= Solution
For the $n$th transverse mode, substitute $q_n=n\pi/a$ into the <membrane waveguide> <dispersion relation>:
$$
k^2=\frac{\sigma\omega^2}{T}-\left(\frac{n\pi}{a}\right)^2.
$$
Longitudinal propagation requires real nonzero $k$, hence
$$
\boxed{\omega>\omega_{c,n}=\frac{n\pi}{a}\sqrt{\frac T\sigma}.}
$$
Below the <cutoff frequency of a membrane waveguide>, $k=i\kappa$, with $\kappa=\sqrt{(n\pi/a)^2-\sigma\omega^2/T}$, and a bounded excitation decays as $e^{-\kappa x}$ rather than carrying a travelling wave along an infinite guide. At equality $k=0$, the displacement has no longitudinal phase advance and the <group velocity> is zero; it is a cutoff oscillation, not a propagating mode carrying finite longitudinal <power>.
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