Boundary integral relations for submerged bodies and free-surface piercing ships and offshore structures

The linear flow model called rigid waterplane flow model, in which a body (ship, offshore structure) that pierces the free surface is treated as the smooth limit of a body that is submerged at a small depth, is considered. The flow within the thin layer between the horizontal rigid lid that closes the (open) body surface and the waterplane above the lid is analyzed. This analysis shows that the seemingly different boundary integral relations obtained by applying Green's basic identity to the boundary value problems associated with the rigid waterplane flow model or a body that pierces the free surface are equivalent. Additionally, the rigid waterplane flow model yields three alternative weakly singular boundary integral relations, which only involve distributions of sources and weak dipoles that are continuous at the body surface. One of these three relations is notable because it does not involve a waterline integral, and only involves the flow potential (not its derivatives along directions tangent to the body surface) and smooth ordinary functions that are easily evaluated and integrated.