<p>For a generalized radiation problem, an infinitely long submerged horizontal cylinder is forced to vibrate periodically in the transverse direction, with a described elastic harmonic motion along its longitudinal direction. A critical frequency corresponds to the described wave number of elastic vibration, and the generalized hydrodynamic coefficients abruptly change in the vicinity of critical frequency. In this work, a numerical examination is carried out to study the characteristics of wave profiles and wave propagation in the vicinity of the critical frequency. Results show that below the critical frequency, the real parts of complex wave profiles have large values in the vicinity of the cylinder and decay to zero with the increasing distance from the cylinder. Meanwhile, the imaginary parts of complex wave profiles are all zero, which explains why the generalized radiation damping is zero when the vibration is less than the critical frequency. At far distances, no radiation wave is observed. When the vibration exceeds the critical frequency, the real and imaginary parts of the wave profiles oscillate harmonically and keep steady amplitudes. In addition, the generated radiation wave propagates obliquely outward. The influence of the cylinder’s submergence depth on the wave profile is also studied, and the results indicate that the amplitude of the wave profile decreases as the submergence depth of the cylinder increases. The 3D wave profiles are graphically presented to show the wave propagation characteristics in the vicinity of the critical frequency for this generalized radiation problem. This study provides a good reference for the interaction between fluid and slender elastic structures.</p>