Bending stiffness in a membrane waveguide

For a plate under uniform tension per length $T$, with surface mass density $\sigma$ and flexural rigidity $D_b>0$, the small-displacement equation is $\sigma z_{tt}=T{\nabla }^{2}z-D_b{\nabla }^{4}z$. If the edge conditions admit a transverse sine mode of wavenumber $q_n$, its longitudinal dispersion relation isBending raises mode cutoff frequencies of a membrane waveguide and changes large-$k$ behaviour from $\omega \propto |k|$ to $\omega \propto k^2$. A genuinely clamped plate also fixes the edge slope, so its transverse mode shapes need not remain simple sine functions. The qualitative increase in restoring force survives that change of boundary conditions.