We consider the internal gravity waves that are produced in an inviscid Boussinesq fluid, whose Brunt–Väisälä frequency N is constant, by the small rectilinear vibrations of a horizontal elliptic cylinder whose major axis is inclined at an arbitrary angle to the horizontal. When the angular frequency ω is greater than N, no waves are produced and the governing elliptic equation is solved using conformal transformations. Analytic continuation in ω to values less than N, when waves are produced, is then used to determine the solution. It exhibits the surprising feature that, apart from certain phase differences, the form of the velocity distributions in each of the beams of waves that occur is the same for all values of the thickness ratio of the ellipse, the inclination of its major axis to the horizontal and the plane in which the vibrations are occurring. The Fourier decomposition of the velocity distribution is found and is used in a sequel, Part 2, to investigate the effects of viscous dissipation.

In an important paper Makarov et al. (1990) have given an approximate solution for a vibrating circular cylinder in a viscous fluid. We show that the limit of this solution as the viscosity tends to zero is not the exact inviscid solution discussed herein. Further comparison of their work and ours will be made in Part 2.