[4] A two-fluid system bounded by two rigid plates, as shown in Figure 1, is used for analysis. The flow in each layer is assumed to be irrotational and inviscid. The symbols Φ_{i}, *U*_{i}, *ρ*_{i}, and *h*_{i} represent the velocity potential for pure wave motions, the current speed, the density, and the undisturbed layer thickness for the upper (*i* = 1) and lower (*i* = 2) fluids, respectively. The horizontal coordinate is denoted by *x*, while *z* indicates the vertical coordinate starting at the undisturbed interface and pointing upward. The displacement of the interface is represented by *η*. The current in each layer is assumed to be uniformly flowing along the *x* direction. The governing equations and boundary conditions for the two-dimensional wave motions are as follows

where *g* and *σ* are the gravity and the surface tension coefficient, respectively. The symbol 〈*F*_{i}〉 is the difference in a quantity across the interface, namely 〈*F*_{i}〉 = *F*_{1} - *F*_{2} where the subscripts 1 and 2 indicate the upper and lower layers, respectively. The velocity potentials and the displacement of interface are then expanded as

where the superscripts indicate the order of magnitude in powers of the wave slope ɛ i.e. Φ_{1}^{(1)} = *O*(ɛ), Φ_{1}^{(2)}, = *O*(*ɛ*^{2}) etc. Inserting equation (5) into equations (1) to (4) and expanding boundary conditions about *z* = 0 using the Taylor-series expansion generate the wave system order by order. The first-order components are represented in the following complex forms

where c.c. denotes the complex conjugate of the preceding term, and. *k* and *ω* are the wave number and frequency of linear waves. The relations between A^{(1)}, B^{(1)} and D^{(1)} are readily obtained

The linear dispersion relation associated with the effects of currents and surface tension is

From equation (11), the critical condition for the well-known Kelvin-Helmholtz instability (for detailed descriptions, see *Drazin and Reid* [1981]) can be recovered and shown as

This critical condition indicates that when the difference of *U*_{1} and *U*_{2} exceeds the critical value, the initial disturbance will grow with time. It is also obvious that the existence of surface tension is advantageous to suppress the unstable phenomenon.

[5] Based on the above first-order solutions, the corresponding governing equations and boundary conditions for second-order components are

In equations (13) to (16), all second-order components appear in the left-hand sides, and the terms on the right denote the nonlinear effects arising from interactions of first-order components. The second-order components, which contain the super- and sub-harmonic parts generated by the interaction of the linear waves (*m*-wave and *n*-wave), are assumed to be:

where *k*_{mn}^{±} = *k*_{m} ± *k*_{n}, *ω*_{mn}^{±} = *ω*_{m} ± *ω*_{n} and the superscripts plus and minus denote the super- and sub-harmonic interactions, respectively. After substituting equations (17) to (19) into equations (13) to (16), the second-order amplitude *D*_{mn}^{(2)±} is solved

where

Note that the magnitude of *V* defined in equation (23) suggests the magnitude of current speed in comparison to the individual wave phase velocity. The coefficients of velocity potentials are solved and given below