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We present exact solutions of the incompressible Navier-Stokes equations in abackground linear shear flow. The method of construction is based on Kelvin'sinvestigations into linearized disturbances in an unbounded Couette flow. Weobtain explicit formulae for all three components of a Kelvin mode in terms ofelementary functions. We then prove that Kelvin modes with parallel (thoughtime-dependent) wave vectors can be superposed to construct the most generalplane transverse shearing wave. An explicit solution is given, with anyspecified initial orientation, profile and polarization structure, with eitherunbounded or shear-periodic boundary conditions.

Plane shearing waves of arbitrary form: exact solutions of the Navier-Stokes equations