  Stationary periodic solutions of the two-dimensional Gross-Pitaevskii
equation are obtained and analyzed for different parameter values in the
context of the problem of a supersonic flow of a Bose-Einstein condensate past
an obstacle. The asymptotic connections with the corresponding periodic
solutions of the Korteweg-de Vries and nonlinear Schr\"odinger equations are
studied and typical spatial wave distributions are discussed.
