Phase (waves)

The phase of an oscillation or wave is the fraction of a complete cycle corresponding to an offset in the displacement from a specified reference point at time t = 0. Phase is a frequency domain or Fourier transform domain concept, and as such, can be readily understood in terms of simple harmonic motion. The same concept applies to wave motion, viewed either at a point in space over an interval of time or across an interval of space at a moment in time. Simple harmonic motion is a displacement that varies cyclically, as depicted below:



and described by the formula:


 * $$x(t) = A\cdot \sin( 2 \pi f t + \theta ),\,$$

where A is the amplitude of oscillation, and f is the frequency. A motion with frequency f has period $$T=\frac{1}{f}.$$   $$t\, $$ is the elapsed time and $$\theta$$ is the phase of the oscillation. It determines or is determined by the initial displacement at time t = 0.

Two potential ambiguities can be noted:
 * One is that the initial displacement of $$\cos( 2 \pi f t + \theta )\,$$  is different than the sine function, yet they appear to have the same "phase".
 * The time-variant angle $$2 \pi f t + \theta,\,$$  or its modulo $$2\pi$$ value, is also commonly referred to as "phase".  Then it is not an initial condition, but rather a continuously-changing condition.

The term instantaneous phase is used to distinguish the time-variant angle from the initial condition. It also has a formal definition that is applicable to more general functions and unambiguously defines a function's initial phase at t=0. I.e., sine and cosine inherently have different initial phases. When not explicitly stated otherwise, cosine should generally be inferred. (also see phasor)

Phase shift


$$\theta$$ is sometimes referred to as a phase-shift, because it represents a "shift" from zero phase. But a change in $$\theta$$ is also referred to as a phase-shift.

For infinitely long sinusoids, a change in $$\theta$$ is the same as a shift in time, such as a time-delay. If $$x(t)\,$$ is delayed (time-shifted) by $$\begin{matrix} \frac{1}{4} \end{matrix}\,$$ of its cycle, it becomes:


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

whose "phase" is now $$\theta - \begin{matrix}\frac{\pi }{2} \end{matrix}.$$ It has been shifted by $$\begin{matrix}\frac{\pi }{2} \end{matrix}$$.

Phase difference


Two oscillators that have the same frequency and different phases have a phase difference, and the oscillators are said to be out of phase with each other. The amount by which such oscillators are out of step with each other can be expressed in degrees from 0° to 360°, or in radians from 0 to 2π. If the phase difference is 180 degrees (π radians), then the two oscillators are said to be in antiphase. If two interacting waves meet at a point where they are in antiphase, then destructive interference will occur. It is common for waves of electromagnetic (light, RF), acoustic (sound) or other energy to become superimposed in their transmission medium. When that happens, the phase difference determines whether they reinforce or weaken each other. Complete cancellation is possible for waves with equal amplitudes.

Time is sometimes used (instead of angle) to express position within the cycle of an oscillation.
 * A phase difference 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 (different frequencies), the phase difference would only reflect different starting positions.
 * We measure the rotation of the earth in hours, instead of radians. And therefore time zones are an example of phase differences.

In-phase and quadrature (I&Q) components
The term in-phase is also found in the context of communication signals:



\begin{align} A(t)\cdot \sin[2\pi ft + \phi(t)] &= I(t)\cdot \sin(2\pi ft) + Q(t)\cdot \cos(2\pi ft) \\ &=I(t)\cdot \sin(2\pi ft) + Q(t)\cdot \sin(2\pi ft + \begin{matrix} \frac{\pi}{2} \end{matrix})\end{align} $$

and:



\begin{align} A(t)\cdot \cos[2\pi ft + \phi(t)] &= 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}), \end{align} $$

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

I(t)\ \stackrel{\mathrm{def}}{=}\ A(t)\cdot \cos[\phi(t)], \, $$

Q(t)\ \stackrel{\mathrm{def}}{=}\ A(t)\cdot \sin[\phi(t)].\, $$

$$A(t)\,$$ and $$\phi(t)\,$$ represent possible modulation of a pure carrier wave, e.g.: $$\sin(2\pi ft).\,$$  The modulation alters the original $$\sin\,$$ component of the carrier, and creates a (new) $$\cos\,$$ 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 with the 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.