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)

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