  A compact and efficient numerical method is described for studying plane
flows of an ideal fluid with a smooth free boundary over a curved and
nonuniformly moving bottom. Exact equations of motion in terms of the so-called
conformal variables are used. In addition to the previously known applications
for shear flows with constant (including zero) vorticity, here a generalization
is made to the case of potential flows in uniformly rotating coordinate
systems, where centrifugal and Coriolis forces are added to the gravity force.
A brief review is given of previous results obtained by this method in a number
of physically interesting problems such as modeling of tsunami waves caused by
the movement of nonuniform bottom, the dynamics of Bragg (gap) solitons over a
spatially periodic bottom profile, the Fermi-Pasta-Ulam (FPU) recurrence
phenomenon for waves in a finite pool, the formation of anomalous waves in an
opposing nonuniform current, and the propagation of a solitary wave in a shear
current and its runup on a depth difference. In addition, a number of new
numerical results are presented concerning the nonlinear dynamics of a free
boundary in closed rotating containers partially filled with a fluid --
centrifuges of complex shape. In this case, the equations of motion differ in
some essential details from those of $x$-periodic systems.
