  Linearized flow past a submerged obstacle with an elastic sheet resting on
the flow surface are studied in the limit that the bending length is small
compared to the obstacle depth, in two and three dimensions. Gravitational
effects are included in the two-dimensional geometry, but absent in the
three-dimensional geometry; the Froude number is chosen so that gravitational
and elastic restoring forces are comparable in size. In each of these problems,
the waves are exponentially small in the asymptotic limit, and can be computed
using exponential asymptotic methods. In the two-dimensional problem, flow past
a submerged step is considered. It is found that the relative strength of the
gravitational and elastic restoring forces produce two distinct classes of
elastic sheet behaviour. In one parameter regime, constant-amplitude elastic
waves and gravity waves extend indefinitely upstream and downstream from the
obstacle. In the other parameter regime, all waves decay exponentially away
from the obstacle. The equivalent nonlinear two-dimensional geometry is then
studied; this asymptotic analysis predicts the existence of a third
intermediate regime in which waves persist indefinitely only in one direction,
depending on whether the submerged step rises or falls. In the
three-dimensional geometry, it is predicted that the elastic waves extend ahead
of the submerged source, decaying algebraically in space. The form of these
elastic waves is computed, and validated by comparison with numerical
computations of the elastic sheet behaviour.
