# Phase (waves) Research Materials

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Phase in sinusoidal functions or in waves has two different, but closely related, meanings. One is the initial angle of a sinusoidal function at its origin and is sometimes called phase offset or phase difference. Another usage is the fraction of the wave cycle which has elapsed relative to the origin.[1]

## Formula

The phase of an oscillation or wave refers to a sinusoidal function such as the following:

\begin{align} x(t) &= A\cdot \cos( 2 \pi f t + \varphi ) \\ y(t) &= A\cdot \sin( 2 \pi f t + \varphi ) = A\cdot \cos\left( 2 \pi f t + \varphi - \frac{\pi}{2}\right) \end{align}

where $\scriptstyle A\,$, $\scriptstyle f\,$, and $\scriptstyle \varphi\,$ are constant parameters called the amplitude, frequency, and phase of the sinusoid. These functions are periodic with period $\scriptstyle T = \frac{1}{f}\,$, and they are identical except for a displacement of $\scriptstyle \frac{T}{4}\,$ along the $\scriptstyle t\,$ axis. The term phase can refer to several different things:

• It can refer to a specified reference, such as $\scriptstyle \cos( 2 \pi f t)\,$, in which case we would say the phase of $\scriptstyle x(t)\,$ is $\scriptstyle \varphi\,$, and the phase of $\scriptstyle y(t)\,$ is $\scriptstyle \varphi\,-\, \frac{\pi}{2}\,$.
• It can refer to $\scriptstyle \varphi\,$, in which case we would say $\scriptstyle x(t)\,$ and $\scriptstyle y(t)\,$ have the same phase but are relative to their own specific references.
• In the context of communication waveforms, the time-variant angle $\scriptstyle 2 \pi f t \,+\, \varphi$, or its principal value, is referred to as instantaneous phase, often just phase.

## Phase shift

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

Phase shift is any change that occurs in the phase of one quantity, or in the phase difference between two or more quantities.[1]

$\scriptstyle \varphi\,$ is sometimes referred to as a phase shift or phase offset, because it represents a "shift" from zero phase. But a change in $\scriptstyle \varphi\,$ is also referred to as a phase shift.

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

\begin{align} x\left(t - \frac{1}{4} T\right) &= A\cdot \cos\left(2 \pi f \left(t - \frac{1}{4}T \right) + \varphi \right) \\ &= A\cdot \cos\left(2 \pi f t - \frac{\pi}{2} + \varphi \right) \end{align}

whose "phase" is now $\scriptstyle \varphi \,-\, \frac{\pi}{2}$. It has been shifted by $\scriptstyle \frac{\pi}{2}$ radians.

## Phase difference

In-phase waves
Out-of-phase waves
Left: the real part of a plane wave moving from top to bottom. Right: the same wave after a central section underwent a phase shift, for example, by passing through a glass of different thickness than the other parts. (The illustration on the right ignores the effect of diffraction, which would make the waveform continuous away from material interfaces and would add increasing distortions with distance.).

Phase difference is the difference, expressed in electrical degrees or time, between two waves having the same frequency and referenced to the same point in time.[1] 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 superposed 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 is undefined and would only reflect different starting positions. Technically, phase difference between two entities at various frequencies is undefined and does not exist.

• Time zones are also analogous to phase differences.

A real-world example of a sonic phase difference occurs in the warble of a Native American flute. The amplitude of different harmonic components of same long-held note on the flute come into dominance at different points in the phase cycle. The phase difference between the different harmonics can be observed on a spectrograph of the sound of a warbling flute.[2]

## In-phase and quadrature (I&Q) components

The term in-phase is also found in the context of communication signals:

$A(t)\cdot \sin[2\pi ft + \varphi(t)] = I(t)\cdot \sin(2\pi ft) + Q(t)\cdot \underbrace{\sin\left(2\pi ft + \begin{matrix} \frac{\pi}{2} \end{matrix} \right)}_{\cos(2\pi ft)}$

and:

$A(t)\cdot \cos[2\pi ft + \varphi(t)] = I(t)\cdot \cos(2\pi ft) + Q(t)\cdot \underbrace{\cos\left(2\pi ft + \begin{matrix} \frac{\pi}{2} \end{matrix}\right)}_{-\sin(2\pi ft)},$

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

\begin{align} I(t) &\equiv A(t)\cdot \cos\left(\varphi(t)\right) \\ Q(t) &\equiv A(t)\cdot \sin\left(\varphi(t)\right) \end{align}

$\scriptstyle A(t)\,$ and $\scriptstyle \varphi(t)\,$ represent possible modulation of a pure carrier wave; e.g.,  $\scriptstyle \sin(2\pi ft)\,$ (or $\scriptstyle \cos(2\pi ft)$). The modulation alters the original $\scriptstyle \sin\,$ (or $\scriptstyle \cos\,$) component of the carrier, and creates a (new) $\scriptstyle \cos\,$ (or $\scriptstyle \sin\,$) 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° ($\scriptstyle \frac{\pi}{2}$ 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.

## References

1. ^ a b c Ballou, Glen (2005). Handbook for sound engineers (3 ed.). Focal Press, Gulf Professional Publishing. p. 1499. ISBN 0-240-80758-8.
2. ^ Clint Goss; Barry Higgins (2013). "The Warble". Flutopedia. Retrieved 2013-03-06.
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