# Phase (waves)

In the context of periodic processes, phase is a word used for:

Instantaneous phase
the current position in the cycle of something that changes cyclically
Phase shift
a constant difference/offset between two instantaneous phases, particularly when one is a standard reference

Waves are amplitudes that change cyclically, often modeled as sinusoidal functions of time ($t$) or some other variable. Consider the ever-changing angle:

$\varphi (t) = (2 \pi f t + \theta) \ \operatorname{mod} \ 2\pi,\,$

and the amplitude wave:

$s(t) = A\cdot \sin[\varphi (t) ] = A\cdot \sin( 2 \pi f t + \theta ),\,$

where $A\,$, $f\,$, and $\theta\,$ are constants, and mod is the Modulo operation.

The initial phase (at $t=0$) of this sinusoid is the initial angle $\varphi(0) = \theta\,$, which is also commonly referred to as just "phase". The instantaneous phase at time $t$ is $2 \pi f t + \theta \,$, which has units of radians. $t$ has units of seconds, so $f$ has units of cycles per second (= 2π radians/sec = 360°/sec), which represents the frequency of the oscillations. Frequency is defined as the rate at which the instantaneous phase changes. The duration of one cycle of the wave, called its period, is given by: $T = 1/f \,$ (seconds per cycle).

When $\theta\,$ is referred to as the phase or phase shift, the implied reference is: $2 \pi f t\,$. For instance, that is what is meant by the "phase" of a Fourier transform at a particular frequency. The term "phase shift" also has a slightly more general usage, described in the next section.

When the frequency of an oscillation is time invariant, then time is sometimes used (instead of angle) to express instantaneous phase. Thus we measure the rotation of the earth in hours, instead of radians. Time zones are actually a good example of phase shifts. Other measures of phase are: (1) distance, and (2) fraction of the wavelength.

##  Phase shift

Image:Phase shift.png
Illustration of phase shift. The horizontal axis represents an angle (phase) that is increasing with time.

Phase shifting describes relative phase shift in superimposing waves. Waves may be of electromagnetic (light, RF), acoustic (sound) or other nature. By superposing waves using different phase shifts the waves can add to (0° shift = "in phase") or cancel out each other (180°).

A phase shift is also a difference or change in the initial phase. If $s(t)\,$ is delayed (time-shifted) by $\begin{matrix} \frac{1}{4} \end{matrix}\,$ of its cycle, it becomes:

 $s(t - \begin{matrix} \frac{1}{4} \end{matrix}T) \,$ $= A\cdot \sin(2 \pi f (t - \begin{matrix} \frac{1}{4} \end{matrix}T) + \theta) \,$ $= A\cdot \sin(2 \pi f t - \begin{matrix}\frac{\pi }{2} \end{matrix} + \theta ),\,$

whose initial phase is $\theta - \begin{matrix}\frac{\pi }{2} \end{matrix}$. Thus, a shift in time is equivalent to a phase shift. Conversely, a change in the initial phase is tantamount to a shift in time.

##  Phase difference

Phase difference is similar to phase shift, but more likely to be applied in the context of two signals, particularly when neither is a standard reference. Two waves that have the same frequency and different initial phases, have a phase difference that is constant (independent of t). So it is referred to simply as the phase difference, rather than the initial phase difference or the phase-shift difference. When the phase difference (modulo 2π) is zero, the waves are said to be in phase with each other. Otherwise, they are out of phase with each other. The terms are also commonly hyphenated, and used as an adjective:   "The out-of-phase signal caused distortion." If the phase difference is 180 degrees (π radians), then the two signals are said to be in antiphase. And if their peak amplitudes are equal, their sum is zero at all values of time, t.

In phase — this is analogous to two athletes running around a race track at the same speed and direction, side by side. They pass a point on the track together (simultaneously). Out of phase — this is analogous to two athletes running around a race track at the same speed and direction but starting at different positions on the track. They pass a point at different instants in time. But the time difference (phase difference) between them is a constant — same for every pass since they are at the same speed and in the same direction. If they were at different speeds, this would be analogous to two waves of different frequencies. Then, the phase (angle) difference measurement would be meaningless and void.

###  In-phase and quadrature components

Communication signals require more complicated forms (than above) of $\varphi(t)\,$. The linear term is often extracted, and written separately, as follows:

 $A(t)\cdot \cos[2\pi ft + \varphi(t)] \,$ $\ \stackrel{\mathrm{def}}{=}\ I(t)\cdot \cos(2\pi ft) - Q(t)\cdot \sin(2\pi ft)\,$ $= I(t)\cdot \cos(2\pi ft) + Q(t)\cdot \cos(2\pi ft + \begin{matrix} \frac{\pi}{2} \end{matrix})\,$

where $f\,$ represents a carrier frequency, and

$I(t) = A(t)\cdot \cos[\varphi(t)],$
$Q(t) = A(t)\cdot \sin[\varphi(t)].$

$A(t)\,$ and $\varphi(t)\,$ represent possible modulation of a pure carrier wave: $\cos(2\pi ft)\,$. The modulation alters the original $\cos\,$ component of the carrier, and creates a (new) $\sin\,$ component, as shown above. The component that is in phase with the original carrier is referred to as the in-phase component. The other component, which is always 90° ($\begin{matrix} \frac{\pi}{2} \end{matrix}$ radians) "out of phase", is referred to as the quadrature component.

##  Phase coherence

Coherence is the quality of a wave to display well defined phase relationship in different regions of its domain of definition.

In physics, quantum mechanics ascribes waves to physical objects. The wave function is complex and since its square modulus is associated to probability of observing the object, the complex character of the wave function is associated to the phase. Since the complex algebra is responsible for the striking interference effect of quantum mechanics, phase of particles is therefore ultimately related to their quantum behavior.