# Phase (waves)

In wave physics, **phase** is the position of a point in time on a waveform cycle, expressed as an angle: one complete cycle corresponds to 360° or, in radians, 2π<sup>[1](https://www.nist.gov/pml/time-and-frequency-division/popular-links/time-frequency-z/time-and-frequency-z-p)</sup>. For a traveling sinusoidal wave written as \( y(x, t) = A \sin(kx \mp \omega t + \phi) \), the quantity \( kx \mp \omega t + \phi \) is the phase of the wave, where \( \phi \) is the initial phase fixed by the wave's starting conditions<sup>[2](https://phys.libretexts.org/Bookshelves/University_Physics/University_Physics_%28OpenStax%29/Book%3A_University_Physics_I_-_Mechanics_Sound_Oscillations_and_Waves_%28OpenStax%29/16%3A_Waves/16.03%3A_Mathematics_of_Waves)</sup>. Phase matters most when two waves are compared: their phase difference determines whether they reinforce or cancel each other, which is the basis of all interference phenomena<sup>[3](https://lipa.physics.oregonstate.edu/sec_phase_diff.html)</sup>.

| Key fact | Detail |
|---|---|
| Definition | Position on a waveform cycle; one cycle = 360° = 2π rad<sup>[1](https://www.nist.gov/pml/time-and-frequency-division/popular-links/time-frequency-z/time-and-frequency-z-p)</sup> |
| SI unit | The coherent SI unit for the phase angle is the radian (rad); the degree is a historical convention, not an SI unit<sup>[4](https://www.nist.gov/pml/special-publication-330/sp-330-section-5)</sup> |
| Wave-function phase | \( kx \mp \omega t + \phi \), with \( k = 2\pi/\lambda \) and \( \omega = 2\pi/T \)<sup>[2](https://phys.libretexts.org/Bookshelves/University_Physics/University_Physics_%28OpenStax%29/Book%3A_University_Physics_I_-_Mechanics_Sound_Oscillations_and_Waves_%28OpenStax%29/16%3A_Waves/16.03%3A_Mathematics_of_Waves)</sup> |
| Path-to-phase conversion | \( \Delta\phi = 2\pi \cdot \Delta D / \lambda \), where \( \Delta D \) is the path length difference<sup>[3](https://lipa.physics.oregonstate.edu/sec_phase_diff.html)</sup> |
| Interference conditions | With zero initial phase difference: constructive when \( \Delta D = m\lambda \), destructive when \( \Delta D = (m + 1/2)\lambda \), for integer \( m \)<sup>[3](https://lipa.physics.oregonstate.edu/sec_phase_diff.html)</sup> |
| Comparison convention | Phase difference between two waveforms is typically expressed as a number greater than −180° and less than or equal to +180°<sup>[1](https://www.nist.gov/pml/time-and-frequency-division/popular-links/time-frequency-z/time-and-frequency-z-p)</sup> |
| Phase velocity | The carrier of a wave group travels at \( c = \omega/k \); the envelope travels at the group velocity<sup>[5](https://ocw.mit.edu/courses/12-510-introduction-to-seismology-spring-2010/c08aed6151a62870da275f098690bd3d_lec12.pdf)</sup> |

## Phase in the wave function

A sinusoidal wave on a string is modeled by \( y(x, t) = A \sin(kx \mp \omega t + \phi) \). The wave number \( k = 2\pi/\lambda \) has units of inverse meters and measures how rapidly phase increases with distance; the angular frequency \( \omega = 2\pi/T \) measures how rapidly phase decreases with time at a fixed position. The sign of the temporal term selects the direction of travel: the minus sign models a wave moving in the positive x-direction, the plus sign one moving in the negative x-direction<sup>[2](https://phys.libretexts.org/Bookshelves/University_Physics/University_Physics_%28OpenStax%29/Book%3A_University_Physics_I_-_Mechanics_Sound_Oscillations_and_Waves_%28OpenStax%29/16%3A_Waves/16.03%3A_Mathematics_of_Waves)</sup>.

For a plane wave in two or three dimensions, the phase is written \( \theta = \vec{k} \cdot \vec{x} - \omega t \). Surfaces of constant phase are planes normal to the wave vector \( \vec{k} \), and they move outward along \( \vec{k} \) as time increases. The wavelength is the distance along the wave vector between two points with the same phase, and \( |\vec{k}| \cdot \lambda = 2\pi \)<sup>[6](https://ocw.mit.edu/courses/12-802-wave-motion-in-the-ocean-and-the-atmosphere-spring-2008/4a2ebfcfa9ec158f1a661499268c9843_MIT12_802S08_lec01.pdf)</sup>.

**Comparing two waves.** If two waveforms share the same frequency, the phase difference (or phase angle) between them is usually given in degrees, as a value exceeding −180° but no greater than +180°. A wave occurring "ahead" of another is said to have leading phase, one occurring "behind" has lagging phase. Two waves differing by −90° or +90° are in phase quadrature; differing by 180° they are in phase opposition, meaning one is inverted relative to the other<sup>[1](https://www.nist.gov/pml/time-and-frequency-division/popular-links/time-frequency-z/time-and-frequency-z-p)</sup>.

## Phase difference and interference

The phase difference between two waves that gives rise to interference has two sources: the initial (natural) phase difference, arising from where in its cycle each wave begins, and the path length difference \( \Delta D \), the difference in distances traveled by the two waves from their sources to the point of interference. The phase difference due to a path length difference is

\[ \Delta\phi = 2\pi \frac{\Delta D}{\lambda} \]

Often the initial phase difference is zero, in which case the interference is entirely due to the path length difference<sup>[3](https://lipa.physics.oregonstate.edu/sec_phase_diff.html)</sup>.

**Constructive and destructive conditions.** If the initial phase difference is zero, maximum constructive interference occurs where the path length difference is an integer multiple of the wavelength, \( \Delta D = m\lambda \) for integer \( m \); complete destructive interference occurs where it is a half-integer multiple, \( \Delta D = (m + 1/2)\lambda \)<sup>[3](https://lipa.physics.oregonstate.edu/sec_phase_diff.html)</sup>. In double-slit interference, waves start from the slits in phase but may arrive out of phase at the screen: paths differing by a whole wavelength arrive crest to crest and interfere constructively, while paths differing by any half-integral number of wavelengths \( (1/2)\lambda, (3/2)\lambda, (5/2)\lambda \) arrive crest to trough and interfere destructively<sup>[7](https://assets.openstax.org/oscms-prodcms/media/documents/UniversityPhysicsVolume3-WEB.pdf)</sup>.

The same logic applies to sound. Two speakers driven by a single signal generator produce waves that are in phase and of a single frequency; when two crests or two troughs coincide there is constructive interference, and when a trough and a crest coincide there is destructive interference. A path length difference of one wavelength gives a resulting amplitude twice the original; a difference of one half wavelength gives zero amplitude<sup>[8](https://nattrass.utk.edu/Phys137Fall2023/UniversityPhysicsVolume1-Ch17.pdf)</sup>.

**Initial phase can reverse the outcome.** The path difference alone does not fix the interference. A listener standing halfway between two identical sources has a path length difference of zero; if the sources are naturally in phase this point shows maximum constructive interference, but if the sources are out of phase to begin with, the same location shows maximum destructive interference<sup>[3](https://lipa.physics.oregonstate.edu/sec_phase_diff.html)</sup>.

## Phase velocity and dispersion

The speed at which surfaces of constant phase move is the phase velocity. For a plane wave it is \( c = \omega/|\vec{k}| \), and it is not a vector; the phase speed in a single coordinate direction, \( c_x = \omega/k \), is likewise defined at fixed values of the other coordinates<sup>[6](https://ocw.mit.edu/courses/12-802-wave-motion-in-the-ocean-and-the-atmosphere-spring-2008/4a2ebfcfa9ec158f1a661499268c9843_MIT12_802S08_lec01.pdf)</sup>.

When two harmonic waves of slightly different wavenumbers and frequencies are superposed, the sum takes the form of a carrier modulated by an envelope: the carrier travels at the phase velocity \( c = \omega/k \), while the envelope, or beat pattern, travels at the group velocity \( u = \delta\omega/\delta k \). In a dispersive medium these two velocities differ; in the limit of high frequency the group velocity approaches the phase velocity<sup>[5](https://ocw.mit.edu/courses/12-510-introduction-to-seismology-spring-2010/c08aed6151a62870da275f098690bd3d_lec12.pdf)</sup>.

## Phase in measurement and engineering

**Time-domain phase.** In time and frequency metrology the phase difference between two signals is usually stated in units of time rather than phase angle. The time interval corresponding to 1° of phase is \( t_{\mathrm{deg}} = 1/(360f) = T/360 \); on a 5 MHz signal, whose period is 200 nanoseconds, a 1° phase shift corresponds to a time shift of 555 picoseconds<sup>[1](https://www.nist.gov/pml/time-and-frequency-division/popular-links/time-frequency-z/time-and-frequency-z-p)</sup>. Phase comparison of two waveforms of the same nominal frequency is generally used to determine the frequency offset of a device under test relative to a reference<sup>[1](https://www.nist.gov/pml/time-and-frequency-division/popular-links/time-frequency-z/time-and-frequency-z-p)</sup>.

**Phase wrapping.** A phase difference between two sine waves can only be uniquely determined over a range of one wavelength, that is 2π radians, so the relationship between phase and time delay is ambiguous by whole cycles. [Phase unwrapping](https://www.edgepedia.org/phase-unwrapping) procedures recover the absolute phase by finding the fringe order for each measurement point and adding the corresponding integer multiples of 2π<sup>[9](https://www.sciencedirect.com/topics/engineering/phase-wrapping)</sup>. This ambiguity is a practical constraint in interferometry, where the measured phase difference is always between −π and π while the absolute phase is what a true measurement requires<sup>[10](https://arxiv.org/abs/2210.10541)</sup>.

**Interferometry.** Interferometers turn path differences into phase differences and phase differences into measurable signals. In a heterodyne interferometer the optical path difference between test and reference beams is converted to a phase difference between electrical signals; in a simple [Michelson interferometer](https://www.edgepedia.org/michelson-interferometer) the phase is related to the displacement of the test reflector, and 1° of phase represents a reflector displacement of \( \lambda/720 \), about 0.9 nm for helium-neon laser wavelengths<sup>[11](https://tsapps.nist.gov/publication/get_pdf.cfm?pub_id=22814)</sup>. [Neutron interferometry](https://www.edgepedia.org/neutron-interferometry) works the same way with [matter waves](https://www.edgepedia.org/matter-wave): a neutron accumulates a phase as it traverses the interferometer, and the difference between the two paths is measured as a phase shift that can be related to electric, magnetic, gravitational, or nuclear interactions<sup>[12](https://www.nist.gov/laboratories/tools-instruments/neutron-interferometry-and-optics-facilities)</sup>.

## References

1. [Time and Frequency from A to Z, P, NIST Time and Frequency Division](https://www.nist.gov/pml/time-and-frequency-division/popular-links/time-frequency-z/time-and-frequency-z-p)
2. [16.3: Mathematics of Waves, Physics LibreTexts (OpenStax University Physics)](https://phys.libretexts.org/Bookshelves/University_Physics/University_Physics_%28OpenStax%29/Book%3A_University_Physics_I_-_Mechanics_Sound_Oscillations_and_Waves_%28OpenStax%29/16%3A_Waves/16.03%3A_Mathematics_of_Waves)
3. [Section 18.3 Path Length Difference, LIPA (Oregon State University physics)](https://lipa.physics.oregonstate.edu/sec_phase_diff.html)
4. [SP 330 - Section 5, The International System of Units, NIST](https://www.nist.gov/pml/special-publication-330/sp-330-section-5)
5. [Lecture 12, Introduction to Seismology 12.510, MIT OpenCourseWare](https://ocw.mit.edu/courses/12-510-introduction-to-seismology-spring-2010/c08aed6151a62870da275f098690bd3d_lec12.pdf)
6. [Waves, Lecture 1, Wave Motion in the Ocean and the Atmosphere 12.802, MIT OpenCourseWare](https://ocw.mit.edu/courses/12-802-wave-motion-in-the-ocean-and-the-atmosphere-spring-2008/4a2ebfcfa9ec158f1a661499268c9843_MIT12_802S08_lec01.pdf)
7. [University Physics Volume 3, Chapter 3: Interference, OpenStax](https://assets.openstax.org/oscms-prodcms/media/documents/UniversityPhysicsVolume3-WEB.pdf)
8. [University Physics Volume 1, Chapter 17, University of Tennessee course text (OpenStax)](https://nattrass.utk.edu/Phys137Fall2023/UniversityPhysicsVolume1-Ch17.pdf)
9. [Phase Wrapping, ScienceDirect Topics overview](https://www.sciencedirect.com/topics/engineering/phase-wrapping)
10. [The phase unwrapping of under-sampled interferograms using radial basis function neural networks, arXiv](https://arxiv.org/abs/2210.10541)
11. [Electronic limitations in phase, NIST publication on heterodyne interferometry phase meters](https://tsapps.nist.gov/publication/get_pdf.cfm?pub_id=22814)
12. [Neutron Interferometry and Optics Facilities, NIST](https://www.nist.gov/laboratories/tools-instruments/neutron-interferometry-and-optics-facilities)

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*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*

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