Nonlinear interfacial progressive waves near a boundary in a Boussinesq fluid

The behavior of nonlinear progressive waves at the interface between two inviscid fluids in the presence of an upper free boundary is studied as a model of waves on the thermocline. A set of relationships between the integral properties of bounded waves in a general two-fluid model is first developed and the Stokes expansion to third order is derived. The exact free boundary problem for the wave profile is then formulated within the Boussinesq approximation as a nonlinear integral equation, which is solved numerically using two different numerical methods. For finite velocity difference across the two-fluid interface bifurcation of solutions into upper and lower branch wave profiles with quite different properties is obtained. Numerically calculated wave shapes and integral properties show good agreement with third-order Stokes expansion predictions in the weakly nonlinear regime for waves which are not too long. Very long waves were found to exhibit distinct solitary wave-like features