The solution is sought of certain unsteady problems in hydrodynamics in which the fluid is bounded by a surface of constant pressure. If no characteristic length parameter is supplied in the initial data, then the position co-ordinates X, Y, Z and the time t occur only in the combinations X/t, Y/t, Z/t. Among the problems discussed are the normal and oblique entry of a thin wedge into a plane surface of water and the corresponding normal entry problem for a thin cone. The method of solution is to linearize the boundary conditions with respect to ɛ, the semi-angle of the wedge or cone. A number of classical boundary value problems arise and, when these are solved, explicit formulae are obtained for the equation of the free boundary.
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A. G. Mackie (1962) studied this question.