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<title>Pendulum wave</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Pendulum wave</span></span>

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</style><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:292px;max-width:292px"><div class="trow"><div class="tsingle" style="width:164px;max-width:164px"><div class="thumbimage" style="height:243px;overflow:hidden"><span typeof="mw:File"><a href="https://upload.wikimedia.org/wikipedia/commons/5/5c/Pendulum_wave_animation.svg" class="external"></a></span></div><div class="thumbcaption text-align-center">Front view<br><a class="external text external" href="https://upload.wikimedia.org/wikipedia/commons/b/b8/Pendulum_wave_animation_half_speed.svg">(half-speed)</a></div></div><div class="tsingle" style="width:124px;max-width:124px"><div class="thumbimage" style="height:243px;overflow:hidden"><span typeof="mw:File"><a href="https://upload.wikimedia.org/wikipedia/commons/4/4a/Pendulum_wave_top_animation.svg" class="external"></a></span></div><div class="thumbcaption text-align-center">Top view<br><a class="external text external" href="https://upload.wikimedia.org/wikipedia/commons/a/a4/Pendulum_wave_top_animation_half_speed.svg">(half-speed)</a></div></div></div><div class="trow" style="display:flex"><div class="thumbcaption"><a href="SVG_animation" title="SVG animation">SVG animation</a> of a pendulum wave with 12 pendulums, the lowest pendulum making 60 oscillations in one minute, the next 61, and so forth – in the animations, tap or hover over a pendulum to pause</div></div></div></div>

<p>A <b>pendulum wave</b> is an elementary physics demonstration and <a href="Kinetic_art" title="Kinetic art">kinetic art</a> comprising a number of uncoupled <a href="Simple_pendulum" class="mw-redirect" title="Simple pendulum">simple pendulums</a> with <a href="Monotonic" class="mw-redirect" title="Monotonic">monotonically</a> increasing lengths. As the pendulums oscillate, they appear to produce travelling and <a href="Standing_wave" title="Standing wave">standing waves</a>, <a href="Beat_(acoustics)" title="Beat (acoustics)">beating</a>, and random motion.<sup id="cite_ref-harvard_1-0" class="reference"><a href="#cite_note-harvard-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-3" class="reference"><a href="#cite_note-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="History">History</h2></div>

<p><a href="Ernst_Mach" title="Ernst Mach">Ernst Mach</a> designed and constructed the first pendulum wave demonstration around 1867 at <a href="Charles-Ferdinand_University" class="mw-redirect" title="Charles-Ferdinand University">Charles-Ferdinand University</a> in Prague. In the <a href="Czech_Republic" title="Czech Republic">Czech Republic</a>, the demonstration is called Mach's wave machine. <a href="Eric_J._Heller" title="Eric J. Heller">Eric J. Heller</a> at <a href="Harvard_University" title="Harvard University">Harvard University</a> suggested the use of the demonstration to simulate quantum revival.<sup id="cite_ref-harvard_1-1" class="reference"><a href="#cite_note-harvard-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

</p><p>In 2001, two <a href="University_of_Minnesota_Morris" title="University of Minnesota Morris">University of Minnesota Morris</a> researchers have derived a continuous function explaining the patterns in the pendulums using an extension to the equation for traveling waves in one dimension, and showed that their cycling arises from aliasing of the underlying continuous function.<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>

</p><p>In 2020, <a href="Illusionist" class="mw-redirect" title="Illusionist">illusionist</a> Kevin McMahon, incorporated a massive pendulum wave apparatus, supposedly with flaming <a href="Cannonball" class="mw-redirect" title="Cannonball">cannonballs</a>, as a stunt in <a href="Britain's_Got_Talent_(series_14)" class="mw-redirect" title="Britain's Got Talent (series 14)">Britain's Got Talent (series 14)</a> under the stage name Kevin Quantum.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>

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

<p>The lengths of the pendulums are set such that in a given time <i>t</i>, the first pendulum completes <i>n</i> oscillations, and each subsequent one completes one more oscillation than the previous. As all pendulums are started together, their relative phases change continuously, but after time <i>t</i>, they come back in sync and the sequence repeats.<sup id="cite_ref-harvard_1-2" class="reference"><a href="#cite_note-harvard-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

</p><p>For small perturbations, the <a href="Frequency#Definitions_and_units" title="Frequency">period</a> of a pendulum is given by

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<annotation encoding="application/x-tex">{\displaystyle T=2\pi {\sqrt {\frac {L}{g}}}}</annotation>

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</math></span><img src="./9a126887719a048aba2314019cef7f7822b6892d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:11.972ex; height:7.509ex;" alt="{\displaystyle T=2\pi {\sqrt {\frac {L}{g}}}}" loading="lazy"></span></dd></dl>

<p>where <i>L</i> is the length of the pendulum and <i>g</i> is the <a href="Standard_gravity" title="Standard gravity">standard acceleration due to gravity</a>.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup>

</p><p>As <style data-mw-deduplicate="TemplateStyles:r1214402035">

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</style><span class="sfrac">⁠<span class="tion"><span class="num"><i>t</i></span><span class="sr-only">/</span><span class="den"><i>n</i></span></span>⁠</span> is the period of a pendulum completing <i>n</i> oscillations in <i>t</i>,

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {t}{n}}&=2\pi {\sqrt {\frac {L}{g}}}\\\therefore L&=g{\Big (}{\frac {t}{2\pi n}}{\Big )}^{2}\\\end{aligned}}}</annotation>

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</math></span><img src="./ab2225724c5247244332c57e75a3c5e3bf82bf0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.701ex; margin-bottom: -0.303ex; width:17.299ex; height:13.176ex;" alt="{\displaystyle {\begin{aligned}{\frac {t}{n}}&=2\pi {\sqrt {\frac {L}{g}}}\\\therefore L&=g{\Big (}{\frac {t}{2\pi n}}{\Big )}^{2}\\\end{aligned}}}" loading="lazy"></span></dd></dl>

<p>A common choice of <i>t</i> is 60 seconds. Thus, for <i>g</i> ≈ 9.8 ms<sup>−2</sup>,

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<annotation encoding="application/x-tex">{\displaystyle L\approx {\frac {894}{n^{2}}}\;{\text{m}}}</annotation>

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</math></span><img src="./da9e31e4cc05be1f94e51bbc7bd9444e39b7f7d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:11.586ex; height:5.509ex;" alt="{\displaystyle L\approx {\frac {894}{n^{2}}}\;{\text{m}}}" loading="lazy"></span></dd></dl>

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<p>Parameters of the<br>pendulum wave in<br>the animation above

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<div class="thumb tnone" style="margin-left:auto;margin-right:auto;overflow:hidden;width:auto;max-width:1210px"><div class="thumbinner"><div class="noresize thumbimage" style="overflow:auto"><span typeof="mw:File"></span></div><div class="thumbcaption">Timeline of the pendulum wave in the animation above</div></div></div>

<ul><li><a href="Newton's_cradle" title="Newton's cradle">Newton's cradle</a> – a set of pendulums constrained to swing along the axis of the apparatus and collide with one another</li></ul>

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

<li id="cite_note-harvard-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-harvard_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-harvard_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-harvard_1-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text">Harvard Natural Sciences Lecture Demonstrations, <a rel="nofollow" class="external text" href="http://sciencedemonstrations.fas.harvard.edu/presentations/pendulum-waves"><i>Pendulum Waves</i></a></span>

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<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">K P Zetie, <a rel="nofollow" class="external text" href="http://iopscience.iop.org/article/10.1088/0031-9120/50/3/285"><i>The pendulum wave machine</i></a>, Physics Education, Volume 50, Number 3, 23 April 2015</span>

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<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">Fyzmatik.pise, <a rel="nofollow" class="external text" href="http://fyzmatik.pise.cz/1331-machuv-vlnostroj.html"><i>Machův vlnostroj</i></a>, 16. červen 2012</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">James A. Flaten and Kevin A. Parendo, <a rel="nofollow" class="external text" href="http://phys.ufl.edu/~mocko/Pendulum_Wave/AJP000778PendulumWaves2(1).pdf"><i>Pendulum waves: A lesson in aliasing</i></a>, <a rel="nofollow" class="external text" href="http://aapt.scitation.org/doi/abs/10.1119/1.1349543">American Journal of Physics 69, 778 (2001)</a></span>

</li>

<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text">BBC News, <a rel="nofollow" class="external text" href="https://bbc.com/news/uk-scotland-edinburgh-east-fife-63991330#piano-inline3"><i>The Faking It magician who now teaches magic</i></a>, 18 December 2020</span>

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</style><cite id="CITEREFHallidayRobert_ResnickJearl_Walker1997" class="citation book cs1">Halliday, David; Robert Resnick; Jearl Walker (1997). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/fundamentalsofp000davi/page/381"><i>Fundamentals of Physics, 5th Ed</i></a></span>. New York: John Wiley & Sons. p. <a rel="nofollow" class="external text" href="https://archive.org/details/fundamentalsofp000davi/page/381">381</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-471-14854-8</bdi>.</cite></span>

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