A ship travels in a straight line at constant deep-water speed

PHYSICS · Multi-part · 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

A ship travels in a straight line at constant deep-water speed UU. Deep-water surface gravity waves are dispersive, ω2=gk\omega^2 = gk, so a wave component making angle θ\theta with the ship's track is steady in the ship's frame only if its phase speed satisfies vp(θ)=Ucos⁡θv_{\rm p}(\theta) = U\cos\theta, and its energy then travels at vg(θ)=12Ucos⁡θv_{\rm g}(\theta) = \tfrac12 U\cos\theta in the direction θ\theta. For wave energy emitted a time τ\tau before the present, this places it, relative to the ship's current position, at x′=−Uτ2(2−cos⁡2θ),y′=Uτ2cos⁡θsin⁡θ.x' = -\frac{U\tau}{2}(2-\cos^2\theta), \qquad y' = \frac{U\tau}{2}\cos\theta\sin\theta. ((a)) Treating x′(θ)x'(\theta) and y′(θ)y'(\theta) (at fixed τ\tau) as a curve traced out as θ\theta ranges over (0,π/2)(0,\pi/2), find the angle φ=arctan⁡(y′/x′)\varphi=\arctan(y'/x') from the track at which ∣φ∣|\varphi| is largest, by finding the θ\theta that extremises it, and hence show that the wake's half-angle is 19.5∘19.5^\circ, independent of UU and τ\tau. ((b)) For a ship with U=12 m s−1U = 12\,\mathrm{m\,s^{-1}}, find the wavelength of the transverse waves running directly behind it, using g=9.8 m s−2g = 9.8\,\mathrm{m\,s^{-2}}.

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