Phase (waves)

In physics and mathematics, the phase (symbol φ or ϕ) of a wave or other periodic function

(such as time) is an angle-like quantity representing the fraction of the cycle covered up to

It may be measured in any angular unit such as degrees or radians, thus increasing by 360° or

, the sine of the phase, multiplied by some factor (the amplitude of the sinusoid).

is an arbitrary "origin" value of the argument, that one considers to be the beginning of a cycle.

is a function of an angle, defined only for a single full turn, that describes the variation of

is a "canonical" function of a phase angle in 0 to 2π, that describes just one cycle of that waveform; and

Since phases are angles, any whole full turns should usually be ignored when performing arithmetic operations on them.

That is, the sum and difference of two phases (in degrees) should be computed by the formulas

The phase difference is particularly important when two signals are added together by a physical process, such as two periodic sound waves emitted by two sources and recorded together by a microphone.

Then the signals have opposite signs, and destructive interference occurs.

The periodic changes from reinforcement and opposition cause a phenomenon called beating.

The phase difference is especially important when comparing a periodic signal

In the clock analogy, this situation corresponds to the two hands turning at the same speed, so that the angle between them is constant.

(in terms of the modulo operation) of the two signals and then scaled to a full turn:

For example, the two signals may be a periodic soundwave recorded by two microphones at separate locations.

Or, conversely, they may be periodic soundwaves created by two separate speakers from the same electrical signal, and recorded by a single microphone.

They may be a radio signal that reaches the receiving antenna in a straight line, and a copy of it that was reflected off a large building nearby.

When two signals with these waveforms, same period, and opposite phases are added together, the sum

A real-world example of a sonic phase difference occurs in the warble of a Native American flute.

[3] A phase comparison can be made by connecting two signals to a two-channel oscilloscope.

The oscilloscope will display two sine signals, as shown in the graphic to the right.

If the two frequencies were exactly the same, their phase relationship would not change and both would appear to be stationary on the oscilloscope display.

Since the two frequencies are not exactly the same, the reference appears to be stationary and the test signal moves.

By measuring the rate of motion of the test signal, the offset between frequencies can be determined.

Vertical lines have been drawn through the points where each sine signal passes through zero.

The bottom of the figure shows bars whose width represents the phase difference between the signals.

In this case the phase difference is increasing, indicating that the test signal is lower in frequency than the reference.

[3] The phase of a simple harmonic oscillation or sinusoidal signal is the value of

are constant parameters called the amplitude, frequency, and phase of the sinusoid.

Plot of one cycle of a sinusoidal function. The phase for each argument value, relative to the start of the cycle, is shown at the bottom, in degrees from 0° to 360° and in radians from 0 to 2π.
Illustration of phase shift. The horizontal axis represents an angle (phase) that is increasing with time.
Phase shifter using IQ modulator
In-phase signals
Out-of-phase signals
Representation of phase comparison. [ 3 ]
Left: the real part of a plane wave moving from top to bottom. Right: the same wave after a central section underwent a phase shift, for example, by passing through a glass of different thickness than the other parts.
Out of phase AE