and described by the formula:
where A is the amplitude of oscillation, f is the frequency, t is the elapsed time, and is the phase of the oscillation. The phase determines or is determined by the initial displacement at time t = 0. A motion with frequency f has period
Two potential ambiguities can be noted:
The term instantaneous phase is used to distinguish the time-variant angle from the initial condition. It also has a formal definition that is applicable to more general functions and unambiguously defines a function's initial phase at t=0. I.e., sine and cosine inherently have different initial phases. When not explicitly stated otherwise, cosine should generally be inferred. (also see phasor)
is sometimes referred to as a phase-shift, because it represents a "shift" from zero phase. But a change in is also referred to as a phase-shift.
For infinitely long sinusoids, a change in is the same as a shift in time, such as a time-delay. If is delayed (time-shifted) by of its cycle, it becomes:
whose "phase" is now It has been shifted by .
Two oscillators that have the same frequency and different phases have a phase difference, and the oscillators are said to be out of phase with each other. The amount by which such oscillators are out of step with each other can be expressed in degrees from 0° to 360°, or in radians from 0 to 2π. If the phase difference is 180 degrees (π radians), then the two oscillators are said to be in antiphase. If two interacting waves meet at a point where they are in antiphase, then destructive interference will occur. It is common for waves of electromagnetic (light, RF), acoustic (sound) or other energy to become superimposed in their transmission medium. When that happens, the phase difference determines whether they reinforce or weaken each other. Complete cancellation is possible for waves with equal amplitudes.
Time is sometimes used (instead of angle) to express position within the cycle of an oscillation.
The term in-phase is also found in the context of communication signals:
where represents a carrier frequency, and
and represent possible modulation of a pure carrier wave, e.g.: The modulation alters the original component of the carrier, and creates a (new) component, as shown above. The component that is in phase with the original carrier is referred to as the in-phase component. The other component, which is always 90° ( radians) "out of phase", is referred to as the quadrature component.
In physics, quantum mechanics ascribes waves to physical objects. The wave function is complex and since its square modulus is associated with the probability of observing the object, the complex character of the wave function is associated to the phase. Since the complex algebra is responsible for the striking interference effect of quantum mechanics, phase of particles is therefore ultimately related to their quantum behavior.