
</div>
<div class="section" id="Envelope-equation-for-round-and-elliptical-beams">
<h2>Envelope equation for round and elliptical beams<a class="headerlink" href="#Envelope-equation-for-round-and-elliptical-beams" title="Permalink to this headline">¶</a></h2>
<p>Given that:</p>
<div class="math notranslate nohighlight">
\[\dot r = \beta c r ',\]</div>
<div class="math notranslate nohighlight">
\[\ddot r = r '' (\dot z) ^ 2 + r '\ddot z \approx r' '\beta ^ 2 c ^ 2 + r' \beta '\beta c ^ 2.\]</div>
<p>And al