Prediction of Low-Frequency Wave Damping: A Simplified Approach
Authors: Kaplan P. , Sankaranarayanan K. 
Published: October 2008

ABSTRACT
An analytical procedure to predict the additional damping of low frequency vessel motions associated with second order drift forces is presented. The method is based on use of first order wave-induced motion responses obtained from strip theory analysis which are applied in a simplified three-dimensional model for determining drift forces. That technique is extended further by inclusion of forward speed effects to allow determination of the additional linear wave drift damping term. Comparison of the theoretical results with model test data demonstrates the utility of the method, which involves simple calculation, in contrast with more complex calculation procedures.

INTRODUCTION
The problem of predicting the low frequency motions of moored vessels has primarily concentrated on the calculation of the second order wave drift forces that provide the excitation for such motions. Various methods for calculating such forces have been developed, with the most successful methods (as determined by agreement between theory and experiment) being three-dimensional hydrodynamic analyses (i.e. see Pinkster [1]).
Evaluation of the response to any excitation is critically dependent on the degree of damping in the system. Since the first order linear damping is small at the low frequencies associated with the slowly varying motions of moored vessels, the damping was considered to be mainly arising from second order viscous drag forces. However, extensive experimental studies carried out by Wichers over a 10 year period ([2]-[5]) has demonstrated the existence of a linear additional low, frequency damping that is associated with the motion of the vessel in waves. This additional damping has been referred to as wave drift damping, and it is associated with the second order wave drift force excitation force. It has been primarily studied for the case of longitudinal low frequency surge motion, where it has exhibited its greatest influence.
Aside from the experimental studies by Wichers, a number of different theoretical analyses have also been applied to predict this low frequency wave damping by many distinguished hydrodynamic analysts (see [6J-[10]). The theoretical methods used there have been primarily by means of three dimensional diffraction theory, which is computationally intensive. The wave drift damping in surge is determined by finding the velocity derivative of the negative of the longitudinal second order steady drift force (see [4]), or alternatively the velocity derivative of the added resistance in waves. This quantity is found by evaluating the second order drift force for a range of low forward speeds about the zero speed reference condition. For any value of frequency, the values of the added resistance as a function of velocity are plotted. The slope of the curve, evaluated at zero speed, is found for each frequency as a measure of this wave drift damping.
In view of the complexity of all of these approaches, another method which is based on a simpler model of the basic second order drift forces [11] is proposed.