<p>Wave diffraction-radiation by a ship that advances through waves or in calm water, or by a stationary body, such as an offshore structure, is widely analyzed within the framework of the potential-flow theory and the classical “Green-function and boundary-integral flow-representation method” based on a Green function that satisfies the linear “Kelvin-Michell” boundary condition at the free surface. A crucial element of this usual theoretical approach is the boundary-integral flow representation that is obtained by applying Green’s basic identity to the flow potential and the Green function in the flow region outside the mean wetted surface of the body (ship, offshore structure). For a ship that advances in calm water or through waves, the boundary-integral flow representation obtained in that classical approach—widely called the Neumann-Kelvin (NK) theory contains a notoriously troublesome line integral around the ship waterline that cannot be reliably evaluated. This fundamental issue is considered based on an alternative linear flow model in which an open free-surface-piercing ship-hull surface is closed by a rigid horizontal lid submerged at an infinitesimally small depth below the free surface. This linear flow model, called the rigid-waterplane (RW) model, yields a <i>remarkably simple new weakly singular integral equation</i> that does not involve the flow potential at the ship waterline and is very well-suited for reliable numerical computations via a typical low-order panel method.</p>