 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 $\sigma z_{tt}=T{\nabla }^{2}z-D_b{\nabla }^{4}z$. When its edge conditions admit the same sine transverse mode, bending stiffness in a membrane waveguide givesThus 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.