Nonlinear wave excitation forces can be determined by the perturbation approach, with the wave amplitude as a small parameter, as discussed in Second Order Wave Excitation Forces.

When the multi-directional wave elevation is generally represented by Equation 2–28 and Equation 2–29, the first order wave excitation force and moment, i.e. the sum of the Froude-Krylov and diffracting forces and moments, can be simply written as

(13–28)

where is the Euler rotation matrix defined by Equation 1–7 at a time , is the relative heading angle of the m-th sub-directional wave (relative to the structure, where is the instantaneous yaw angle of the structure), and is the total first order wave excitation force induced by an unit amplitude incident wave with frequency and wave direction . The components of this excitation force are expressed in Equation 4–48 and are calculated by a hydrodynamic diffraction analysis (Aqwa-Line) prior to the present time domain analysis.

From Equation 5–20, the second order wave excitation force in the fixed reference axes is given by

(13–29)

where and .

Substituting the first and second order wave forces into Equation 13–3, the following equation of motion is solved:

(13–30)

in which the total stiffness matrix includes the linear hydrostatic stiffness as well as all other stiffnesses, for example from mooring lines or articulations, is the current hull drag force, is the wind drag force, is the nonlinear bilge roll damping force, is the mooring and articulation force, and represents additional external forces. The wave drift damping coefficients are optionally included in the damping matrix .

The structure responses due to mean and slowly-varying wave loads can be estimated from Equation 13–30 by including only the low frequency drift forces, i.e. the difference frequency terms of in Equation 13–29.

If only the first order wave excitation force is involved in Equation 13–30, the wave frequency responses can be calculated.

Wave frequency response with slow drift can be analyzed from Equation 13–30 by including the first order wave excitation force together with the difference frequency second order drift forces.

The sum frequency second order force and moment components without directional coupling effects can be considered when the first order wave excitation force and both the difference frequency and sum frequency second order forces are included. The sum frequency second order force may only be included if the full quadratic transfer function (QTF) option has been applied in the hydrodynamic diffraction analysis (Aqwa-Line) prior to this time domain analysis.