Equation 13–30 represents the second order differential equations of motion for each structure. This is solved using the semi-implicit two stage predictor-corrector integration scheme described in Integration in Time of Motion Equation. The resulting accelerations are then integrated to form a time history of structure motions. In order to begin this integration, the solver requires initial conditions for the total and slow positions at time . It is also important that the transient experienced by the structure is as small as possible.

The initial conditions for the slow motion are relatively intuitive, since these relate to the general motion of the structure about its equilibrium position. The wave frequency motions, however, are in response to randomly-phased wave frequency forces that generally cannot be specified. In this case the program automatically computes an initial wave frequency position from Equation 13–34 at time , which is added to the defined slow position to form an initial total position. The initial wave frequency velocity is estimated from Equation 13–35. These treatments ensure that the total initial conditions contain a fast (wave frequency) component equal to the steady state solution in response to the wave frequency forces at that instant; the transients can thus be minimized.