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# Phase (waves)

In physics and mathematics, the phase of a periodic function F {displaystyle F} of some real variable t {displaystyle t} is the relative value of that variable within the span of each full period. In physics and mathematics, the phase of a periodic function F {displaystyle F} of some real variable t {displaystyle t} is the relative value of that variable within the span of each full period. The phase is typically expressed as an angle ϕ ( t ) {displaystyle phi (t)} , in such a scale that it varies by one full turn as the variable t {displaystyle t} goes through each period (and F ( t ) {displaystyle F(t)} goes through each complete cycle). Thus, if the phase is expressed in degrees, it will increase by 360° as t {displaystyle t} increases by one period. If it is expressed in radians, the same increase in t {displaystyle t} will increase the phase by 2 π {displaystyle 2pi } . This convention is especially appropriate for a sinusoidal function, since its value at any argument t {displaystyle t} then can be expressed as the sine of the phase ϕ ( t ) {displaystyle phi (t)} , multiplied by some factor (the amplitude of the sinusoid). (The cosine may be used instead of sine, depending on where one considers each period to start.) Usually, whole turns are ignored when expressing the phase; so that ϕ ( t ) {displaystyle phi (t)} is also aperiodic function, with the same period as F {displaystyle F} , that repeatedly scans the same range of angles as t {displaystyle t} goes through each period. Then, F {displaystyle F} is said to be 'at the same phase' at two argument values t 1 {displaystyle t_{1}} and t 2 {displaystyle t_{2}} (that is, ϕ ( t 1 ) = ϕ ( t 2 ) {displaystyle phi (t_{1})=phi (t_{2})} ) if the difference between them is a whole number of periods. The numeric value of the phase ϕ ( t ) {displaystyle phi (t)} depends on the arbitrary choice of the start of each period, and on the interval of angles that each period is to be mapped to. The term 'phase' is also used when comparing a periodic function F {displaystyle F} with a shifted version G {displaystyle G} of it. If the shift in t {displaystyle t} is expressed as a fraction of the period, and then scaled to an angle φ {displaystyle varphi } spanning a whole turn, one gets the phase shift, phase offset, or phase difference of G {displaystyle G} relative to F {displaystyle F} . If F {displaystyle F} is a 'canonical' function for a class of signals, like sin ⁡ ( t ) {displaystyle sin(t)} is for all sinusoidal signals, then φ {displaystyle varphi } is called the initial phase of G {displaystyle G} . Let F {displaystyle F} be a periodic signal (that is, a function of one real variable), and T {displaystyle T} be its period (that is, the smallest positive real number such that F ( t + T ) = F ( t ) {displaystyle F(t+T)=F(t)} for all t {displaystyle t} ) . Then the phase of F {displaystyle F} at any argument t {displaystyle t} is Here [ [ ⋅ ] ] {displaystyle !]} denotes the fractional part of a real number, discarding its integer part; that is, [ [ x ] ] = x − ⌊ x ⌋ {displaystyle !]=x-leftlfloor x ight floor } ; and t 0 {displaystyle t_{0}} is an arbitrary 'origin' value of the argument, that one considers to be the beginning of a cycle. This concept can be visualized by imagining a clock with a hand that turns at constant speed, making a full turn every T {displaystyle T} seconds, and is pointing straight up at time t 0 {displaystyle t_{0}} . The phase ϕ ( t ) {displaystyle phi (t)} is then the angle from the 12:00 position to the current position of the hand, at time t {displaystyle t} , measured clockwise.

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