Past exam of the physics A course of the University of Cambridge / 2015 / Paper 2 / Section B / B7

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 $\omega =k{v}_p$ and $\lambda =2\pi /k$,For the gravity-capillary wave, put $s=\sigma /\rho$ and take $k>0$. Direct differentiation givesThus $v_g=v_p$ when $s{k}_{\ast }^2=g$, orAt 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, $v_{p,\min }=(4g\sigma /\rho {)}^{1/4}=0.214\mathrm{m}{\mathrm{s}}^{-1}$. Locally the carrier crests and narrow envelope travel together. The medium has not become nondispersive: $d^2\omega /d{k}^2=k_{\ast }{d}^2{v}_p/d{k}^2\ne 0$ there, so a finite wave packet can still spread. In the gravity-dominated limit $v_g/v_p\to 1/2$; in the capillary-dominated limit it tends to $3/2$.