Locally forced critical surface waves in channels of arbitrary cross section
نویسنده
چکیده
Critical long surface waves forced by locally distributed external pressure applied on the freesurface in channels of arbitrary cross section are studied in this paper. The fluid under consideration isinviscid and has constant density. The upstream flow is uniform and the upstream velocity is assumedto be near critical, i.e., uo = u c + e2 + 0(e2), where 0 < e ~ 1 and u c is the critical velocity determined bythe geometry of the channel. The external pressure applied on the free surface as the forcing is e2P,~(x).Then the first order perturbation of the free surface elevation satisfies a forced Kor teweg-de Vriesequation ( fK-dV) . It is shown in this paper that: (i) If 2 > 2~ = (3b2pZot2/(-41~m12)) 1/3 > 0 (supercrit-ical), the stationary f K d V has two cusped solitary wave solutions; (ii) if 2 < 2L = (3bZPZo:2/~m~) 1/3 <0 (subcritical), the stationary f K d V has a downstream cnoidal wave solution; (iii) when 2 = 2 L, theunique stationary solution of the f K d V is a wave free hydraulic fall; (iv) if 2 = 2 a = 2 L , the f K d Vhas a jump solution; and (v) if 2L < 2 < 2c, the f K d V does not have stationary solutions. Some freesurface profiles and bifurcation diagrams are presented.(Received: November 30, 1989; revised: July 16, 1990)
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