On the planing of a flat plate at high Froude numbers in a two-dimensional case

We seek the solution of the planing of a flat plate at high Froude numbers by a perturbation procedure. The angle of attack of the plate is assumed to vary with the speed of the plate in the present study. A harmonic function K is introduced for the solution of the first-order disturbance potential which becomes the Green function in the limiting case when the Froude number tends to infinity. We get the solution of the first-order potential from Green's theorem applied to K and the first-order potential. Then we obtain the asymptotic solutions of the angle of attack α, lift L and drag D as follows: α∼WρLW1U,LW∼1-2π[1-ln(2sinα)]α,DW∼α, where α≪1. Here W, LW, and U are the weight of the plate per unit width, wetted length, and speed of the plate, respectively.