Interfacial viscous ship waves near the cusp lines

The interfacial waves due to a steady Oseenlet in a system of two semi-infinite immiscible fluids of different densities are investigated analytically. The two-fluid system which consists of an upper inviscid and a lower viscous fluid is assumed to be incompressible, homogenous and stable. The interfacial elevation is given by a double Fourier integral, which involves a generic amplitude function, a complex dispersion function and an elementary phase function. With the aid of four kinds of expansion of the phase function, the asymptotic representations of the far-field wave profiles are explicitly derived for four regions as the azimuth angle is less than, less than but close to, at, and greater than but close to, the maximal angle of the V-wedge for Kelvin ship waves. Graphical representations of wave profiles show the consistency of four asymptotic schemes. A combination of the Stokes and Scorer methods of stationary phase gives uniformly valid results over the panoramic region.