  We present a comprehensive theory for linear gravity-driven ship waves in the
presence of a shear current with uniform vorticity, including the effects of
finite water depth. The wave resistance in the presence of shear current is
calculated for the first time, containing in general a non-zero lateral
component. While formally apparently a straightforward extension of existing
deep water theory, the introduction of finite water depth is physically
non-trivial, since the surface waves are now affected by a subtle interplay of
the effects of the current and the sea bed. This becomes particularly
pronounced when considering the phenomenon of critical velocity, the velocity
at which transversely propagating waves become unable to keep up with the
moving source. The phenomenon is well known for shallow water, and was recently
shown 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 criterion
for 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 angle
between the shear current and the ship's motion. Formulae for both the normal
and lateral wave resistance force are derived, and we analyse its dependence on
the source velocity (or Froude number $Fr$) for different amounts of shear and
different directions of motion. The effect of the shear current is to increase
wave 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 and
increased for downstream directions of ship motion. For oblique angles between
ship motion and current there is a lateral wave resistance component which can
amount to $10$-$20\%$ of the normal wave resistance for side-on shear and
$S\sqrt{b/g}$ of order unity. (Continues...)
