As well as being excited by drift forces, the structure will be subjected to first order wave frequency forces. Since the added mass and damping are not constant over the wave frequency range, Equation 13–30 should be employed directly.

When both drift and wave frequency forces are present, the structure will still experience drift oscillations, but these will be accompanied by wave frequency oscillations about the slow position. These latter oscillations are termed the wave frequency motion, with a corresponding wave frequency position. The sum of the slow position and the wave frequency position is called the total position, which is referred to simply as the position.

In cases where both drift and wave frequency motions exist, the current drag and wave drift forces are applied to the structure in an axis system that follows the slow position. However, wind forces are applied using an axis system that follows the total position.

The slow position is obtained by applying a low-pass filter to the total position, which removes the high frequency (fast) oscillations. This is achieved by integrating the following equation at each time step:

(13–32)

where is the slow position, is the filtering frequency, is the filter damping, and is the total position.

The filtering frequency is chosen automatically to eliminate the wave frequency effects. The damping is set to be 20% of the critical damping.

The difference between the unfiltered (total) and filtered (slow) positions is the wave frequency position , i.e.

(13–33)

The response amplitude operator-based (RAO-based) position can also be estimated in a time domain analysis. As discussed in Response Amplitude Operators, an RAO is determined from a frequency domain hydrodynamic analysis. A time history of the RAO-based position can be constructed by combining RAOs at a series of frequencies with the wave spectrum. This is done for each degree of freedom as follows:

(13–34)

where is the motion RAO at frequency and relative wave direction , as calculated by a hydrodynamic diffraction analysis.

This RAO-based position is used to calculate the initial fast position to minimize transients.

A similar expression is used to calculate the RAO-based velocity and acceleration, such as

(13–35)

For simple cases, the RAO-based position will be very similar to the wave frequency position defined by Equation 13–33. This can provide a useful check on the wave frequency position in runs where wave frequency forces are included.