Internal waves caused by the interaction of three harmonics under no restoring forces

The physical system under consideration consists of two fluids in uniform horizontal laminar motion parallel to their interface. All external restoring forces are absent. We find that nevertheless, three-wave interaction between different harmonics of the motion can take place. Attention is focused on the situation when this resonance is among the lowest three harmonics. Using the method of strained coordinates, a system of three coupled nonlinear partial differential equations is derived. These model the propagation of the interface, correct to second order. Solutions are found whose stability is briefly considered.