Abstract: An
exact analytical solution to the Navier-Stokes
equations is constructed for an unsteady
viscous incompressible flow governing the
motion of a monochromatic surface wave on deep
water by applying boundary conditions on an a
priori unknown surface profile y = h.It
is shown that, contrary to Lamb
[Hydrodynamics, 6th ed., Cambridge University
Press, 1932], even in the presence of
viscosity the vorticity is zero and thus the
motion is exactly irrotational. It is argued
that the appearance of vorticity in Lamb’s
solution is solely due to mathematical
approximations for the problem. It is shown
that the free surface profile obtained here
damps down exponentially in time at the rate
half of that obtained previously by Lamb.
Keywords and phrases: water waves, Navier-Stokes equations, surface wave.
