  We consider two-dimensional hydroelastic waves, in which a free fluid surface
separates two fluids of infinite vertical extent. Elastic effects are accounted
for at the interface, with a parameter measuring the elastic bending force and
another parameter measuring the mass of the elastic sheet. In prior work, the
authors have demonstrated well-posedness of this initial value problem in
Sobolev spaces. We now take the limit as these two parameters vanish. Since the
size of the time interval of existence given by this prior theory vanishes as
the mass and bending parameters go to zero, we now establish estimates which
are uniform with respect to these parameters. We may then make an additional
estimate which demonstrates that the solutions form a Cauchy sequence as the
parameters go to zero, so that the limit may be taken. This demonstrates that
the vortex sheet with surface tension is the zero mass, zero bending limit of
hydroelastic waves in two spatial dimensions.
