  Two-dimensional free-surface flow over localised topography is examined with
the emphasis on the stability of hydraulic-fall solutions. A Gaussian
topography profile is assumed with a positive or negative amplitude modelling a
bump or a dip, respectively. Steady hydraulic-fall solutions to the full
incompressible, irrotational Euler equations are computed, and their linear and
nonlinear stability is analysed by computing eigenspectra of the pertinent
linearised operator and by solving an initial value problem. The computations
are carried out numerically using a specially developed computational framework
based on the finite element method. The Hamiltonian structure of the problem is
demonstrated and stability is determined by computing eigenspectra of the
pertinent linearised operator. It is found that a hydraulic-fall flow over a
bump is spectrally stable. The corresponding flow over a dip is found to be
linearly unstable. In the latter case, time-dependent simulations show that the
flow ultimately settles into a time-periodic motion that corresponds to an
invariant solution in an appropriately defined phase space. Physically, the
solution consists of a localised large amplitude wave that pulsates above the
dip while simultaneously emitting nonlinear cnoidal waves in the upstream
direction and multi-harmonic linear waves in the downstream direction.
