Locally forced critical surface waves in channels of arbitrary cross section

Critical long surface waves forced by locally distributed external pressure applied on the free surface in channels of arbitrary cross section are studied in this paper. The fluid under consideration is inviscid and has constant density. The upstream flow is uniform and the upstream velocity is assumed to be near critical, i.e.,u₀=u_c++0(²), where 0<1 andu_c is the critical velocity determined by the geometry of the channel. The external pressure applied on the free surface as the forcing is²P(x). Then the first order perturbation of the free surface elevation satisfies a forced Korteweg-de Vries equation (fK-dV). It is shown in this paper that: (i) If (supercritical), the stationaryfK-dV has two cusped solitary wave solutions; (ii) if (subcritical), the stationaryfK-dV has a downstream cnoidal wave solution; (iii) when=_L, the unique stationary solution of thefK-dV is a wave free hydraulic fall; (iv) if=_d=–_L, thefK-dV has a jump solution; and (v) if_L<<_c, thefK-dV does not have stationary solutions. Some free surface profiles and bifurcation diagrams are presented.