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circular_plate_vibrations.html
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circular_plate_vibrations.html
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<!doctype html>
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<head>
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<meta charset="utf-8">
<title>Sam Bellows, Acoustician</title>
<meta name="description" content="Sam Bellows acoustics BYU leishman music acoustician bio contact research projects">
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<!--
<center><h2>Vibrations of Circular Plates</h2></center>
In polar coordinates, the eigenfunctions
<br><br>
<center>$\Psi_n(\rho,\phi) = J_n(k_{nm}\rho)\cos(n\phi - \gamma)$</center>
<br>
where $J_n$ are the Bessel functions of order $n$.
For Dirichlet boundary conditions at $\rho = a$, $\Psi_n(a, \phi) = 0$ and $k_{nm} = \alpha_{nm}/a$, where $\alpha_{nm}$ are the $m$th zero of $J_n$.
These eigenfunctions are used, for example, to describe the vibrations of a circular membrane like a drum head.
-->
<br><br>
<center><img src="./img/simply_supported_circular_plate.gif" width="350" height="350"/></center>
<center>Eigenfunctions for a simply supported circular plate.</center>
<br><br>
<!--
For Neumann boundary conditions at $\rho = a$, $\Psi_n^\prime(a, \phi) = 0$ and $k_{nm} = \alpha_{nm}^\prime/a$, where $\alpha_{nm}^\prime$ are the $m$th zero of $J_n^\prime$.
These eigenfunctions are used, for example, to describe the radiation of sound in a cylindrical duct.
-->
<br><br>
<center><img src="./img/clamped_circular_plate.gif" width="350" height="350"/></center>
<center>Eigenfunctions for a clamped circular plate.</center>
<br><br>
<br><br>
<center><img src="./img/free_circular_plate.gif" width="350" height="350"/></center>
<center>Eigenfunctions for a free circular plate.</center>
<br><br>
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