= Solution
For <group velocity from a period-dependent phase velocity>, start with $k=\omega/c_n(\omega)$ rather than equating <group velocity> to <phase velocity>. Differentiating,
$$
\frac1{u_n}=\frac{dk}{d\omega}=\frac1{c_n}-\frac{\omega}{c_n^2}\frac{dc_n}{d\omega}.
$$
For the stated inverse-frequency term, $dc_n/d\omega=-a/\omega^2$, so
$$
\boxed{\frac{u_n}{c_n}=\frac1{1+a/(\omega c_n)}
=\frac{c_n}{c_n+a/\omega},\qquad
u_n=\frac{c_n^2}{c_n+a/\omega}<c_n.}
$$
The inequality assumes positive <frequency> and physical positive <phase velocity>, with $a>0$. The symbol $a$ here has units of acceleration and is unrelated to the plate thickness in Question 1.
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