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A compact and efficient numerical method is described for studying planeflows of an ideal fluid with a smooth free boundary over a curved andnonuniformly moving bottom. Exact equations of motion in terms of the so-calledconformal variables are used. In addition to the previously known applicationsfor shear flows with constant (including zero) vorticity, here a generalizationis made to the case of potential flows in uniformly rotating coordinatesystems, 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 numberof physically interesting problems such as modeling of tsunami waves caused bythe movement of nonuniform bottom, the dynamics of Bragg (gap) solitons over aspatially periodic bottom profile, the Fermi-Pasta-Ulam (FPU) recurrencephenomenon for waves in a finite pool, the formation of anomalous waves in anopposing nonuniform current, and the propagation of a solitary wave in a shearcurrent and its runup on a depth difference. In addition, a number of newnumerical results are presented concerning the nonlinear dynamics of a freeboundary in closed rotating containers partially filled with a fluid --centrifuges of complex shape. In this case, the equations of motion differ insome essential details from those of $x$-periodic systems.

Waves over Curved Bottom: The Method of Composite Conformal Mapping