  We show that interfacial gravity waves comprising strong hydraulic jumps
(bores) can be described by a two-layer hydrostatic shallow-water (SW)
approximation without invoking additional front conditions. The theory is based
on a new SW momentum equation which is derived in locally conservative form
containing a free parameter $\alpha.$ This parameter, which defines the
relative contribution of each layer to the pressure at the interface, affects
only hydraulic jumps but not continuous waves. The Rankine-Hugoniot jump
conditions for the momentum and mass conservation equations are found to be
mathematically equivalent to the classical front conditions, which were
previously thought to be outside the scope of SW approximation. Dimensional
arguments suggest that $\alpha$ depends on the density ratio. For nearly equal
densities, both layers are expected to affect interfacial pressure with
approximately equal weight coefficients, which corresponds to $\alpha\approx0.$
The front propagation velocity for $\alpha=0$ agrees well with experimental and
numerical results in a wide range of bore strengths. A remarkably better
agreement with high-accuracy numerical results is achieved by
$\alpha=\sqrt{5}-2,$ which yields the largest height that a stable gravity
current can have.
