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We present a comprehensive theory for linear gravity-driven ship waves in thepresence of a shear current with uniform vorticity, including the effects offinite water depth. The wave resistance in the presence of shear current iscalculated for the first time, containing in general a non-zero lateralcomponent. While formally apparently a straightforward extension of existingdeep water theory, the introduction of finite water depth is physicallynon-trivial, since the surface waves are now affected by a subtle interplay ofthe effects of the current and the sea bed. This becomes particularlypronounced when considering the phenomenon of critical velocity, the velocityat which transversely propagating waves become unable to keep up with themoving source. The phenomenon is well known for shallow water, and was recentlyshown to exist also in deep water in the presence of a shear current[Ellingsen, J.~Fluid Mech.\ {\bf 742} R2 (2014)]. We derive the exact criterionfor criticality as a function of an intrinsic shear Froude number $S\sqrt{b/g}$($S$ is uniform vorticity, $b$ size of source), the water depth, and the anglebetween the shear current and the ship's motion. Formulae for both the normaland lateral wave resistance force are derived, and we analyse its dependence onthe source velocity (or Froude number $Fr$) for different amounts of shear anddifferent directions of motion. The effect of the shear current is to increasewave resistance for upstream ship motion and decrease it for downstream motion.Also the value of $Fr$ at which $R$ is maximal is lowered for upstream andincreased for downstream directions of ship motion. For oblique angles betweenship motion and current there is a lateral wave resistance component which canamount to $10$-$20\%$ of the normal wave resistance for side-on shear and$S\sqrt{b/g}$ of order unity. (Continues...)

Ship waves on uniform shear current at finite depth: wave resistance and critical velocity