  The propagation and transformation of water waves over varying bathymetries
is a subject of fundamental interest to ocean, coastal and harbor engineers.
The specific bathymetry considered in this paper consists of one or two,
naturally formed or man-made, trenches. The problem we focus on is the
transformation of an incoming solitary wave by the trench(es), and the impact
of the resulting wave system on a vertical wall located after the trench(es).
The maximum run-up and the maximum force exerted on the wall are calculated for
various lengths and heights of the trench(es), and are compared with the
corresponding quantities in the absence of them. The calculations have been
performed by using the fully nonlinear water-wave equations, in the form of the
Hamiltonian coupled-mode theory, recently developed in Papoutsellis et al (Eur.
J. Mech. B/Fluids, Vol. 72, 2018, pp. 199-224). Comparisons of the calculated
free-surface elevation with existing experimental results indicate that the
effect of the vortical flow, inevitably developed within and near the
trench(es) but not captured by any potential theory, is not important
concerning the frontal wave flow regime. This suggests that the predictions of
the run-up and the force on the wall by nonlinear potential theory are expected
to be nearly realistic. The main conclusion of our investigation is that the
presence of two tandem trenches in front of the wall may reduce the run-up from
(about) 20% to 45% and the force from 15% to 38%, depending on the trench
dimensions and the wave amplitude. The percentage reduction is greater for
higher waves. The presence of only one trench leads to reductions 1.4 - 1.7
times smaller.
