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<title>Phase (waves)</title>

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Phase (waves)

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In <a href="Physics" title="Physics">physics</a> and <a href="Mathematics" title="Mathematics">mathematics</a>, the phase (symbol φ or ϕ) of a <a href="Wave_(physics)" class="mw-redirect" title="Wave (physics)">wave</a> or other <a href="Periodic_function" title="Periodic function">periodic function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

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<annotation encoding="application/x-tex">{\displaystyle F}</annotation>

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</math><img src="./545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"> of some <a href="Real_number" title="Real number">real</a> variable <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

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<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

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</math><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"> (such as time) is an <a href="Angle" title="Angle">angle</a>-like quantity representing the fraction of the cycle covered up to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

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<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

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</math><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy">. It is expressed in such a <a href="Scale_(ratio)" title="Scale (ratio)">scale</a> that it varies by one full <a href="Turn_(geometry)" class="mw-redirect" title="Turn (geometry)">turn</a> as the variable <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

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<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

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</math><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"> goes through each <a href="Period_(physics)" class="mw-redirect" title="Period (physics)">period</a> (and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(t)}">

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<annotation encoding="application/x-tex">{\displaystyle F(t)}</annotation>

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</math><img src="./5b57ed3bbf501fb3c7f4bc5c4eafa96bb9e32165.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.39ex; height:2.843ex;" alt="{\displaystyle F(t)}" loading="lazy"> goes through each complete cycle). It may be <a href="Measure_(mathematics)" title="Measure (mathematics)">measured</a> in any <a href="Angular_unit" class="mw-redirect" title="Angular unit">angular unit</a> such as <a href="Degree_(angle)" title="Degree (angle)">degrees</a> or <a href="Radians" class="mw-redirect" title="Radians">radians</a>, thus increasing by 360° or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi }">

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<annotation encoding="application/x-tex">{\displaystyle 2\pi }</annotation>

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</math><img src="./73efd1f6493490b058097060a572606d2c550a06.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.494ex; height:2.176ex;" alt="{\displaystyle 2\pi }" loading="lazy"> as the variable <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

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<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

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</math><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"> completes a full period.<a href="#cite_note-Ballou2005-1">[1]</a>

This convention is especially appropriate for a <a href="Sinusoid" class="mw-redirect" title="Sinusoid">sinusoidal</a> function, since its value at any argument <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

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<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

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</math><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"> then can be expressed as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (t)}">

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<annotation encoding="application/x-tex">{\displaystyle \varphi (t)}</annotation>

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</math><img src="./cd7776ed1f48fe8d39d9852e6ed6aa8a61a93d28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.169ex; height:2.843ex;" alt="{\displaystyle \varphi (t)}" loading="lazy">, the <a href="Sine" class="mw-redirect" title="Sine">sine</a> of the phase, multiplied by some factor (the <a href="Amplitude" title="Amplitude">amplitude</a> of the sinusoid). (The <a href="Cosine" class="mw-redirect" title="Cosine">cosine</a> 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 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (t)}">

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<annotation encoding="application/x-tex">{\displaystyle \varphi (t)}</annotation>

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</math><img src="./cd7776ed1f48fe8d39d9852e6ed6aa8a61a93d28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.169ex; height:2.843ex;" alt="{\displaystyle \varphi (t)}" loading="lazy"> is also a periodic function, with the same period as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

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<annotation encoding="application/x-tex">{\displaystyle F}</annotation>

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</math><img src="./545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy">, that repeatedly scans the same range of angles as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

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<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

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</math><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"> goes through each period. Then, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

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<annotation encoding="application/x-tex">{\displaystyle F}</annotation>

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</math><img src="./545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"> is said to be "at the same phase" at two argument values <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{1}}">

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<annotation encoding="application/x-tex">{\displaystyle t_{1}}</annotation>

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</math><img src="./cb0768c0bd659f2f84fb5ef9f4b74f336123d915.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.894ex; height:2.343ex;" alt="{\displaystyle t_{1}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle t_{2}}</annotation>

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</math><img src="./749fee708b41e7079eabd50d61c8bf3e965db16f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.894ex; height:2.343ex;" alt="{\displaystyle t_{2}}" loading="lazy"> (that is, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (t_{1})=\varphi (t_{2})}">

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<annotation encoding="application/x-tex">{\displaystyle \varphi (t_{1})=\varphi (t_{2})}</annotation>

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</math><img src="./daa0702e9cf824af2343a910335ed79e62636013.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.545ex; height:2.843ex;" alt="{\displaystyle \varphi (t_{1})=\varphi (t_{2})}" loading="lazy">) if the difference between them is a whole number of periods.

The numeric value of the phase <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (t)}">

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<annotation encoding="application/x-tex">{\displaystyle \varphi (t)}</annotation>

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</math><img src="./cd7776ed1f48fe8d39d9852e6ed6aa8a61a93d28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.169ex; height:2.843ex;" alt="{\displaystyle \varphi (t)}" loading="lazy"> 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 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

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<annotation encoding="application/x-tex">{\displaystyle F}</annotation>

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</math><img src="./545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"> with a shifted version <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

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<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

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</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> of it. If the shift in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

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<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

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</math><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"> is expressed as a fraction of the period, and then scaled to an angle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }">

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<annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation>

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</math><img src="./33ee699558d09cf9d653f6351f9fda0b2f4aaa3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" loading="lazy"> spanning a whole turn, one gets the phase shift, phase offset, or phase difference of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

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<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

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</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> relative to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

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<annotation encoding="application/x-tex">{\displaystyle F}</annotation>

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</math><img src="./545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy">. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

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<annotation encoding="application/x-tex">{\displaystyle F}</annotation>

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</math><img src="./545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"> is a "canonical" function for a class of signals, like <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(t)}">

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<annotation encoding="application/x-tex">{\displaystyle \sin(t)}</annotation>

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</math><img src="./0eb63cccca21b1edef50f707888e3204ab5fda1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.505ex; height:2.843ex;" alt="{\displaystyle \sin(t)}" loading="lazy"> is for all sinusoidal signals, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }">

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<annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation>

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</math><img src="./33ee699558d09cf9d653f6351f9fda0b2f4aaa3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" loading="lazy"> is called the initial phase of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

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<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

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</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">.

<div class="mw-heading mw-heading2"><h2 id="Mathematical_definition">Mathematical definition</h2></div>

Let the signal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

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<annotation encoding="application/x-tex">{\displaystyle F}</annotation>

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</math><img src="./545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"> be a periodic function of one real variable, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

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<annotation encoding="application/x-tex">{\displaystyle T}</annotation>

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</math><img src="./ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy"> be its period (that is, the smallest positive <a href="Real_number" title="Real number">real number</a> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(t+T)=F(t)}">

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<annotation encoding="application/x-tex">{\displaystyle F(t+T)=F(t)}</annotation>

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</math><img src="./fa3df005e781ba0045ecebf95f452d2b77976de8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.355ex; height:2.843ex;" alt="{\displaystyle F(t+T)=F(t)}" loading="lazy"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

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<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

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</math><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy">). Then the phase of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

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<annotation encoding="application/x-tex">{\displaystyle F}</annotation>

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</math><img src="./545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"> at any argument <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

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<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

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</math><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"> is

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<annotation encoding="application/x-tex">{\displaystyle \varphi (t)=2\pi \left[\!\!\left[{\frac {t-t_{0}}{T}}\right]\!\!\right]}</annotation>

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Here <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [\![\,\cdot \,]\!]\!\,}">

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<annotation encoding="application/x-tex">{\displaystyle [\![\,\cdot \,]\!]\!\,}</annotation>

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</math><img src="./8536aff14a69f60ce6d112201324f74f85a061b4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.234ex; height:2.843ex;" alt="{\displaystyle [\![\,\cdot \,]\!]\!\,}" loading="lazy"> denotes the fractional part of a real number, discarding its integer part; that is, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [\![x]\!]=x-\left\lfloor x\right\rfloor \!\,}">

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<annotation encoding="application/x-tex">{\displaystyle [\![x]\!]=x-\left\lfloor x\right\rfloor \!\,}</annotation>

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</math><img src="./7841bc0e7dc0ad82ff00be9dfa5c3dd4728f0ef6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.806ex; height:2.843ex;" alt="{\displaystyle [\![x]\!]=x-\left\lfloor x\right\rfloor \!\,}" loading="lazy">; and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{0}}">

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<annotation encoding="application/x-tex">{\displaystyle t_{0}}</annotation>

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</math><img src="./02d3006c4190b1939b04d9b9bb21006fb4e6fa4a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.894ex; height:2.343ex;" alt="{\displaystyle t_{0}}" loading="lazy"> 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 <a href="Clock" title="Clock">clock</a> with a hand that turns at constant speed, making a full turn every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

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<annotation encoding="application/x-tex">{\displaystyle T}</annotation>

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</math><img src="./ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy"> seconds, and is pointing straight up at time <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{0}}">

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<annotation encoding="application/x-tex">{\displaystyle t_{0}}</annotation>

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</math><img src="./02d3006c4190b1939b04d9b9bb21006fb4e6fa4a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.894ex; height:2.343ex;" alt="{\displaystyle t_{0}}" loading="lazy">. The phase <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (t)}">

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<annotation encoding="application/x-tex">{\displaystyle \varphi (t)}</annotation>

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</math><img src="./cd7776ed1f48fe8d39d9852e6ed6aa8a61a93d28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.169ex; height:2.843ex;" alt="{\displaystyle \varphi (t)}" loading="lazy"> is then the angle from the 12:00 position to the current position of the hand, at time <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

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<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

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</math><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy">, measured <a href="Clockwise" title="Clockwise">clockwise</a>.

The phase concept is most useful when the origin <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t_{0}}</annotation>

</semantics>

</math><img src="./02d3006c4190b1939b04d9b9bb21006fb4e6fa4a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.894ex; height:2.343ex;" alt="{\displaystyle t_{0}}" loading="lazy"> is chosen based on features of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F}</annotation>

</semantics>

</math><img src="./545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy">. For example, for a sinusoid, a convenient choice is any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

</semantics>

</math><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"> where the function's value changes from zero to positive.

The formula above gives the phase as an angle in radians between 0 and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2\pi }</annotation>

</semantics>

</math><img src="./73efd1f6493490b058097060a572606d2c550a06.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.494ex; height:2.176ex;" alt="{\displaystyle 2\pi }" loading="lazy">. To get the phase as an angle between <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\pi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -\pi }</annotation>

</semantics>

</math><img src="./f2359073fe90a84a705e02f0c1e63b32df850a60.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.14ex; height:2.176ex;" alt="{\displaystyle -\pi }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +\pi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle +\pi }</annotation>

</semantics>

</math><img src="./cbae717a544919265768ad63c73919176113fb12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.14ex; height:2.176ex;" alt="{\displaystyle +\pi }" loading="lazy">, one uses instead

<mrow>

<mrow>

<mrow>

<mrow>

<mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi (t)=2\pi \left(\left[\!\!\left[{\frac {t-t_{0}}{T}}+{\frac {1}{2}}\right]\!\!\right]-{\frac {1}{2}}\right)}</annotation>

</semantics>

</math>

The phase expressed in degrees (from 0° to 360°, or from −180° to +180°) is defined the same way, except with "360°" in place of "2π".

<div class="mw-heading mw-heading3"><h3 id="Consequences">Consequences</h3></div>

With any of the above definitions, the phase <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (t)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi (t)}</annotation>

</semantics>

</math><img src="./cd7776ed1f48fe8d39d9852e6ed6aa8a61a93d28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.169ex; height:2.843ex;" alt="{\displaystyle \varphi (t)}" loading="lazy"> of a periodic signal is periodic too, with the same period <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T}</annotation>

</semantics>

</math><img src="./ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy">:

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi (t+T)=\varphi (t)\quad \quad {\text{ for all }}t.}</annotation>

</semantics>

</math>

The phase is zero at the start of each period; that is

<msub>

</mrow>

</msub>

<mtext> for any integer </mtext>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi (t_{0}+kT)=0\quad \quad {\text{ for any integer }}k.}</annotation>

</semantics>

</math>

Moreover, for any given choice of the origin <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t_{0}}</annotation>

</semantics>

</math><img src="./02d3006c4190b1939b04d9b9bb21006fb4e6fa4a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.894ex; height:2.343ex;" alt="{\displaystyle t_{0}}" loading="lazy">, the value of the signal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F}</annotation>

</semantics>

</math><img src="./545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"> for any argument <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

</semantics>

</math><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"> depends only on its phase at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

</semantics>

</math><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy">. Namely, one can write <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(t)=f(\varphi (t))}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F(t)=f(\varphi (t))}</annotation>

</semantics>

</math><img src="./37bff0d2f17879f532070c0560b69190c72cf10a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.745ex; height:2.843ex;" alt="{\displaystyle F(t)=f(\varphi (t))}" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> is a function of an angle, defined only for a single full turn, that describes the variation of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F}</annotation>

</semantics>

</math><img src="./545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"> as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

</semantics>

</math><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"> ranges over a single period.

In fact, every periodic signal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F}</annotation>

</semantics>

</math><img src="./545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"> with a specific <a href="Waveform" title="Waveform">waveform</a> can be expressed as

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F(t)=A\,w(\varphi (t))}</annotation>

</semantics>

</math>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle w}</annotation>

</semantics>

</math><img src="./88b1e0c8e1be5ebe69d18a8010676fa42d7961e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.664ex; height:1.676ex;" alt="{\displaystyle w}" loading="lazy"> is a "canonical" function of a phase angle in 0 to 2π, that describes just one cycle of that waveform; and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> is a scaling factor for the amplitude. (This claim assumes that the starting time <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t_{0}}</annotation>

</semantics>

</math><img src="./02d3006c4190b1939b04d9b9bb21006fb4e6fa4a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.894ex; height:2.343ex;" alt="{\displaystyle t_{0}}" loading="lazy"> chosen to compute the phase of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F}</annotation>

</semantics>

</math><img src="./545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"> corresponds to argument 0 of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle w}</annotation>

</semantics>

</math><img src="./88b1e0c8e1be5ebe69d18a8010676fa42d7961e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.664ex; height:1.676ex;" alt="{\displaystyle w}" loading="lazy">.)

<div class="mw-heading mw-heading2"><h2 id="Adding_and_comparing_phases">Adding and comparing phases</h2></div>

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

<mrow>

<mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

<mrow>

<mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 360\,\left[\!\!\left[{\frac {\alpha +\beta }{360}}\right]\!\!\right]\quad \quad {\text{ and }}\quad \quad 360\,\left[\!\!\left[{\frac {\alpha -\beta }{360}}\right]\!\!\right]}</annotation>

</semantics>

</math>

respectively. Thus, for example, the sum of phase angles 190° + 200° is 30° (190 + 200 = 390, minus one full turn), and subtracting 50° from 30° gives a phase of 340° (30 − 50 = −20, plus one full turn).

Similar formulas hold for radians, with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2\pi }</annotation>

</semantics>

</math><img src="./73efd1f6493490b058097060a572606d2c550a06.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.494ex; height:2.176ex;" alt="{\displaystyle 2\pi }" loading="lazy"> instead of 360.

<div class="mw-heading mw-heading2"><h2 id="Phase_shift">Phase shift </h2></div>

The difference <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (t)=\varphi _{G}(t)-\varphi _{F}(t)}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi (t)=\varphi _{G}(t)-\varphi _{F}(t)}</annotation>

</semantics>

</math><img src="./bf955e90c71d7b9a643ffd898dd9b34ab6856551.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.433ex; height:2.843ex;" alt="{\displaystyle \varphi (t)=\varphi _{G}(t)-\varphi _{F}(t)}" loading="lazy"> between the phases of two periodic signals <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F}</annotation>

</semantics>

</math><img src="./545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> is called the phase difference or phase shift of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> relative to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F}</annotation>

</semantics>

</math><img src="./545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy">.<a href="#cite_note-Ballou2005-1">[1]</a> At values of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

</semantics>

</math><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"> when the difference is zero, the two signals are said to be in phase; otherwise, they are out of phase with each other.

In the clock analogy, each signal is represented by a hand (or pointer) of the same clock, both turning at constant but possibly different speeds. The phase difference is then the angle between the two hands, measured clockwise.

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. This is usually the case in <a href="Linear_algebra" title="Linear algebra">linear</a> systems, when the <a href="Superposition_principle" title="Superposition principle">superposition principle</a> holds.

For arguments <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

</semantics>

</math><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"> when the phase difference is zero, the two signals will have the same sign and will be reinforcing each other. One says that <a href="Interference_(wave_propagation)" class="mw-redirect" title="Interference (wave propagation)">constructive interference</a> is occurring. At arguments <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

</semantics>

</math><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"> when the phases are different, the value of the sum depends on the waveform.

<div class="mw-heading mw-heading3"><h3 id="For_sinusoids">For sinusoids</h3></div>

For sinusoidal signals, when the phase difference <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (t)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi (t)}</annotation>

</semantics>

</math><img src="./cd7776ed1f48fe8d39d9852e6ed6aa8a61a93d28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.169ex; height:2.843ex;" alt="{\displaystyle \varphi (t)}" loading="lazy"> is 180° (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi }</annotation>

</semantics>

</math><img src="./9be4ba0bb8df3af72e90a0535fabcc17431e540a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }" loading="lazy"> radians), one says that the phases are opposite, and that the signals are in antiphase. Then the signals have opposite signs, and <a href="Interference_(wave_propagation)" class="mw-redirect" title="Interference (wave propagation)">destructive interference</a> occurs.

Conversely, a phase reversal or phase inversion implies a 180-degree phase shift.<a href="#cite_note-2">[2]</a>

When the phase difference <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (t)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi (t)}</annotation>

</semantics>

</math><img src="./cd7776ed1f48fe8d39d9852e6ed6aa8a61a93d28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.169ex; height:2.843ex;" alt="{\displaystyle \varphi (t)}" loading="lazy"> is a quarter of turn (a right angle, +90° = π/2 or −90° = 270° = −π/2 = 3π/2), sinusoidal signals are sometimes said to be in quadrature, e.g., <a href="In-phase_and_quadrature_components" title="In-phase and quadrature components">in-phase and quadrature components</a> of a composite signal or even different signals (e.g., voltage and current).

If the frequencies are different, the phase difference <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (t)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi (t)}</annotation>

</semantics>

</math><img src="./cd7776ed1f48fe8d39d9852e6ed6aa8a61a93d28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.169ex; height:2.843ex;" alt="{\displaystyle \varphi (t)}" loading="lazy"> increases linearly with the argument <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

</semantics>

</math><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy">. The periodic changes from reinforcement and opposition cause a phenomenon called <a href="Beat_(acoustics)" title="Beat (acoustics)">beating</a>.

<div class="mw-heading mw-heading3"><h3 id="For_shifted_signals">For shifted signals</h3></div>

The phase difference is especially important when comparing a periodic signal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F}</annotation>

</semantics>

</math><img src="./545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"> with a shifted and possibly scaled version <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> of it. That is, suppose that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G(t)=\alpha \,F(t+\tau )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G(t)=\alpha \,F(t+\tau )}</annotation>

</semantics>

</math><img src="./808875860e58d0a88eecd0ebbb594becdc5bc078.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.881ex; height:2.843ex;" alt="{\displaystyle G(t)=\alpha \,F(t+\tau )}" loading="lazy"> for some constants <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ,\tau }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha ,\tau }</annotation>

</semantics>

</math><img src="./9b01038dc9a7502139bb0657ef89468c32145f6a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.723ex; height:2.009ex;" alt="{\displaystyle \alpha ,\tau }" loading="lazy"> and all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

</semantics>

</math><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy">. Suppose also that the origin for computing the phase of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> has been shifted too. In that case, the phase difference <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation>

</semantics>

</math><img src="./33ee699558d09cf9d653f6351f9fda0b2f4aaa3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" loading="lazy"> is a constant (independent of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

</semantics>

</math><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy">), called the 'phase shift' or 'phase offset' of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> relative to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F}</annotation>

</semantics>

</math><img src="./545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy">. 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 this case, the phase shift is simply the argument shift <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tau }</annotation>

</semantics>

</math><img src="./38a7dcde9730ef0853809fefc18d88771f95206c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }" loading="lazy">, expressed as a fraction of the common period <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T}</annotation>

</semantics>

</math><img src="./ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy"> (in terms of the <a href="Modulo_operation" class="mw-redirect" title="Modulo operation">modulo operation</a>) of the two signals and then scaled to a full turn:

<mrow>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi =2\pi \left[\!\!\left[{\frac {\tau }{T}}\right]\!\!\right].}</annotation>

</semantics>

</math>

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F}</annotation>

</semantics>

</math><img src="./545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"> is a "canonical" representative for a class of signals, like <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(t)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sin(t)}</annotation>

</semantics>

</math><img src="./0eb63cccca21b1edef50f707888e3204ab5fda1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.505ex; height:2.843ex;" alt="{\displaystyle \sin(t)}" loading="lazy"> is for all sinusoidal signals, then the phase shift <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation>

</semantics>

</math><img src="./33ee699558d09cf9d653f6351f9fda0b2f4aaa3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" loading="lazy"> called simply the initial phase of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">.

Therefore, when two periodic signals have the same frequency, they are always in phase, or always out of phase. Physically, this situation commonly occurs, for many reasons. 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 <a href="Radio" title="Radio">radio</a> signal that reaches the receiving antenna in a straight line, and a copy of it that was reflected off a large building nearby.

A well-known example of phase difference is the length of shadows seen at different points of Earth. To a first approximation, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(t)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F(t)}</annotation>

</semantics>

</math><img src="./5b57ed3bbf501fb3c7f4bc5c4eafa96bb9e32165.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.39ex; height:2.843ex;" alt="{\displaystyle F(t)}" loading="lazy"> is the length seen at time <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

</semantics>

</math><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"> at one spot, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> is the length seen at the same time at a <a href="Longitude" title="Longitude">longitude</a> 30° west of that point, then the phase difference between the two signals will be 30° (assuming that, in each signal, each period starts when the shadow is shortest).

<div class="mw-heading mw-heading3"><h3 id="For_sinusoids_with_same_frequency">For sinusoids with same frequency</h3></div>

For sinusoidal signals (and a few other waveforms, like square or symmetric triangular), a phase shift of 180° is equivalent to a phase shift of 0° with negation of the amplitude. When two signals with these waveforms, same period, and opposite phases are added together, the sum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F+G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F+G}</annotation>

</semantics>

</math><img src="./cbdfe6e6306ae2363bd55c55e11a4341a3fbe98a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.408ex; height:2.343ex;" alt="{\displaystyle F+G}" loading="lazy"> is either identically zero, or is a sinusoidal signal with the same period and phase, whose amplitude is the difference of the original amplitudes.

The phase shift of the co-sine function relative to the sine function is +90°. It follows that, for two sinusoidal signals <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F}</annotation>

</semantics>

</math><img src="./545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> with same frequency and amplitudes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B}</annotation>

</semantics>

</math><img src="./47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy">, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> has phase shift +90° relative to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F}</annotation>

</semantics>

</math><img src="./545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy">, the sum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F+G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F+G}</annotation>

</semantics>

</math><img src="./cbdfe6e6306ae2363bd55c55e11a4341a3fbe98a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.408ex; height:2.343ex;" alt="{\displaystyle F+G}" loading="lazy"> is a sinusoidal signal with the same frequency, with amplitude <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C}</annotation>

</semantics>

</math><img src="./4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"> and phase shift <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -90^{\circ }<\varphi <+90^{\circ }}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -90^{\circ }<\varphi <+90^{\circ }}</annotation>

</semantics>

</math><img src="./28c62b7a8ac1e193ca18465831fc25d9fcf70477.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.092ex; height:2.843ex;" alt="{\displaystyle -90^{\circ }<\varphi <+90^{\circ }}" loading="lazy"> from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F}</annotation>

</semantics>

</math><img src="./545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy">, such that

<msqrt>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C={\sqrt {A^{2}+B^{2}}}\quad \quad {\text{ and }}\quad \quad \sin(\varphi )=B/C.}</annotation>

</semantics>

</math>

A real-world example of a sonic phase difference occurs in the <a href="Native_American_flute#The_Warble" title="Native American flute">warble of a Native American flute</a>. The amplitude of different <a href="Harmonics" class="mw-redirect" title="Harmonics">harmonic components</a> of same long-held note on the flute come into dominance at different points in the phase cycle. The phase difference between the different harmonics can be observed on a <a href="Spectrogram" title="Spectrogram">spectrogram</a> of the sound of a warbling flute.<a href="#cite_note-4">[4]</a>

<div class="mw-heading mw-heading2"><h2 id="Phase_comparison">Phase comparison</h2></div>

Phase comparison is a comparison of the phase of two waveforms, usually of the same nominal frequency. In time and frequency, the purpose of a phase comparison is generally to determine the frequency offset (difference between signal cycles) with respect to a reference.<a href="#cite_note-phnist-3">[3]</a>

A phase comparison can be made by connecting two signals to a <a href="Oscilloscope" title="Oscilloscope">two-channel oscilloscope</a>. The oscilloscope will display two sine signals, as shown in the graphic to the right. In the adjacent image, the top sine signal is the <a href="Frequency" title="Frequency">test frequency</a>, and the bottom sine signal represents a signal from the reference.

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.<a href="#cite_note-phnist-3">[3]</a>

<div class="mw-heading mw-heading2"><h2 id="Formula_for_phase_of_an_oscillation_or_a_periodic_signal">Formula for phase of an oscillation or a periodic signal</h2></div>

The phase of a <a href="Simple_harmonic_motion" title="Simple harmonic motion">simple harmonic oscillation</a> or <a href="Sine" class="mw-redirect" title="Sine">sinusoidal signal</a> is the value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \varphi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \varphi }</annotation>

</semantics>

</math><img src="./99015519246670af1cb5592e439ad64a27fb4830.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\textstyle \varphi }" loading="lazy"> in the following functions:

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}x(t)&=A\cos(2\pi ft+\varphi )\\y(t)&=A\sin(2\pi ft+\varphi )=A\cos \left(2\pi ft+\varphi -{\tfrac {\pi }{2}}\right)\end{aligned}}}</annotation>

</semantics>

</math>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle A}</annotation>

</semantics>

</math><img src="./a118c6ad00742b3f5dccd2f0e74b5e369df6fd31.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\textstyle A}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle f}</annotation>

</semantics>

</math><img src="./e1b77076edca76caf3331d0551d1645b8f678283.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\textstyle f}" loading="lazy">, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \varphi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \varphi }</annotation>

</semantics>

</math><img src="./99015519246670af1cb5592e439ad64a27fb4830.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\textstyle \varphi }" loading="lazy"> are constant parameters called the amplitude, frequency, and phase of the sinusoid. These signals are periodic with period <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle T={\frac {1}{f}}}">

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle T={\frac {1}{f}}}</annotation>

</semantics>

</math><img src="./30764aaa3a9b1baff598f62e7889a2e09003c349.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:6.475ex; height:4.009ex;" alt="{\textstyle T={\frac {1}{f}}}" loading="lazy">, and they are identical except for a displacement of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {T}{4}}}">

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle {\frac {T}{4}}}</annotation>

</semantics>

</math><img src="./f5622bb7af934e590446c570ab2fcc4a68372cc9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:1.993ex; height:3.509ex;" alt="{\textstyle {\frac {T}{4}}}" loading="lazy"> along the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle t}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle t}</annotation>

</semantics>

</math><img src="./b2bc926f90178739fccd01a96c6fa778ab3535d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\textstyle t}" loading="lazy"> axis. The term phase can refer to several different things:

<ul><li>It can refer to a specified reference, such as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \cos(2\pi ft)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \cos(2\pi ft)}</annotation>

</semantics>

</math><img src="./6bb5b6e4add551c8a9b35e50b0ad2b921098daae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.533ex; height:2.843ex;" alt="{\textstyle \cos(2\pi ft)}" loading="lazy">, in which case we would say the phase of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x(t)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle x(t)}</annotation>

</semantics>

</math><img src="./a57de678103d8f8a5493032d13d8bc36ce6f7efe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.979ex; height:2.843ex;" alt="{\textstyle x(t)}" loading="lazy"> is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \varphi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \varphi }</annotation>

</semantics>

</math><img src="./99015519246670af1cb5592e439ad64a27fb4830.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\textstyle \varphi }" loading="lazy">, and the phase of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle y(t)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle y(t)}</annotation>

</semantics>

</math><img src="./ae51d741938ad306415a19662048397457005849.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.804ex; height:2.843ex;" alt="{\textstyle y(t)}" loading="lazy"> is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \varphi -{\frac {\pi }{2}}}">

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \varphi -{\frac {\pi }{2}}}</annotation>

</semantics>

</math><img src="./8ab8e457474687213d3e488582594b05c5358b95.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:6.139ex; height:3.176ex;" alt="{\textstyle \varphi -{\frac {\pi }{2}}}" loading="lazy">.</li>

<li>It can refer to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \varphi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \varphi }</annotation>

</semantics>

</math><img src="./99015519246670af1cb5592e439ad64a27fb4830.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\textstyle \varphi }" loading="lazy">, in which case we would say <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x(t)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle x(t)}</annotation>

</semantics>

</math><img src="./a57de678103d8f8a5493032d13d8bc36ce6f7efe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.979ex; height:2.843ex;" alt="{\textstyle x(t)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle y(t)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle y(t)}</annotation>

</semantics>

</math><img src="./ae51d741938ad306415a19662048397457005849.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.804ex; height:2.843ex;" alt="{\textstyle y(t)}" loading="lazy"> have the same phase but are relative to their own specific references.</li>

<li>In the context of communication waveforms, the time-variant angle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle 2\pi ft+\varphi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle 2\pi ft+\varphi }</annotation>

</semantics>

</math><img src="./8e2d9e844db7bd940b93990b919b8e31041ba25d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.973ex; height:2.676ex;" alt="{\textstyle 2\pi ft+\varphi }" loading="lazy">, or its <a href="Principal_value" title="Principal value">principal value</a>, is referred to as <a href="Instantaneous_phase" class="mw-redirect" title="Instantaneous phase">instantaneous phase</a>, often just phase.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Absolute_phase">Absolute phase</h2></div>

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</style><div role="note" class="hatnote navigation-not-searchable dablink excerpt-hat selfref">This section is an excerpt from <a href="Absolute_phase" title="Absolute phase">Absolute phase</a>.[<a class="external text external" href="https://en.wikipedia.org/w/index.php?title=Absolute_phase&action=edit">edit</a>]</div><div class="excerpt">

<a href="Absolute_phase" title="Absolute phase">Absolute phase</a> is the phase of a <a href="Waveform" title="Waveform">waveform</a> relative to some standard (strictly speaking, phase is always relative). To the extent that this standard is accepted by all parties, one can speak of an absolute phase in a particular field of application.</div></div>

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<ul><li><a href="Absolute_phase" title="Absolute phase">Absolute phase</a></li>

<li><a href="AC_phase" class="mw-redirect" title="AC phase">AC phase</a></li>

<li><a href="In-phase_and_quadrature_components" title="In-phase and quadrature components">In-phase and quadrature components</a></li>

<li><a href="Instantaneous_phase" class="mw-redirect" title="Instantaneous phase">Instantaneous phase</a></li>

<li><a href="Lissajous_curve" title="Lissajous curve">Lissajous curve</a></li>

<li><a href="Phase_cancellation" class="mw-redirect" title="Phase cancellation">Phase cancellation</a></li>

<li><a href="Phase_problem" title="Phase problem">Phase problem</a></li>

<li><a href="Phase_spectrum" class="mw-redirect" title="Phase spectrum">Phase spectrum</a></li>

<li><a href="Phase_velocity" title="Phase velocity">Phase velocity</a></li>

<li><a href="Phasor" title="Phasor">Phasor</a></li>

<li><a href="Polarization_(waves)" title="Polarization (waves)">Polarization (waves)</a></li>

<li><a href="Coherence_(physics)" title="Coherence (physics)">Coherence (physics)</a>, the quality of a wave to display a well defined phase relationship in different regions of its domain of definition</li>

<li><a href="Hilbert_transform" title="Hilbert transform">Hilbert transform</a>, a method of changing phase by 90°</li>

<li><a href="Reflection_phase_shift" class="mw-redirect" title="Reflection phase shift">Reflection phase shift</a>, a phase change that happens when a wave is reflected off of a boundary from fast medium to slow medium</li></ul>

</div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFBallou2005" class="citation book cs1">Ballou, Glen (2005). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=y0d9VA0lkogC&pg=PA1499">Handbook for sound engineers</a> (3 ed.). Focal Press, Gulf Professional Publishing. p. 1499. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-240-80758-4</bdi>.</cite>

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.its.bldrdoc.gov/fs-1037/fs-1037c.htm">"Federal Standard 1037C: Glossary of Telecommunications Terms"</a>.</cite>

</li>

<cite id="CITEREFTime_and_Frequency_from_A_to_Z2010" class="citation journal cs1">Time and Frequency from A to Z (2010-05-12). <a rel="nofollow" class="external text" href="https://www.nist.gov/pml/div688/grp40/enc-p.cfm">"Phase"</a>. NIST. <a href="National_Institute_of_Standards_and_Technology" title="National Institute of Standards and Technology">National Institute of Standards and Technology</a> (NIST). Retrieved 12 June 2016.</cite> This content has been copied and pasted from an NIST web page and is in the public domain.

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> <cite id="CITEREFClint_GossBarry_Higgins2013" class="citation web cs1">Clint Goss; Barry Higgins (2013). <a rel="nofollow" class="external text" href="http://Flutopedia.com/warble.htm">"The Warble"</a>. Flutopedia. Retrieved 2013-03-06.</cite>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

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<div class="side-box-text plainlist">Wikimedia Commons has media related to <a href="https://commons.wikimedia.org/wiki/Category:Phase_(waves)" class="extiw external" title="commons:Category:Phase (waves)">Phase (waves)</a>.</div></div>

</div>

<ul><li>"<a rel="nofollow" class="external text" href="http://www.indiana.edu/~emusic/acoustics/phase.htm">What is a phase?</a>". Prof. Jeffrey Hass. "An Acoustics Primer", Section 8. <a href="Indiana_University" title="Indiana University">Indiana University</a>, 2003. See also: (<a rel="nofollow" class="external text" href="http://www.indiana.edu/~emusic/etext/acoustics/chapter1_phase.shtml">pages 1 thru 3</a>, 2013)</li>

<li><a rel="nofollow" class="external text" href="http://www.sengpielaudio.com/calculator-timedelayphase.htm">Phase angle, phase difference, time delay, and frequency</a></li>

<li><a rel="nofollow" class="external text" href="http://www.tedpavlic.com/teaching/osu/ece209/lab3_opamp_FO/lab3_opamp_FO_phase_shift.pdf">ECE 209: Sources of Phase Shift</a> — Discusses the time-domain sources of phase shift in simple linear time-invariant circuits.</li>

<li><a rel="nofollow" class="external text" href="https://iwant2study.org/ospsg/index.php/interactive-resources/physics/04-waves/02-general-waves/113-transverse-wave-model">Open Source Physics JavaScript HTML5</a></li>

<li><a rel="nofollow" class="external text" href="http://phy.hk/wiki/englishhtm/phase.htm">Phase Difference</a> Java Applet</li></ul></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-02-27" href="https://en.wikipedia.org/wiki/?title=Phase_(waves)&oldid=1277873292">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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