An exact expression for transient forced internal gravity waves in a Boussinesq fluid

An exact expression is derived for the propagation of transient forced internal gravity waves in a Boussinesq fluid with constant horizontal mean velocity. The horizontal wavelength of the forcing is assumed to be large relative to the vertical wavelength of the perturbation, and so the long-wave limit is taken. The solution consists of a part with steady amplitude and a transient part that goes to zero in the limit of infinite time. Two independent time scales are identified in the evolution of the transient term, one connected to the height of the wave above the source level and the other to the horizontal velocity of the background flow. The amplitude of the transient term is determined by the vertical component of the group velocity of the wave propagation. Because of the exact nature of the solution, it can be used as a starting point for further analytical and numerical studies of gravity-wave propagation.