Phase (waves)

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.

Formula

The phase of an oscillation or wave refers to a sinusoidal function such as the following:

where , , and are constant parameters called the amplitude, frequency, and phase of the sinusoid. These functions are periodic with period , and they are identical except for a displacement of along the axis. The term phase can refer to several different things:

It can refer to a specified reference, such as , in which case we would say the phase of is , and the phase of is .

It can refer to , in which case we would say and have the same phase but are relative to their own specific references.

In the context of communication waveforms, the time-variant angle , or its principal value, is referred to as instantaneous phase, often just phase.

Phase music

Phase music is a form of music that uses phasing as a primary compositional process. It is an approach to musical composition that is often associated with minimal music, as it shares similar characteristics, but some commentators prefer to treat phase music as a separate category. Phasing is a compositional technique in which the same part (a repetitive phrase) is played on two musical instruments, in steady but not identical tempi. Thus, the two instruments gradually shift out of unison, creating first a slight echo as one instrument plays a little behind the other, then a doubling with each note heard twice, then a complex ringing effect, and eventually coming back through doubling and echo into unison. Phasing is the rhythmic equivalent of cycling through the phase of two waveforms as in phasing. Note that the tempi of the two instruments are almost identical, so that both parts are perceived as being in the same tempo: the changes only separate the parts gradually. In some cases, especially live performance where gradual separation is extremely difficult, phasing is accomplished by periodically inserting an extra note (or temporarily removing one) into the phrase of one of the two players playing the same repeated phrase, thus shifting the phase by a single beat at a time, rather than gradually.