A second-order solution for the spiral wave field produced by a rotating cylinder in a basin of constant depth is presented. The analysis is limited to cases in which the wave amplitude is small compared with the wave maker radius, which is in turn small compared to the wavelength. Under these circumstances, we show by an order-of magnitude argument that one may consistently simplify the second-order analysis by neglecting the quadratic terms in the free surface boundary conditions while retaining the leading terms in the wave maker boundary condition. The resulting solution indicates the presence of free secondary waves which do not propagate at the same speed or in the same direction as the first-order waves. Laboratory measurements are presented which demonstrate the ability of the solution to reproduce measurements of the surface displacement with reasonable accuracy. At large distances from the wave maker, the free waves completely dominate the second-order waves forced by the nonlinear terms in the free surface boundary conditions. Consequently, a cylindrical spiral wave maker does not simulate correctly the second-order component of regular plane waves.