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Phase describes the current state of something that changes cyclically (aka ''oscillates'').

Examples:
  • The phase of the earth in its yearly revolution around the sun is measured in months and seasons.

  • The phase of the earth's daily rotation is measured in hours ( GMT ) or perhaps longitude.

  • ---The convenient correspondence of time with angle is due to the constant rate (aka Frequency ) of rotation.

  • ---If the rotation were in the other direction, mathematically it could be described as a Negative Frequency to distinguish one type of rotation from the other.



Waves are often modeled as sinusoidal functions (of time), whose amplitude changes cyclically. So they have a '''phase''' that changes with time. It describes where the wave is in its cycle of amplitude change. Different waves oscillate at different frequencies. So time is often not a convenient measure of phase. Consider the sinusoid''':'''

:\sin(2 \pi F t + \phi)\,

where t\, represents time, and F\, is the frequency. If 2 \pi F t\, has units of radians, and t\, has units of seconds, then F\, has units of cycles 2 \pi radians = 360 degrees '''per second'''. The '''angle''', (2 \pi F t + \phi)\, is called the '''phase''' of the sinusoid. \phi\, is the phase at t=0, sometimes called the ''initial phase''.

The duration (aka ''period'') of one cycle of the wave is given by: T = \begin{matrix} rac{1}{F} \end{matrix}\, (seconds per cycle). If the sinusoid above is delayed (time-shifted) by \begin{matrix} rac{1}{4} \end{matrix}\, of its cycle, it becomes:

:\sin(2 \pi F (t - \begin{matrix} rac{1}{4} \end{matrix}T) +\phi) = \sin(2 \pi F t - \begin{matrix} rac{\pi }{2} \end{matrix} +\phi )\,

So a shift in time is equivalent to a change in the initial phase. Conversely, a change in the initial phase is tantamount to a shift in time.




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