On narrow V-like ship wakes

This paper is concerned with the generation of short gravity waves and their radiation from the outer edge of the turbulent boundary layer and wake of a ship. They arise primarily near the ship's stern. The wave spectrum in the direction of wavenumber vector at an angle (90° – δ) to the ship's track is: \[\Phi_{\delta}(\omega) = \Psi\left(\frac{UT_d}{2l},\frac{U\sin\delta}{c_g},\frac{R}{UT_d}\right)\frac{1}{k_0R}\frac{2l\omega^2}{g^2}\gamma\left(0,\frac{\pi}{l};0,\omega \right),\] where Ψ is dimensionless and a function of three dimensionless parameters. γ is the spectrum of the oscillating motion at the boundary, U the ship speed, Td the decay timescale of the oscillating motion, 2l the lengthscale of the eddies, and R the distance away from the boundary along the wavenumber vector. Generally, Φδ has large values near δ = 0 and small values at large δ; it behaves as 1/R at distances not far from the ship, then may vary slower than 1/R at intermediate distances, and finally behaves as 1/R again at distances far from the ship. These are consistent with the pattern found in SAR images of narrow V-like ship wakes. The method developed here is also applicable to various problems of surface wave generation by turbulence in water.