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 that 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  \tfrac{\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 timevariant 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.
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 (timeshifted) by \scriptstyle \frac{1}{4}\, of its cycle, it becomes:

\begin{align} x\left(t  \tfrac{1}{4} T\right) &= A\cdot \cos\left(2 \pi f \left(t  \tfrac{1}{4}T \right) + \varphi \right) \\ &= A\cdot \cos\left(2 \pi f t  \tfrac{\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
Inphase waves
Outofphase 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 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 no phase difference are said to be in phase. 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 phase 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 realworld example of a sonic phase difference occurs in the warble of a Native American flute. The amplitude of different harmonic components of same longheld 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]}
See also
References

^ ^{a} ^{b} ^{c} Ballou, Glen (2005). Handbook for sound engineers (3 ed.). Focal Press, Gulf Professional Publishing. p. 1499.

^ Clint Goss; Barry Higgins (2013). "The Warble". Flutopedia. Retrieved 20130306.
External links

Relationship of phase difference and timedelay

Phase angle, phase difference, time delay, and frequency

ECE 209: Sources of Phase Shift — Discusses the timedomain sources of phase shift in simple linear timeinvariant circuits.

Phase Difference Java Applet
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