Interfacial progressive water waves - a singularity-theoretic view (Q1360368)

This study presents a mathematical theory of bifurcation of capillary-gravity surface waves. Previous works indicate that the capillary-gravity surface waves of infinite depth can be described by two parameters, which correspond generically to the existence of double bifurcation points. On the other hand, numerical computations show that the actual bifurcation diagrams for surface waves are more complicated than those predicted by using the double bifurcation points, thus indicating the possible degeneracy. However, previous studies showed that degeneracy does not exist when one varies the depth of the flow. This paper establishes that interfacial waves can be considered as generalizations of the surface waves, and degeneracy does exist when a new parameter is introduced. The description of interfacial wave between two fluids of different densities contains a new parameter, \(b= m_uc^2_u/(m_lc^2_l)\), where \(m\) is the mass density, \(c\) is the mean speed of the fluid, the subscripts \(u\) and \(l\) denote the upper and the lower fluids, respectively. As \(b\to 0\), the problem reduces to the surface wave problem. It is demonstrated that a degenerate bifurcation point exists when the ratio of the propagation speeds is varied. The computations show that the aspect ratio of the flow, i.e., the depths of the upper and the lower fluids, play a little role in the bifurcation formation from the singularity-theoretic viewpoint. When two layers are simultaneously infinitely deep, the degenerate bifurcation point appears when \(b=1\). Consequently, the complicated structure of the surface waves can be explained by regarding surface waves as special cases of interfacial waves.