A note on the existence of pure gravity waves at the interface of two fluids

Abstract

© 2019 Elsevier B.V. Studying numerically traveling gravity waves of large amplitude at the interface of two fluids of infinite extent, Meiron & Saffman (J. Fluid Mech., vol. 129, 1983, pp. 213–218) demonstrated the existence of overhanging waves, for which some portions of the heavier fluid lie above the lighter one. By making use of a global bifurcation theorem, Sun (SIAM J. Math. Anal, vol. 32, No. 5, 2001, pp. 1014–1031) proved that the interfacial traveling waves of large amplitude exist until either a bifurcation parameter, introduced by him, goes to infinity or the function of the wave profile looses its smoothness. In the present paper, we provide numerical evidence that none of the waves whose existence was proved by Sun is overhanging, moreover, as the density ratio of the fluids tends to unity, the amplitudes of these waves tend to zero. We also discuss the limiting passage from the interfacial waves to the surface ones when the density ratio tends to zero.