The hydrodynamic forces associated with the motion of ship or hydrodynamic reaction forces were included for the reasons stated above. Here, the frequency dependent coefficients obtained in the frequency domain using linear theory and transformed to the time domain through the use of the impulse response functions.

　

 When a body performs an irregular motion around its mean position, it is appropriate to express the hydrodynamic force acting on the body in the time domain. Following the work by Cummins [12] and others, the radiation force in the time domain is written as:

　

　

 Where the first term is the infinite frequency added mass and the second term is the impulse response function or the so called "memory effect". The retardation (Kernel) function of equation (16) (Kij) is the real part of Fourier transform of the frequency domain damping function. In addition, the retardation can be described in terms of damping as

　

　

where Bij is damping coefficients. These equations are standard relations in linear system theory. The impulse response function (Kij) will be solved from added mass and damping data and the convolution integral (16) then evaluated for each term in the equations of motion at each time step during the simulation.

　

3.1 Numerical Solution of Kernel Functions

　

 To solve the Kernel functions, use is made of the Discrete Fourier transforms (DFT). The DFT is particularly suitable for describing phenomena related to a discrete time series. It can be developed from the Fourier transform of the continuing waveform samples of which are taken to form the time series. Hence, the retardation (Kernel) function for any number of sample values is written as

　

　

where B(ω) is damping coefficient and dω. is frequency range. Note that the ωt expression can be written by means of general physical description as

　

　

where t(N) indicates the each time step.

　

 In order to carry out parametrical study using the developed code, a container ship and a purse seiner trawler for which very extensive input and seakeeping test data for a following-sea environment is available, are used. The developed numerical model already has been validated using this test data. Those validation results and all details regarding the model tests can be found in Ayaz et al. [6],[7]. Lines plans and particulars of the ships are given in Figure 2, 3 and Table 1:

　

 The GM selected is not the design GM value, but one that only just satisfies the IMO regulations. In the numerical simulations parameters and conditions were chosen to be similar to previous ITTC Benchmark tests [4], [5], [6].

　

 Here,the numerical calculations were carried out in order to compare the simulation of ship motions in different degrees of freedom and with and without the memory effects included.

　

　

　

　

4.1 Results and Discussion of Parametrical Study

　

Effects of degree of freedom: In the previous studies such as Umeda et al. [5], it was shown that difference between 6 DOF and 4 DOF numerical model with static equilibrium in heave and pitch that has traditionally been used in the simulation of ship motions in astern seas, is not so significant. Based on this conclusion both ships have been tested in different wave steepness for the parameters and conditions given in Table 2 and 3. Maximum rudder angle is also set 10 and 35 degrees, respectively. Results for the numerical simulations are given in Figures 4-11. Here, A indicates the 6 DOF numerical model and B indicates the 4 DOF numerical model with static equilibrium in heave and pitch.