= 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 $\sigma z_{tt}=T\nabla^2z-D_b\nabla^4z$. If the edge conditions admit a transverse sine mode of <wavenumber> $q_n$, its longitudinal <dispersion relation> is
$$
\omega^2=\frac T\sigma(k^2+q_n^2)+\frac{D_b}\sigma(k^2+q_n^2)^2.
$$
Bending 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>.
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