Phase (waves, $\varphi$)

The wave phase specifies position within an oscillation cycle. For a sinusoidal travelling wave,where $k$ is the wavenumber, $\omega$ the angular frequency and ${\varphi }_0$ an initial phase shift. Adding $2\pi$ to $\Phi$ leaves the field unchanged; one complete cycle advances the wave phase by $2\pi$ radians.

A surface of equal wave phase is a wavefront. Tracking one such surface gives the phase velocity, whereas comparing two fields at the same point gives their phase difference. Two equal-frequency disturbances add constructively when their wave phases agree and can cancel when their phase difference is $\pi$. The sign assigned to a phase lag depends on the chosen time convention. In a complex amplitude, the complex argument carries this information while the modulus sets the wave amplitude.

A phase shift is a change in an oscillation's wave phase, specified modulo $2\pi$. It can arise from propagation, reflection or an imposed response. With the time convention $e^{-i\omega t}$, a delay $\tau$ multiplies the complex amplitude by $e^{i\omega \tau }$. A sign reversal is a $\pi$ phase shift.

A phase factor is a unit-modulus complex number $e^{i\varphi }$. Multiplying a complex amplitude by it changes its wave phase by $\varphi$ while preserving its squared modulus. A common phase factor does not change light intensity, while different phase factors of contributions can change their interference.

A phase difference compares two wave phases at the same position and time. For $u_1=A_1\cos (\omega t+{\varphi }_1)$ and $u_2=A_2\cos (\omega t+{\varphi }_2)$, set $\Delta \varphi ={\varphi }_2-{\varphi }_1$, modulo $2\pi$. Adding the two complex amplitudes gives the squared resultant amplitudeEqual amplitudes reinforce at $\Delta \varphi =0$ and cancel at $\Delta \varphi =\pi$; at phase quadrature the cross term vanishes in the cycle average.

For propagation through a uniform nondispersive path difference $\Delta L$, the magnitude of the accumulated phase difference is $2\pi \Delta L/\lambda$, with its sign fixed by the comparison convention. A two-path interference measurement therefore converts a small path change into an observable change of wave intensity. A stable fringe pattern requires that the relative wave phase remain sufficiently stable during the measurement.

For an oscillation written $A\cos (\omega t-\varphi )$ relative to $A\cos (\omega t)$, a positive $\varphi$ represents delayed maxima. The corresponding unwrapped delay is $\varphi /\omega$, while a single periodic observation determines the phase difference only modulo $2\pi$.

Two sinusoidal signals are in phase quadrature when their phase difference is a quarter cycle, for example $\cos \varphi$ and $\sin \varphi$. Their time averages have zero correlation over a cycle. In a plane wave, this separates fields such as displacement and its time derivative, velocity.