Firstly, the problem of separation of flow at transom sterns is presented. A Virtual Appendage Method is suggested and applied to remedy this problem by smoothly extending the hull at the transom stern such that it encloses the separated flow. Secondly, the problem of irregular frequencies that all boundary integral methods - with pulsating source formulations - suffer from is presented, together with more than one technique for prediction and elimination. A Multiple Green Function Expression Method is suggested and applied by adding singularities in the interior of the hull at the free surface and modifying the Green function to account for their influence on solving the boundary value problem. Finally, the influence of the steady wave system is accounted for in the body boundary condition of the unsteady wave pattern for a more accurate representation of the forward speed effects. A Kelvin Wave Source Potential (Kwsp) based code is used for the numerical evaluation of the steady-state potential. The KWSP code is extended for calculating the second derivatives of the steady state potential by numerically differentiating the first derivatives and using relationships which the second derivatives satisfy (e.g. Laplace's equation and symmetry relationships). Subsequently, the so-called m-terms are calculated as a function of the obtained derivatives and the complete form of the linear body-boundary condition of the unsteady problem is evaluated.