= Solution
Write the cosine in <2/c/i> as two complex exponentials. On a ray $x=vt$, the oscillatory phases are $kv\pm\omega(k)$. The <stationary phase method> requires $v=\mp\omega'(k)$. By <2/b/ii>, the largest absolute horizontal <group velocity> for the occupied $n=1$ mode is
$$
\boxed{\frac{|x|}{t}\leq\frac{N}{b_1}=\frac{2h\sqrt{\alpha g}}{\sqrt{4\pi^2+\alpha^2h^2}}}.
$$
Inside this region there are stationary wavenumbers supporting appreciable <wave packet> contributions. Outside it no stationary wavenumber exists; oscillatory cancellation suppresses the large-time response. This explains "mainly confined": the initial Lorentzian profile already has nonzero algebraic tails, so the displayed region is not the exact support of the solution. Near its boundary the stationary points meet $k=0$ and the usual nondegenerate stationary-phase formula needs a transition approximation.
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