An infinitely long circular cylinder is fixed with its generators horizontal so that it is half-immersed, with its axis lying in the free surface of water. A regular train of water waves is incident on the cylinder from an arbitrary horizontal direction, and is partly reflected and partly transmitted under the cylinder. In the present paper we are concerned with the vertical component of the wave acting on the cylinder. It is assumed that the fluid is inviscid, that the fluid motion is irrotational, and that the depth of water is infinite. The equations of motion are linearized, and surface tension is neglected.
We shall find it convenient to use the fact that the required vertical force component can be inferred from the solution of a related problem, which we shall call the generalized heaving problem. In this latter problem a certain normal velocity is prescribed on the cylinder so that water waves which travel obliquely outwards are generated. There are no waves incident from infinity. When the prescribed velocity has the same phase everywhere on the cylinder the waves travel normally outwards, and in this case the generalized heaving problem reduces to the ordinary heaving problem, on which much information is already available. The generalized problem is solved here by a method which is a generalization of the known method (Ursell 1949) for ordinary heaving (when the wave crests are parallel to the cylinder axis). Generalized-added-mass coefficients and generalized-wave-making parameters for generalized heaving are computed for a range of wavenumbers and angles of travel, and are extended to larger wave-numbers by means of asymptotic analysis. Reciprocity relations (the Haskind relations) are then used to obtain the vertical force component in the original transmission problem from the wave-making parameters of the generalized heaving problem.