First- and second-order forces on a cylinder submerged under a free surface

Several years ago Dean (1948) showed how to solve the linearized potential problem of water waves passing over a submerged circular cylinder. He discovered the remarkable fact that there is no reflexion from the cylinder; the transmitted waves have the same amplitude as the incident waves, but they suffer a phase shift in passing the cylinder. Soon after the publication of Dean's paper, Ursell (1950) investigated the problem anew. He placed the solution on a rigorous basis, supplied a uniqueness proof, and developed a form of the solution with which it was reasonable to perform calculations.