= Solution
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, $\omega^2=\omega_p^2+c^2k^2$.
The fixed-edge <membrane waveguide> has transverse profiles $\sin(n\pi y/b)$ with
$$
\boxed{k_y=n\pi/b,\quad n=1,2,\ldots,\qquad
\omega_n^2=v^2[k_x^2+(n\pi/b)^2].}
$$
There 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,
$$
\boxed{v_{p,x}=\frac\omega{k_x},\quad v_{g,x}=\frac{v^2k_x}{\omega},\quad
v_{p,x}v_{g,x}=v^2,\qquad f_{c,n}=\frac{nv}{2b}.}
$$
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:
$$
\boxed{k_x=\sqrt{(30\pi)^2-(20\pi)^2}=10\pi\sqrt5=70.2\,\mathrm{m^{-1}}.}
$$
The negative value represents travel in the opposite direction; higher transverse modes are evanescent.
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