A solitary group of two-dimensional deep-water waves

If the magnitude of a line of concentrated force moving with a constant velocity on the surface of deep water oscillates with a constant frequency, a single group of two-dimensional gravity waves is created. The group moves with the velocity of the concentrated force, and the waves in it have a wave number and phase velocity which are determined by the velocity of the concentrated force and the frequency of oscillation of its magnitude. If these latter quantities are such that the velocity of the force is near the group velocity, in the classical sense, of the waves created, the length of the group will be very long, but in general it can take on any value. The phase velocity of the waves created is larger than that of classical deep-water waves of the same wave number, whereas the velocity of the group is less than the group velocity of these classical waves. The respective differences, as well as the magnitude of the concentrated force, tend to zero as the length of the group increases indefinitely. The solution given provides the elemental solution from which gravity wave groups caused by any travelling oscillating pressure distribution can be found.