  The potential flow of an incompressible inviscid heavy fluid over a light one
is considered. The integral version of the method of matched asymptotic
expansion is applied to the construction of the solution over long intervals of
time. The asymptotic solution describes the flow in which a bubble rises with
constant speed and the "tongue" is in free fall. The outer expansion is
stationary, but the inner one depends on time. It is shown that the solution
exists within the same range of Froude number obtained previously by
Vanden-Broeck (1984a,b). The Froude number and the solution depend on the
initial energy of the disturbance. At the top of the bubble, the derivative of
the free-surface curvature has a discontinuity when the Froude number is not
equal to 0.23. This makes it possible to identify the choice of the solution
obtained in a number of studies with the presence of an artificial numerical
surface tension. The first correction term in the neighborhood of the tongue is
obtained when large surface tension is included.
