Linearization of Free Surface Boundary Conditions - Birk Lothar (Fundamentals of Ship Hydrodynamics, 2019)
This chapter formally derives the linearized versions of the kinematic and dynamic free surface conditions used in wave theory. The procedure is known as a perturbation approach and is applicable to the linearization of nonlinear equations in general. For the potential wave flow problem, the authors derive two free surface boundary conditions. Two boundary conditions are required because people have two unknown functions: the velocity potential and the actual position of the free surface relative to the calm water level. Unfortunately these boundary conditions are: nonlinear and implicit. The authors linearize the free surface boundary conditions based on a perturbation approach. The perturbation approach follows the general concept of mathematical series like the Taylor series or the Fourier series. The idea is that an approximate, basic solution for the unknown function is improved with additional terms that become smaller and smaller.