= Solution
For a <monochromatic wave>, a constant phase $kx-\omega t$ moves at <phase velocity> $v_p=\omega/k$. For a narrow <wave packet> around $k$, expansion $\omega(k+\Delta k)\simeq\omega(k)+(d\omega/dk)\Delta k$ shows that its envelope moves at <group velocity> $v_g=d\omega/dk$. The envelope interpretation requires a sufficiently narrow spectrum and weak distortion; it is not a general identification with a causal signal front. Since $\omega=kv_p$ and $\lambda=2\pi/k$,
$$
v_g=v_p+k\frac{dv_p}{dk}
=\boxed{v_p-\lambda\frac{dv_p}{d\lambda}}.
$$
For the <gravity-capillary wave>, put $s=\sigma/\rho$ and take $k>0$. Direct differentiation gives
$$
v_p=\sqrt{g/k+sk},\qquad
v_g=\frac{g+3sk^2}{2\sqrt{gk+sk^3}},\qquad
\boxed{\frac{v_g}{v_p}=\frac{g+3sk^2}{2(g+sk^2)}
=1-\frac{g/k^2-s}{2(g/k^2+s)}.}
$$
Thus $v_g=v_p$ when $sk_*^2=g$, or
$$
\boxed{\lambda^*=2\pi\sqrt{\frac{\sigma}{\rho g}}=2.89\times10^{-2}\,\mathrm m.}
$$
At this <wavelength> the gravity and surface-tension restoring terms contribute equally. Also $d(v_p^2)/dk=-g/k^2+s=0$ with positive <second derivative>: this is the <minimum phase speed of a gravity-capillary wave>, $v_{p,\min}=(4g\sigma/\rho)^{1/4}=0.214\,\mathrm{m\,s^{-1}}$. Locally the carrier crests and narrow envelope travel together. \b[The medium has not become nondispersive]: $d^2\omega/dk^2=k_*d^2v_p/dk^2\ne0$ there, so a finite <wave packet> can still spread. In the gravity-dominated limit $v_g/v_p\to1/2$; in the capillary-dominated limit it tends to $3/2$.
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