A FORMAL EXPANSION PROCEDURE FOR THE INTERNAL SOLITARY WAVE PROBLEM IN A 2-FLUID SYSTEM OF CONSTANT TOPOGRAPHY

Equations are derived for a two-dimensional internal solitary wave between two immiscible superimposed, inviscid and incompressible liquids bounded by two rigid planes in a channel of infinite length analogous to those derived by Epstein (Q. Appl. Math. 1974) for the case of two-dimensional free surface solitary water waves. The effect of surface tension at the surface of separation has been neglected. A pair of ordinary differential equations, of an "infinite" order, as well as an algebraic equation has been obtained and solved by a power series expansion in terms of a parameter "a" which depends mainly on the reciprocal of Froude number. Comparing similar powers of "a" leads to a set of non-linear, second-order differential equations. The analytical form of the wave profile has been found, up to a fourth order of the parameter "a", in terms of the Froude number, density ratio between the two liquids, and the distance between the two planes bounding the two fluids. The effect of the Froude number, density ratio and the distance between the two planes, on the wave profile, maximum amplitude of the wave and the speed of a particle just above the interface, has been studied and illustrated.