= Solution
Resistance to bending adds a restoring contribution that grows more rapidly with spatial curvature than the tension contribution. Short-<wavelength> modes are therefore stiffened especially strongly, raising their <frequencies> and changing the large-<wavenumber> behaviour.
For example, a thin plate of <flexural rigidity> $D_b$ has $\sigma z_{tt}=T\nabla^2z-D_b\nabla^4z$. When its edge conditions admit the same sine transverse mode, <bending stiffness in a membrane waveguide> gives
$$
\omega_n^2=\frac T\sigma(k^2+q_n^2)+\frac{D_b}\sigma(k^2+q_n^2)^2.
$$
Thus the cutoffs increase, and bending-dominated propagation has $\omega\propto k^2$ at large $|k|$, rather than approaching $c_m|k|$. Its <phase velocity> and <group velocity> then grow with $|k|$. If “rigidly fixed” means a genuinely clamped plate edge, the slope must also vanish and the mode shapes change; the qualitative stiffening and stronger <wave dispersion> remain, but simple $q_n=n\pi/a$ need not remain exact.
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