**4. Results**

The added mass and damping coefficients for the wave experiments were obtained by two different approaches. In the "Absolute model" approach, Equation (3) is used in the least squares evaluation. This model is identical to the still water case, and all wave-induced variations were visible in the trends of the coefficients with the phase angle. In the "Relative model" approach, Equation (8) which incorporates relative kinematics is used. Sample added mass and damping results using the two equation models are shown in Figures 6 and 7 for different phase angles at *H*/*L* = 0.02 and *KC* = 0.84. At this *KC* value, still water added mass and damping values are 1.42 and 0.85, respectively.

**Figure 6.** Added mass coefficient vs. phase angle for *H*/*L* = 0.02, *KC* = 0.84.

**Figure 7.** Damping coefficient vs. phase angle for *H*/*L* = 0.02, *KC* = 0.84.

When using the absolute model approach, a clear sinusoidal trend is observed with respect to the phase angle. Interestingly, the mean value of this sinusoidal variation matches with the corresponding still water added mass and damping values to within 3%. When the relative flow approach is used, the trend of both coefficients with the phase is much flatter, tending towards a constant value that matches the still water value to within 4%.

The relative phase between the plate and the wave gives rise to a relative change in the *KC*, although the amplitude of oscillation is kept constant. Figures 8 and 9 present the added mass and damping coefficients obtained using the relative model against *KCw*. Additionally shown are the results obtained in still water for the added mass and damping coefficients vs. *KC*. The results are presented for the cases *KC* = 0.84 and *KC* = 0.5 for a frequency of oscillation of 1Hz and for *H*/*L* = 0.018 and *H*/*L* = 0.02. The observed linear trend in the coefficients is remarkable. It can be seen that the added mass and damping coefficients increase as the relative displacement between the plate and the wave particles increases. The added mass coefficients in waves show a steeper linear trend when compared with the still water coefficients. For small *KC*, the added mass coefficients in still water are higher. As *KC* increases, the coefficients in waves become slightly higher than the ones in still water. The damping coefficients in still water and in waves are very similar in slope, with the zero offset showing a difference.

**Figure 8.** Comparison of the added mass coefficients in waves vs. *KCw* with those in still water vs. *KC*.

**Figure 9.** Comparison of the damping coefficients in waves vs. *KCw* with those in still water vs. *KC*.

#### **5. Discussion and Conclusions**

This paper has focused on the effect of ambient wave motion on the hydrodynamic forces acting on an oscillating heave plate. When compared to the plate oscillating in still water, large differences in the values of the added mass and damping coefficients are observed. These differences are quite pronounced when the relative motion between the water and the plate are not taken into account. The results from Figures 6 and 7 tend to indicate that applying the added mass and damping coefficients obtained from still water experiments for simulating the motion of a structure in waves may lead to inconsistent results. However, due to scarcity of data on oscillating plates in waves, one method of getting reliable added mass and damping values would be by using the newly defined *KCw*, which depends on the relative amplitude of motion with respect to the wave. As seen in Figures 8 and 9, the trends between the results in waves are somewhat closer to those that were obtained in still water.

Because damping values are more critical in estimating the maximum motions around resonance, a relative phase angle of *π*/2 may be used for *KCw*. This could be used iteratively along with motion magnitude to find the optimum damping coefficient. On the other hand, it is seen in Figure 6 that, at around a phase angle of 90◦, the added mass coefficients in waves and in still water are similar in magnitude. However, added mass coefficients are of relevance in all motion ranges. From Figure 8, it is seen that, at lower *KC* values, the added mass coefficients could differ by 30%, which can affect inertial load calculations. Thus, caution needs to be exerted in selection of hydrodynamic

coefficients for heave plates oscillating in proximity to the free surface. More data would support better estimates of hydrodynamic coefficients for use in simulation of offshore wind turbine platform motions. Future work by the researchers would include a broader range of wave parameters and oscillation ranges. It is also envisaged that currents could be added to the environment in order to understand the combined effect of waves and currents.

**Author Contributions:** Conceptualization, K.T.; methodology, K.T. and J.M.; experimentation and validation, J.M.; analysis, J.M. and K.T.; resources, K.T.; writing–original draft preparation, K.T.; writing–review and editing, K.T. and J.M.; project administration, K.T.; All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Acknowledgments:** The paper is a development of an earlier version presented at the 36th International Workshop on Water Waves and Floating Bodies (IWWWFB36) held in Plymouth, MI. The authors acknowledge the support of Stephen Abbadessa, Matthew Cameron and Raul Urbina, Department of Mechanical Engineering, University of Maine in support of the MOOR laboratory. The second author acknowledges the financial support of Iberdrola Foundation for his fellowship during the study.

**Conflicts of Interest:** The authors declare no conflict of interest.
