Membrane waveguide (${\omega }^2=v^2[k_x^2+(n\pi /b{)}^2]$)

A strip of membrane with fixed edges $y=0,b$ supports transverse normal modes proportional to $\sin (n\pi y/b)$, $n\ge 1$. Separation of the two-dimensional wave equation gives the displayed dispersion relation. Confinement produces longitudinal wave dispersion even when the membrane's underlying wave speed $v$ is independent of frequency.

A fixed-edge membrane strip has dispersion ${\omega }_n^2=v^2[k_x^2+(n\pi /b{)}^2]$, n≥1. Differentiating gives $v_{g,x}=v^2{k}_x/{\omega }_n$, whereas $v_{p,x}={\omega }_n/k_x$. Their product is $v^2$. In this lossless scalar mode the integrated energy flux divided by energy density agrees with group speed. At cutoff the longitudinal group speed vanishes while formal phase speed diverges; an infinite phase speed therefore indicates no superfast energy transport.

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.

The cutoff frequency of a membrane waveguide is the lowest frequency at which a specified transverse normal mode can propagate longitudinally. For fixed edges separated by $b$, ${\omega }_{c,n}=n\pi v/b$ and $f_{c,n}={\omega }_{c,n}/(2\pi )$. Below cutoff the longitudinal wavenumber is imaginary and a bounded excitation is an evanescent wave. Above it, $v_g=v\sqrt{1-({\omega }_{c,n}/\omega {)}^2}$ and $v_p{v}_g=v^2$.

For a membrane waveguide branch ${\omega }^2=v^2[k_x^2+(n\pi /b{)}^2]$, the exact cutoff has zero longitudinal wavevector and zero group velocity. It is not a traveling channel carrying finite longitudinal energy flux. Modes strictly below cutoff are evanescent; amplitudes of the permitted above-cutoff modes depend on the transverse excitation profile.