Phase (waves)

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Phase in waves is the fraction of a wave cycle which has elapsed relative to an arbitrary point.[1]

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Formula

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

where A, f, and $\scriptstyle \theta$ are constant parameters. These functions are periodic with period $\scriptstyle T = 1/f$, and they are identical except for a displacement of $\scriptstyle T/4$ 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 $\scriptstyle \theta$, and the phase of $\scriptstyle y(t)$ is $\scriptstyle \theta -\pi/2$.
• It can refer to $\scriptstyle \theta$, in which case we would say $\scriptstyle x(t)$ and $\scriptstyle y(t)$ have the same phase but are relative to different references.
• In the context of communication waveforms, the time-variant angle  $\scriptstyle 2 \pi f t + \theta,\,$  or its modulo $\scriptstyle 2\pi$ value, is referred to as instantaneous phase, but often just phase.  Instantaneous phase has a formal definition that is applicable to more general functions and unambiguously defines a function's initial phase at t=0.  Accordingly, it is $\scriptstyle \theta$ for $\scriptstyle x(t)$  and  $\scriptstyle \theta -\pi/2$  for $\scriptstyle y(t)$.  (also see phasor)