Phase (waves)
From Free net encyclopedia
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:
- <math>\sin(2 \pi F t + \phi)\,</math>
where <math>t\,</math> represents time, and <math>F\,</math> is the frequency. If <math>2 \pi F t\,</math> has units of radians, and <math>t\,</math> has units of seconds, then <math>F\,</math> has units of cycles <math>[= 2 \pi </math> radians <math>= 360</math> degrees<math>]</math> per second. The angle, <math>(2 \pi F t + \phi)\,</math> is called the phase of the sinusoid. <math>\phi\,</math> is the phase at <math>t=0</math>, sometimes called the initial phase.
The duration (aka period) of one cycle of the wave is given by: <math>T = \begin{matrix} \frac{1}{F} \end{matrix}\,</math> (seconds per cycle). If the sinusoid above is delayed (time-shifted) by <math>\begin{matrix} \frac{1}{4} \end{matrix}\,</math> of its cycle, it becomes:
- <math>\sin(2 \pi F (t - \begin{matrix} \frac{1}{4} \end{matrix}T) +\phi) = \sin(2 \pi F t - \begin{matrix}\frac{\pi }{2} \end{matrix} +\phi )\,</math>
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.
See also
- Phase, for other meanings of the word phase.
- Frequencyda:Fase (svingning)
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