The conformal mappings are described in greater detail in section Conformal Mapping on page 9, but the general form of the mapping equation is given below in
Equation ( 46 ).

The work of Ursell (1949) develops a formulation for the added mass and damping of a heaving circular cylinder in a free surface. The presence of the free surface gives rise to the frequency dependence of the hydrodynamic coefficients. Ursell used a multipole expansion of the stream function and velocity potential to determine the flow around the cylinder, and hence derive the hydrodynamic coefficients.

The principle steps of the method are given in the following sections.