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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.

A dispersive wave has phase speed depending on frequency. Its phase velocity is $v_p=\omega /k$, whereas the group velocity $v_g=d\omega /dk$ describes a narrow wave packet in a lossless single branch. Examples are deep-water gravity waves, ${\omega }^2=gk$, and electromagnetic waves in a cold plasma, ${\omega }^2={\omega }_p^2+c^2{k}^2$.

The fixed-edge membrane waveguide has transverse profiles $\sin (n\pi y/b)$ withThere is no nonzero n=0 displacement satisfying both fixed-edge conditions. Even though the unconfined membrane is nondispersive, transverse confinement makes propagation along x dispersive. The membrane-guide phase and energy speeds are, for a propagating mode,For positive $k_x$, the phase speed exceeds v while group speed is below v. At cutoff $k_x=0$ the formal phase speed diverges, but group speed vanishes; no finite longitudinal energy flux is implied.

With the supplied width and speed, $f_{c,n}=10n\mathrm{kHz}$. At15kHz only n=1 propagates:The negative value represents travel in the opposite direction; higher transverse modes are evanescent.

Reflection at the fixed edges combines waves with opposite $k_y$ into a transverse standing wave while retaining their common longitudinal propagation factor. The membrane waveguide scalar transverse displacement satisfies the wave equation and vanishes at both edges:The $n=0$ sine gives no displacement. separation of variables givesFor a propagating branch with $k_x>0$,Their product is $v_m^2$. The phase velocity can exceed the material wave speed; energy transport follows the group velocity.

Here the cutoff sequence is $10n\mathrm{kHz}$. At $15\mathrm{kHz}$ only $n=1$ propagates, with $k_x=175.6{\mathrm{m}}^{-1}$ for travel toward positive $x$.

For the $n$th transverse mode, substitute $q_n=n\pi /a$ into the membrane waveguide dispersion relation:Longitudinal propagation requires real nonzero $k$, henceBelow the cutoff frequency of a membrane waveguide, $k=i\kappa$, with $\kappa =\sqrt{(n\pi /a{)}^2-\sigma {\omega }^2/T}$, and a bounded excitation decays as $e^{-\kappa x}$ rather than carrying a travelling wave along an infinite guide. At equality $k=0$, the displacement has no longitudinal phase advance and the group velocity is zero; it is a cutoff oscillation, not a propagating mode carrying finite longitudinal power.

A nondispersive wave has a linear dispersion relation $\omega =vk$ over the relevant range, so every Fourier transform component has the same phase velocity $v_p=\omega /k=v$ and group velocity $v_g=d\omega /dk=v$. A wave packet then translates without spreading in an ideal homogeneous lossless medium.

For wave dispersion, $v_p$ depends on frequency or wavenumber. Different components accumulate different phases, and in general the group velocity differs from the phase velocity; variation of group velocity across the spectrum produces spreading of a wave packet. The membrane material can be nondispersive while confinement in a membrane waveguide introduces wave dispersion along its length.

A waveguide confines a wave transversely while allowing propagation along a longitudinal direction. Material properties and boundary conditions select transverse normal modes, whose longitudinal dispersion relations may have cutoffs even when the bulk material is nondispersive. Examples include an optical fiber, a hollow electromagnetic waveguide, and a membrane waveguide. Guiding reduces geometric spreading along the chosen route but does not eliminate absorption, scattering or leakage.