ABSTRACT
The eigenfunction expansion techniques described in Chapter 2 rely on the ability to expand the potential in terms of the vertical eigenfunctions constructed in 52.1. This can only be done when the fluid is made up of regions of constant finite depth and when the boundaries of all subregions coincide with coordinate lines or surfaces. In this chapter a different approach will be considered, in which the potential in the fluid is represented as a sum of singularities placed within any structures that are present. These singularities, called multipoles, are constructed to satisfy the field equation, the free-surface and bed boundary conditions (infinite and constant finite depth can be considered), and a radiation condition which demands that they behave like outgoing waves in the far field. A linear combination of these multipoles is then constructed and made to satisfy the appropriate body boundary condition. This leads to an infinite system of linear algebraic equations for the unknown coefficients in the multipole sum which can be solved numerically by truncation. Experience shows that the systems of equations that result from using a multipole method possess excellent convergence characteristics and only a few equations are needed to obtain accurate numerical answers.
