The radiation problem for an oscillating cylinder advancing under the free surface with a constant horizontal velocity is studied using the Green integral equation in the frequency domain. The Green function expressed in terms of the complex exponential, is derived using the damped free surface condition. Special attention is given to the behavior of the numerical solution in the vicinity of the critical Brard number ${gamma}_c=omega{cdot}u/g=0.25$ where $omega$ is the circular frequency of encounter, u the advancing speed and g the gravitational acceleration. It is shown that the solution is finite in the vicinity of ${gamma}_c$ although the Green function becomes singular at ${gamma}_c$. It is also shown that the computed hydrodynamic coefficients agree well with those obtained from the solution of the same problem formulated in the time domain.