Deep-water gravity waves: nonlinear theory of wave groups

Abstract: Nonlinear initial-boundary value problem on deep-water gravity waves of finite amplitude is solved approximately (up to small terms of higher order) assuming that the waves are generated by an initial disturbance to the water and the horizontal dimensions of the initially disturbed body of the water are much larger than the magnitude of the free surface displacement. A numerable set of specific free surface waves is obtained in closed form and it is shown that free surface waves produced by an arbitrary initial disturbance to the water is a combination (not superposition: the waves are nonlinear) of the specific waves. A set of dispersive wave packets is found with one-to-one correspondence between the packets and positive integers, say, packet numbers, such that any initial free surface displacement gradually disintegrates into a number (limited or unlimited, depending on initial conditions) of the wave packets. The greater the packet number, the shorter the wavelength of the packet's carrier wave component, the slower the packet travels, the slower the packet disperses; evolution of any of the packets is not influenced by evolution of any other one. It is found that in case of infinitesimal wave amplitude the present theory is in agreement with linear wave theory. On the other hand, the behaviour of wave packets of large numbers and asymptotic behaviour of solutions of Schr\"odinger equation for weakly nonlinear waves are found to be similar, except for dispersion The theory is tested against experiments performed in a water tank.