A two-dimensional wave motion is set up in an ideal fluid under gravity by an infinitely long cylinder with horizontal generators which is oscillating vertically with constant frequency and small amplitude. Surface tension is neglected. At a distance from the cylinder the motion approximates to a wave-train travelling away from the cylinder. The amplitude at infinity depends on the shape of the cylinder and on the frequency, and this dependence is relevant to the design of wave-makers and to the wave-damping of ships' motions. Approximate methods for calculating it have been given by several writers, but their results do not agree with a recent exact calculation for short waves. A critical comparison between exact and approximate methods is now made for very long and very short waves. The rigorous method for short waves is very laborious, and is replaced here by plausible assumptions about the velocity potential. The principal result is that short waves generated on one side of the body cannot pass under the body and therefore cannot contribute to the wave amplitude on the other side. This is intuitively obvious but is not obtained by the earlier methods. Green's theorem on potential functions plays a central part and is used to obtain the leading term in the wave amplitude when the wavelength is short. The representation of bodies by a source distribution is criticized.