Internal Waves of a Two-Layer Fluid with Free Surface over a Semi-circular bump

In this paper we study steady capillary-gravity waves in a two-layer fluid bounded above by a free surface and below by a horizontal rigid boundary with a small obstruction, Two critical speeds for the waves are obtained. Near the smaller critical speed, the derivation of the usual forced KdV equation (FKdV) fails when the coefficient of the nonlinear term in the FKdV vanishes. To overcome this difficulty, a new equation called a forced extended KdV equation (FEKdV) governing interfacial wave forms is obtained by a refined asymptotic method. Various solutions and numerical results of this equation are presented.