Resonant oscillations of water waves I. Theory

A theory is presented to describe the oscillations of a liquid in a tank near a resonant frequency, where linearized theory is invalid. It is shown that although the oscillations are described adequately by the classical wave equation, the boundary conditions cannot be properly satisfied unless the non-linear terms are included. The effects of dissipation and dispersion are also significant in the determination of the oscillations, even though the terms to which they give rise in the equations are multiplied by small parameters under normal laboratory conditions. When the former is dominant a weak bore is formed which travels to and fro in the tank and is continually reflected at either end. When dispersion is significant the surface profile can be likened to a series of cnoidal waves which also travel along the tank and suffer reflexion. Several novel features appear. The amplitude does not increase monotonically as the nominal resonant frequency is approached. There are several distinct frequencies at which there is a sharp change in amplitude and in the form of the profile. More than one stable oscillation is possible at some frequencies. Near a resonant frequency higher than the fundamental, subharmonic oscillations are possible over part of the range.