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<title>Reflection phase change</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Reflection phase change</span></span>

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<p>A <a href="Phase_change_(waves)" class="mw-redirect" title="Phase change (waves)">phase change</a> sometimes occurs when a <a href="Wave" title="Wave">wave</a> is <a href="Reflection_(physics)" title="Reflection (physics)">reflected</a>, specifically from a medium with faster wave speed to the boundary of a medium with slower wave speed.<sup id="cite_ref-hypRPC_1-0" class="reference"><a href="#cite_note-hypRPC-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-hypSound_2-0" class="reference"><a href="#cite_note-hypSound-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Such reflections occur for many types of wave, including <a href="Light_wave" class="mw-redirect" title="Light wave">light waves</a>, <a href="Sound_wave" class="mw-redirect" title="Sound wave">sound waves</a>, and waves on vibrating strings.<sup id="cite_ref-Anim_3-0" class="reference"><a href="#cite_note-Anim-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading2"><h2 id="General_theory">General theory</h2></div>

<p>For an incident wave traveling from one medium (where the wave speed is <span class="texhtml"><i>c</i><sub>1</sub></span>) to another medium (where the wave speed is <span class="texhtml"><i>c</i><sub>2</sub></span>), one part of the wave will transmit into the second medium, while another part reflects back into the other direction and stays in the first medium. The amplitude of the transmitted wave and the reflected wave can be calculated by using the continuity condition at the boundary.

</p><p>Consider the component of the incident wave with an <a href="Angular_frequency" title="Angular frequency">angular frequency</a> of <span class="texhtml"><i>ω</i></span>, which has the waveform<span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u^{inc}(x,t)=Ae^{i(k_{1}x-\omega t)};\ A\in \mathbb {C} }">

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<annotation encoding="application/x-tex">{\displaystyle u^{inc}(x,t)=Ae^{i(k_{1}x-\omega t)};\ A\in \mathbb {C} }</annotation>

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</math></span></span>At t=0, the incident reaches the boundary between the two mediums at x=0. Therefore, the corresponding reflected wave and the transmitted wave will have the waveforms<span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u^{\mathrm {ref} }(x,t)=Be^{i(-k_{1}x-\omega t)};\ u^{\mathrm {trans} }(x,t)=Ce^{i(k_{2}x-\omega t)};\ B,C\in \mathbb {C} }">

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<annotation encoding="application/x-tex">{\displaystyle u^{\mathrm {ref} }(x,t)=Be^{i(-k_{1}x-\omega t)};\ u^{\mathrm {trans} }(x,t)=Ce^{i(k_{2}x-\omega t)};\ B,C\in \mathbb {C} }</annotation>

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</math></span></span>The continuity condition at the boundary is<span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u^{\mathrm {inc} }(0,t)+u^{\mathrm {ref} }(0,t)=u^{\mathrm {trans} }(0,t);\ {\frac {\partial }{\partial x}}u^{\mathrm {inc} }(0,t)+{\frac {\partial }{\partial x}}u^{\mathrm {ref} }(0,t)={\frac {\partial }{\partial x}}u^{\mathrm {trans} }(0,t)}">

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<annotation encoding="application/x-tex">{\displaystyle u^{\mathrm {inc} }(0,t)+u^{\mathrm {ref} }(0,t)=u^{\mathrm {trans} }(0,t);\ {\frac {\partial }{\partial x}}u^{\mathrm {inc} }(0,t)+{\frac {\partial }{\partial x}}u^{\mathrm {ref} }(0,t)={\frac {\partial }{\partial x}}u^{\mathrm {trans} }(0,t)}</annotation>

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</math></span></span>This gives the equations<span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A+B=C;\ A-B={\frac {k_{2}}{k_{1}}}C={\frac {c_{1}}{c_{2}}}C}">

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<annotation encoding="application/x-tex">{\displaystyle A+B=C;\ A-B={\frac {k_{2}}{k_{1}}}C={\frac {c_{1}}{c_{2}}}C}</annotation>

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</math></span></span>And we have the reflectivity and transmissivity<span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {B}{A}}={\frac {c_{2}-c_{1}}{c_{2}+c_{1}}};\ {\frac {C}{A}}={\frac {2c_{2}}{c_{2}+c_{1}}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\frac {B}{A}}={\frac {c_{2}-c_{1}}{c_{2}+c_{1}}};\ {\frac {C}{A}}={\frac {2c_{2}}{c_{2}+c_{1}}}}</annotation>

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</math></span></span>When <span class="texhtml"><i>c</i><sub>2</sub> < <i>c</i><sub>1</sub></span>, the reflected wave has a reflection phase change of 180°, since <span class="texhtml">B/A < 0</span>. The energy conservation can be verified by<span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {B^{2}}{c_{1}}}+{\frac {C^{2}}{c_{2}}}={\frac {A^{2}}{c_{1}}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\frac {B^{2}}{c_{1}}}+{\frac {C^{2}}{c_{2}}}={\frac {A^{2}}{c_{1}}}}</annotation>

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</math></span></span>The above discussion holds true for any component, regardless of its angular frequency of <span class="texhtml"><i>ω</i></span>.

</p><p>The limiting case of <span class="texhtml"><i>c</i><sub>2</sub> = 0</span> corresponds to a "fixed end" that doesn't move, whereas the limiting case of <span class="texhtml"><i>c</i><sub>2</sub> → ∞</span> corresponds to a "free end".

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

<p>Light waves change phase by 180° when they reflect from the surface of a <a href="Medium_(optics)" class="mw-redirect" title="Medium (optics)">medium</a> with higher <a href="Refractive_index" title="Refractive index">refractive index</a> than that of the medium in which they are travelling.<sup id="cite_ref-hypRPC_1-1" class="reference"><a href="#cite_note-hypRPC-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> A light wave travelling in air that is reflected by a glass barrier will undergo a 180° phase change, while light travelling in glass <i>will not</i> undergo a phase change if it is reflected by a boundary with air. For this reason, optical boundaries are normally specified as an <a href="Ordered_pair" title="Ordered pair">ordered pair</a> (air-glass, glass-air); indicating which material the light is moving out of, and in to, respectively.

</p><p>"Phase" here is the phase of the <a href="Electric_field" title="Electric field">electric field</a> oscillations, not the <a href="Magnetic_field" title="Magnetic field">magnetic field</a> oscillations (while the electric field will undergo 180° phase change, the magnetic field will undergo 0° phase change. Vice versa is true when reflection occurs at lower refractive index interface.)<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> Also, this is referring to near-<a href="Normal_(geometry)" title="Normal (geometry)">normal</a> incidence—for p-polarized light reflecting off glass at <i>glancing</i> angle, beyond the <a href="Brewster_angle" class="mw-redirect" title="Brewster angle">Brewster angle</a>, the phase change is 0°. The phase changes that take place upon reflection play an important part in <a href="Thin_film_interference" class="mw-redirect" title="Thin film interference">thin film interference</a>.

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<div class="mw-heading mw-heading2"><h2 id="Sound_waves">Sound waves</h2></div>

<p>Sound waves in a solid experience a phase reversal (a 180° change) when they reflect from a boundary with air.<sup id="cite_ref-hypSound_2-1" class="reference"><a href="#cite_note-hypSound-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Sound waves in air do not experience a phase change when they reflect from a solid, but they do exhibit a 180° change when reflecting from a region with lower <a href="Acoustic_impedance" title="Acoustic impedance">acoustic impedance</a>. An example of this is when a sound wave in a hollow tube encounters the open end of the tube. The phase change on reflection is important in the physics of <a href="Wind_instrument" title="Wind instrument">wind instruments</a>.

</p>

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

<p>A <a href="Vibrating_string" class="mw-redirect" title="Vibrating string">wave on a string</a> experiences a 180° phase change when it reflects from a point where the string is fixed.<sup id="cite_ref-hypSound_2-2" class="reference"><a href="#cite_note-hypSound-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Anim_3-1" class="reference"><a href="#cite_note-Anim-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Reflections from the free end of a string exhibit no phase change. The phase change when reflecting from a fixed point contributes to the formation of <a href="Standing_wave" title="Standing wave">standing waves</a> on strings, which produce the sound from <a href="Stringed_instrument" class="mw-redirect" title="Stringed instrument">stringed instruments</a>.

</p><p>The same 180° phase change happens when the wave traveling in a lighter string (lower linear <a href="Density" title="Density">mass density</a>) reflects off of the boundary of a heavier string (higher linear mass density). This happens because the heavier string doesn't respond as quickly to the tension force as the lighter string, and therefore the amplitude of the oscillation at the boundary point is less than the incoming wave. By the <a href="Superposition_principle" title="Superposition principle">superposition principle</a>, the reflected wave must cancel part of the incoming wave, and therefore it is phase shifted. Note that when the wave traveling in a heavier string reflects off of the boundary of a lighter string, since the boundary point has the freedom to move as quickly as possible, no such phase shift would occur in the reflected wave.

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<div class="mw-heading mw-heading2"><h2 id="Electrical_transmission_lines">Electrical transmission lines</h2></div>

<p><a href="Reflections_of_signals_on_conducting_lines" title="Reflections of signals on conducting lines">Reflections of signals on conducting lines</a> typically exhibit a phase change from the incident signal. There are two extreme cases of termination: short circuit (closed line), and open circuit (broken line). In both cases the full amplitude of the wave is reflected.

</p>

<dl><dt>short circuit</dt>

<dd>The voltage wave reflection on a line terminated with a short circuit is 180° phase shifted. This is analogous (by the <a href="Mobility_analogy" title="Mobility analogy">mobility analogy</a>) to a string where the end is fixed in position, or a sound wave in a tube with a blocked off end. The current wave, on the other hand, is not phase shifted.</dd>

<dt>broken / open line</dt>

<dd>A <a href="Transmission_line" title="Transmission line">transmission line</a> terminated with an open circuit is the <a href="Duality_(electrical_circuits)" title="Duality (electrical circuits)">dual</a> case; the voltage wave is shifted by 0° and the current wave is shifted by 180°.</dd>

<dt>reactive termination</dt>

<dd>A transmission line terminated with a pure <a href="Capacitance" title="Capacitance">capacitance</a> or <a href="Inductance" title="Inductance">inductance</a> will also give rise to a phase shifted wave at full amplitude. The voltage phase shift is given by<sup id="cite_ref-Bleaney_Bleaney2013_5-0" class="reference"><a href="#cite_note-Bleaney_Bleaney2013-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page: 275">: 275 </span></sup> <span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi =2\tan ^{-1}{Z_{0} \over X}}">

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<annotation encoding="application/x-tex">{\displaystyle \varphi =2\tan ^{-1}{Z_{0} \over X}}</annotation>

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</math></span></span> where

<ul><li><i>Z</i><sub>0</sub> is the <a href="Characteristic_impedance" title="Characteristic impedance">characteristic impedance</a> of the line</li>

<li><i>X</i> is the <a href="Electrical_reactance" title="Electrical reactance">reactance</a> of the inductance or capacitance, given respectively by <i>ωL</i> or <style data-mw-deduplicate="TemplateStyles:r1154941027">

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</style><span class="frac"><span class="num">−1</span>⁄<span class="den"><i>ωC</i></span></span></li>

<li><i>L</i> and <i>C</i> are, respectively, inductance and capacitance, and</li>

<li><i>ω</i> is the <a href="Angular_frequency" title="Angular frequency">angular frequency</a>.</li></ul></dd></dl>

<p>In the case of reactive termination the phase shift will be between 0 and +180° for <a href="Inductor" title="Inductor">inductors</a> and between 0 and −180° for <a href="Capacitor" title="Capacitor">capacitors</a>. The phase shift will be exactly ±90° when <big>|</big><i>X</i><big>|</big> = <i>Z</i><sub>0</sub>.

</p><p>For the general case when the line is terminated with some arbitrary <a href="Electrical_impedance" title="Electrical impedance">impedance</a>, <i>Z</i>, the reflected wave is generally less than the incident wave. The full expression for phase shift needs to be used,<sup id="cite_ref-Bleaney_Bleaney2013_5-1" class="reference"><a href="#cite_note-Bleaney_Bleaney2013-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 273">: 273 </span></sup>

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<annotation encoding="application/x-tex">{\displaystyle \varphi =\tan ^{-1}\left({\frac {2\sin(\arg Z)}{\left({\frac {|Z|}{Z_{0}}}-{\frac {Z_{0}}{|Z|}}\right)}}\right)}</annotation>

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</p><p>This expression assumes the characteristic impedance is purely <a href="Resistive" class="mw-redirect" title="Resistive">resistive</a>.

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<ul><li><a href="Reflection_coefficient" title="Reflection coefficient">Reflection coefficient</a></li></ul>

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

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</style><cite id="CITEREFNave" class="citation web cs1">Nave, C.R. <a rel="nofollow" class="external text" href="http://hyperphysics.phy-astr.gsu.edu/hbase/phyopt/interf.html#c2">"Reflection Phase Change"</a>. <i>Hyperphysics</i>. Georgia State University<span class="reference-accessdate">. Retrieved <span class="nowrap">2016-03-28</span></span>.</cite></span>

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<li id="cite_note-hypSound-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-hypSound_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-hypSound_2-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-hypSound_2-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFNave" class="citation web cs1">Nave, C.R. <a rel="nofollow" class="external text" href="http://hyperphysics.phy-astr.gsu.edu/hbase/sound/reflec.html">"Reflection of Sound"</a>. <i>Hyperphysics</i>. Georgia State University<span class="reference-accessdate">. Retrieved <span class="nowrap">2016-03-28</span></span>.</cite></span>

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<li id="cite_note-Anim-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-Anim_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Anim_3-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFRussell" class="citation web cs1">Russell, Daniel A. <a rel="nofollow" class="external text" href="https://www.acs.psu.edu/drussell/Demos/reflect/reflect.html">"Reflection of Waves from Boundaries"</a>. <i>Graduate Program in Acoustics</i>. Pennsylvania State University<span class="reference-accessdate">. Retrieved <span class="nowrap">2021-05-12</span></span>.</cite></span>

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<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><cite id="CITEREFByrnes2016" class="citation arxiv cs1">Byrnes, Steven J. (2016). "Multilayer optical calculations". <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1603.02720">1603.02720</a></span> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/physics.comp-ph">physics.comp-ph</a>].</cite> Appendix A</span>

</li>

<li id="cite_note-Bleaney_Bleaney2013-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-Bleaney_Bleaney2013_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Bleaney_Bleaney2013_5-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFBleaneyBleaney2013" class="citation book cs1">Bleaney, B.I. & Bleaney, Brebis (2013). <i>Electricity and Magnetism</i>. Vol. 1. Oxford University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0199645428</bdi>.</cite></span>

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