 Cutoff frequency of a membrane waveguide

The cutoff frequency of a membrane waveguide is the lowest frequency at which a specified transverse normal mode can propagate longitudinally. For fixed edges separated by $b$, ${\omega }_{c,n}=n\pi v/b$ and $f_{c,n}={\omega }_{c,n}/(2\pi )$. Below cutoff the longitudinal wavenumber is imaginary and a bounded excitation is an evanescent wave. Above it, $v_g=v\sqrt{1-({\omega }_{c,n}/\omega {)}^2}$ and $v_p{v}_g=v^2$.