A note on a Galerkin technique for integral equations in potential flows

The properties are studied of a Galerkin numerical solution of integral equations for an assumed singularity distribution or a velocity potential arising in potential flows around rigid bodies in incompressible aerodynamics, acoustics and surface waves. The body boundary is approximated by a collection of panels and the integral equation is averaged over each panel instead of being enforced at a collocation point. For the resulting Galerkin synthesis the matrix equation obtained for the source distribution is the exact transpose of the corresponding equation obtained for the velocity potential on the body boundary, a property known to hold for the continuous operators. Moreover, the integrated hydrodynamic forces experienced by the body are shown to be identically predicted by the source-distribution method or by directly solving for the velocity potential.