Someone argues as follows for a ship moving at across deep

PHYSICS · Subjective · Fermi University Physics · Oscillations, Waves & Acoustics · Waves in two and three dimensions, dispersion and wave packets · Waves in two and three dimensions, dispersion and wave packets

Modes, Packets and Media, Ch Water waves and de Broglie waves

Problem

Someone argues as follows for a ship moving at U=9.0 m s−1U = 9.0\,\mathrm{m\,s^{-1}} across deep, still water: "The transverse waves right behind the ship set the relevant wave speed, exactly as a fixed sound speed sets the Mach cone of a supersonic aircraft. Taking that speed to be the group velocity of those transverse waves, vg=U/2=4.5 m s−1v_{\rm g} = U/2 = 4.5\,\mathrm{m\,s^{-1}}, the wake should be a Mach cone of half-angle θ=arcsin⁡(vg/U)=arcsin⁡(0.5)=30∘\theta = \arcsin(v_{\rm g}/U) = \arcsin(0.5) = 30^\circ. A faster ship, with a larger UU in the denominator, would then give a smaller θ\theta: the wake narrows as the ship speeds up." Find the error in this reasoning, and give the correct half-angle of the wake.

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