The oscillatory flexural displacement of the riser can be extended well below the wave zone where the orbital movements of the water particles cease to exist. Under this condition, the flow condition is nearly two dimensional. The inertia force exerted by the fluid is 180 degrees out of phase and linearly proportional to the body acceleration. The corresponding constant has a dimension of mass and is known as the added mass of the fluid. The drag force is due to the dissipation of kinetic energy of the fluid motion in the wake and therefore can be treated as the damping force. The added mass and drag coefficient are defined by the following:
Added mass force = ...(17)
Viscous Damping Forces = ...(18)
where and are the velocity and acceleration of the riser.
The values of Ca and Cd are governed by two dimensionless parameters, namely Re and KC, and the surface roughness. The value of Ca for a smooth circular cylinder tends to approach the theoretical value of 1.0 if the motion amplitude is less than one diameter. The magnitude of damping is determined by the loss of fluid momentum due mainly to flow separation and vortex formation. The corresponding flow condition is sensitive to the surface roughness. For large amplitude motion, the hydrodynamic forces may experience hysteretic effects due to the reversal of the wake in each motion cycle.13
...(18)
and 
are the velocity and acceleration of the riser.