Cylindrical waves are fundamental building blocks in constructing the solution of two-dimensional scattering problems. The reflection problem of a cylindrical wave of any integer order from a generic plane surface of discontinuity for the electromagnetic constants is faced making use of the plane-wave decomposition approach. The resulting numerical problem, involving the quadrature of highly oscillating functions is solved by means of suitable techniques. Numerical results are presented for some cases of practical interest. A comparison with a particular case for which a closed-form solution is available is also presented.