  We study the propagation of two-dimensional tsunami waves triggered by a
seaquake in the open sea in the presence of underlying wind-generated currents,
corresponding to background flows of constant vorticity. A suitable scaling of
the governing equations introduces dimensionless parameters, of particular
interest being the setting of linear waves that only depend on the vertical
movement of the sea bed. We use Fourier analysis methods to extract formulae
for the function $f$ which describes the vertical displacement of the water's
free surface. We show that the results are particularly useful in the
physically relevant shallow-water regime: in the irrotational case the
predictions fit well with the observed behaviour of some historical tsunamis.
In other situations, the stationary-phase principle gives insight into the
asymptotic behaviour of $f$.
