The instabilities of a free surface shear flow are considered, with special emphasis on
the shear flow with the velocity profile
U* = U*0sech2 (by*). This velocity profile, which
is found to model very well the shear flow in the wake of a hydrofoil, has been focused
on in previous studies, for instance by Dimas & Triantyfallou who made a purely
numerical investigation of this problem, and by Longuet-Higgins who simplified the
problem by approximating the velocity profile with a piecewise-linear profile to make
it amenable to an analytical treatment. However, none has so far recognized that this
problem in fact has a very simple solution which can be found analytically; that is, the
stability boundaries, i.e. the boundaries between the stable and the unstable regions
in the wavenumber (k)–Froude number (F)-plane, are given by simple algebraic
equations in k and F. This applies also when surface tension is included. With no
surface tension present there exist two distinct regimes of unstable waves for all values
of the Froude number F > 0. If 0 < F [Lt ] 1, then one of the regimes is given by
0 < k < (1 − F2/6), the other by
F−2 < k < 9F−2, which is a
very extended region on the k-axis. When F [Gt ] 1 there is one small unstable
region close to k = 0, i.e. 0 < k < 9/(4F2),
the other unstable region being
(3/2)1/2F−1 < k < 2 + 27/(8F2).
When surface tension is included there may be one, two or even three distinct regimes
of unstable modes depending on the value of the Froude number. For small F there
is only one instability region, for intermediate values of F there are two regimes of
unstable modes, and when F is large enough there are three distinct instability regions.
A suggested analytical treatment of the orbit of a close satellite.