= Solution
The cutoff occurs at $k=0$:
$$
\boxed{\omega_{n,c}=c_Sq_n
=\frac{c_S(n+1/2)\pi}{h}.}
$$
For $k>0$, the <phase velocity> and <group velocity> are
$$
\boxed{c_{p,n}=\frac{\omega_n}{k}
=c_S\sqrt{1+\frac{q_n^2}{k^2}},
\qquad
c_{g,n}=\frac{d\omega_n}{dk}
=\frac{c_Sk}{\sqrt{k^2+q_n^2}}.}
$$
Their product is $c_{p,n}c_{g,n}=c_S^2$. On a plot against positive $k$, $c_{p,n}$ decreases from $+\infty$ to $c_S$, while $c_{g,n}$ increases from zero to $c_S$; larger mode number shifts both transitions to larger wavenumber.
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