Ethan's Notes

    Rectangular Wave Guides

    #Physics

    $\displaystyle B_{z}=B_{0}\cos\left( \frac{m\pi x}{a} \right)\cos\left( \frac{n\pi y}{b} \right)$

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

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

    • 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
    • $\displaystyle m$ corresponds to x and $\displaystyle a$, which is the longer length
    • $\displaystyle n$ corresponds to $\displaystyle y$ and $\displaystyle b$, which is the shorter length

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

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

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

    $\displaystyle v_{g}=c\sqrt{ 1-(\frac{\omega_{mn}}{\omega})^{2} }$