Phase (waves)
Angle-like quantity representing the fraction of a cycle covered.
In physics and mathematics, the phase (symbol φ or ϕ) of a wave or other periodic function F of some real variable t (such as time) is an angle-like quantity representing the fraction of the cycle covered up to t. It is expressed in a scale that varies by one full turn as t goes through each period, and may be measured in any angular unit such as degrees or radians. The concept is especially appropriate for sinusoidal functions, where the value at any argument t can be expressed as the sine of the phase, multiplied by the amplitude. The numeric value of the phase depends on an arbitrary choice of the start of each period and on the interval of angles to which each period is mapped. Usually, whole turns are ignored when expressing the phase, so that it is also a periodic function that repeatedly scans the same range of angles. Two argument values are said to be at the same phase if their difference is a whole number of periods. Mathematically, for a periodic signal F with period T, the phase φ at any argument t is given by φ(t) = 2π((t - t₀)/T - floor((t - t₀)/T)), where t₀ is an arbitrary origin marking the beginning of a cycle. This can be visualized as a clock hand turning at constant speed, making a full turn every T seconds, with the phase being the clockwise angle from the 12:00 position. The phase is most useful when the origin is chosen based on features of F, such as for a sinusoid where the function changes from zero to positive. The phase difference between two periodic signals is called the phase shift; when zero, the signals are in phase, and when 180° (π radians), they are in antiphase, leading to destructive interference for sinusoids.
- field
- Physics and mathematics
- symbol
- φ or ϕ
- units
- Degrees or radians
- key_use
- Describing periodic functions and phase shifts
Lore & Background
The phase of a periodic function F with period T is defined mathematically as φ(t) = 2π [[(t − t₀)/T]], where [[·]] denotes the fractional part of a real number, discarding its integer part, and t₀ is an arbitrary origin value marking the beginning of a cycle. This can be visualized as a clock hand turning at constant speed, making a full turn every T seconds, with the phase being the clockwise angle from the 12:00 position at time t₀ to the current position at time t. The numeric value of the phase depends on the arbitrary choice of the start of each period and the interval of angles to which each period is mapped.
Reader's Guide
The phase concept is fundamental for comparing periodic functions. When a periodic function F is compared with a shifted version G, the shift in t expressed as a fraction of the period and scaled to an angle φ spanning a whole turn gives the phase shift, phase offset, or phase difference of G relative to F. If F is a canonical function for a class of signals, such as sin(t) for all sinusoidal signals, then φ is called the initial phase of G. The phase is most useful when the origin t₀ is chosen based on features of F; for a sinusoid, a convenient choice is any t where the function's value changes from zero to positive. The phase can be expressed as an angle between 0 and 2π, or between −π and +π, using an alternative formula that subtracts half a turn.
Did You Know?
- The phase φ(t) is a periodic function with the same period as F, repeatedly scanning the same range of angles as t goes through each period.
- Two argument values t₁ and t₂ are said to be at the same phase if the difference between them is a whole number of periods.
- The concept can be visualized by imagining a clock hand that turns at constant speed, making a full turn every T seconds.
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