Rectangular Wave Guides

$B_z=B_0\cos \left(\frac{m\pi x}{a}\right)\cos \left(\frac{n\pi y}{b}\right)$

• $m,n\in \mathbb{N}+\left\{0\right\}$
• $a\ge b$ which are both dimensions

${\omega }_{mn}\equiv c\pi \sqrt{{\left(\frac{m}{a}\right)}^2+{\left(\frac{n}{b}\right)}^2}$

• Cutoff frequency for which waves traveling at a lower frequency will exponentially decay whereas those at a higher frequency will travel in an oscillatory fashion
• $m$ corresponds to x and $a$, which is the longer length
• $n$ corresponds to $y$ and $b$, which is the shorter length

$k=\frac{1}{c}\sqrt{{\omega }^2-{\omega }_{mn}^2}$

• Wavenumber

$\omega =\sqrt{(ck{)}^2+({\omega }_{mn}{)}^2}$

• Angular frequency

$v=\frac{c}{\sqrt{1-{\left(\frac{{\omega }_{mn}}{\omega }\right)}^2}}$

• Phase velocity of a wave traveling through

$v_g=c\sqrt{1-(\frac{{\omega }_{mn}}{\omega }{)}^2}$

• Group velocity of a wave traveling through a rectangular wave guide