  We present exact solutions of the incompressible Navier-Stokes equations in a
background linear shear flow. The method of construction is based on Kelvin's
investigations into linearized disturbances in an unbounded Couette flow. We
obtain explicit formulae for all three components of a Kelvin mode in terms of
elementary functions. We then prove that Kelvin modes with parallel (though
time-dependent) wave vectors can be superposed to construct the most general
plane transverse shearing wave. An explicit solution is given, with any
specified initial orientation, profile and polarization structure, with either
unbounded or shear-periodic boundary conditions.
