www.edgepedia.org/physical-review) paper that treated the zones from the point of view of group theory, associating a representation of the space group with every energy value.<sup>[21](https://journals.aps.org/pr/abstract/10.1103/PhysRev.50.58)</sup> For a lattice of spacing \( a \), the first zone boundaries occur at \( k = \pm\pi/a \).<sup>[8](https://roboculator.com/calculator/wave-number-calculator)</sup>

**Matter waves.** ISO 80000-12 defines the angular wavenumber as the quotient of momentum and the reduced [Planck constant](https://www.edgepedia.org/planck-constant), with the corresponding vector quantity called the wave vector: \( \mathbf{k} = \mathbf{p}/\hbar \) for quasi-free electrons.<sup>[7](https://cdn.standards.iteh.ai/samples/102847/4936259ea5e3457788627e7389c18634/ISO-80000-12-2019.pdf)</sup>

## Scattering, diffraction, and modern wave-vector engineering

In wave optics the Descartes sphere, also known as the Ewald sphere or simply the k-sphere, is the surface on which the wave-vector magnitude is constrained to be proportional to the refractive index \( n \).<sup>[17](https://ocw.mit.edu/courses/2-71-optics-spring-2009/339199275a4ea3997df533690f65270b_MIT2_71S09_lec13.pdf)</sup>

Wave vectors are also measured directly. Fourier imaging resolves the wave-vector \( \mathbf{k} \) dependence of optical reflection: unlike standard angle-resolved reflectivity, many wave vectors from a high-numerical-aperture objective are examined simultaneously, yielding a three-dimensional dataset of intensity as a function of wavenumber \( \tilde{\nu} \), and wave-vector components \( k_y \) and \( k_z \).<sup>[22](https://nano-cops.com/app/uploads/2025/10/optex34vreman2026momentum-resolved-2D-reflectivity_2D-phot-xtals_near-IR.pdf)</sup>

**Engineering \( \mathbf{k} \) itself.** Recent photonic structures treat the wave vector as a design variable. A metasurface of elliptical resonators with near-field coupling exhibits wave-vector-dependent eigenstates, so that tuning the incident \( \mathbf{k} \)-vector strongly reflects certain polarization-energy eigenstates.<sup>[23](https://pubs.acs.org/doi/pdf/10.1021/acsphotonics.5c02865)</sup> A misaligned bilayer metagrating achieves spatio-spectral selectivity, singling out one photonic mode with a specific wavevector (angle) and wavelength, with Fano-like reflection in angular (±5°) and wavelength (14 nm) bandwidths; conventional periodic structures instead lock wavevectors and wavelengths together through their continuous band dispersion.<sup>[24](https://link.springer.com/article/10.1186/s43593-025-00092-y)</sup> Spatiotemporally modulated metasurfaces go further, dynamically transforming both the momentum (wave vector) and the frequency of light, demonstrated as momentum conversion followed by frequency conversion along the dispersion curve.<sup>[25](https://pubs.acs.org/doi/abs/10.1021/acs.nanolett.5c06362)</sup>

## References

1. [Wave Vector, RP Photonics Encyclopedia](https://www.rp-photonics.com/wave_vector.html)
2. [A Primer on Dispersion in Waveguides, University of Toronto](https://www.physics.utoronto.ca/~phy485/ModOpt/dloads/DispPrimer.pdf)
3. [Plane Waves and Wave Propagation, electrodynamics lecture notes, LSU](https://www.phys.lsu.edu/~jarrell/COURSES/ELECTRODYNAMICS/Chap7/chap7.pdf)
4. [Dispersion Relations and Wave Energy, UCL](https://uclnatsci.github.io/2021/Electromagnetism-Fluids-and-Waves/waves/dispersionrelations.html)
5. [Wave vector, IUPAC Gold Book](https://goldbook.iupac.org/terms/view/09441)
6. [Wavenumber, IUPAC Gold Book](https://goldbook.iupac.org/terms/view/W06664)
7. [ISO 80000-12:2019, Quantities and units, condensed matter physics (sample)](https://cdn.standards.iteh.ai/samples/102847/4936259ea5e3457788627e7389c18634/ISO-80000-12-2019.pdf)
8. [Wave Number Calculator](https://roboculator.com/calculator/wave-number-calculator)
9. [Vector Wavenumber, Julius O. Smith III, Stanford CCRMA](https://ccrma.stanford.edu/~jos/waveguide/Vector_Wavenumber.html)
10. [Wave basics, EAPS 53400, Purdue University](https://web.ics.purdue.edu/~dchavas/download/teaching/EAPS53400_S24/Chavas_EAPS53400_M05.2cPRIMER_WavesBasics_PUBLIC.pdf)
11. [A Unit of Wave-number, Nature (1952)](https://preview-www.nature.com/articles/170043b0)
12. [Plane Waves, Physics LibreTexts](https://phys.libretexts.org/Bookshelves/University_Physics/Radically_Modern_Introductory_Physics_Text_I_(Raymond)/02%3A_Waves_in_Two_and_Three_Dimensions/2.02%3A_Plane_Waves)
13. [Chapter 5. Wave Propagation, Western University](https://physics.uwo.ca/~mhoude2/courses/PDF%20files/astro9620/Ch5-Wave_propagation.pdf)
14. [Types of Waves, Physics LibreTexts](https://phys.libretexts.org/Bookshelves/University_Physics/Radically_Modern_Introductory_Physics_Text_I_(Raymond)/01%3A_Waves_in_One_Dimension/1.03%3A_Types_of_Waves)
15. [Waves and Dispersion, David Morin, Harvard](https://scholar.harvard.edu/files/david-morin/files/waves_dispersion.pdf)
16. [Phase and Group Velocities, UT Austin](https://web2.ph.utexas.edu/~vadim/Classes/2024f-emt/dispersion.pdf)
17. [MIT 2.71 Optics, Lecture 13](https://ocw.mit.edu/courses/2-71-optics-spring-2009/339199275a4ea3997df533690f65270b_MIT2_71S09_lec13.pdf)
18. [Missing harmonic dynamics in generalized Snell's law, Light: Science & Applications (2025)](https://www.nature.com/articles/s41377-025-02009-3)
19. [Anisotropic Medium, University of Virginia](https://galileoandeinstein.phys.virginia.edu/Elec_Mag/2022_Lectures/EM_52_Anisotropic_Medium.html)
20. [History of the reciprocal lattice, Powder Diffraction (2019)](https://www.cambridge.org/core/journals/powder-diffraction/article/abs/history-of-the-reciprocal-lattice/005E3423D62834CF941C070A3EABFC36)
21. [Theory of Brillouin Zones and Symmetry Properties of Wave Functions in Crystals, Phys. Rev. 50, 58 (1936)](https://journals.aps.org/pr/abstract/10.1103/PhysRev.50.58)
22. [Momentum-resolved reflectivity of 2D photonic crystals in the near-infrared, Optics Express](https://nano-cops.com/app/uploads/2025/10/optex34vreman2026momentum-resolved-2D-reflectivity_2D-phot-xtals_near-IR.pdf)
23. [Polarization- and Wave-Vector Selective Optical Metasurface with Near-Field Coupling, ACS Photonics (2025)](https://pubs.acs.org/doi/pdf/10.1021/acsphotonics.5c02865)
24. [Overcoming intrinsic dispersion locking with misaligned bilayer metagratings, eLight (2025)](https://link.springer.com/article/10.1186/s43593-025-00092-y)
25. [Spatiotemporally Modulated Nonlocal Metasurfaces: Walking the Dispersion Curve, Nano Letters (2025)](https://pubs.acs.org/doi/abs/10.1021/acs.nanolett.5c06362)

---
*Topic: Encyclopedia › Physical world and mathematics › Physics › Classical physics › Waves and optics › Wave phenomena and acoustics › Wave propagation and interaction with media › Dispersion and wave velocity in media*

*Initially written Sep 17, 2026 · Reviewed: Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*

*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*

License: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license