THE WATER ENTRY PROBLEM

Abstract

An investigation is made of the well-known boundary value problem posed by the symmetric entry of a wedge into an incompressible non-viscous fluid. The use of similarity variables makes this difficult problem somewhat more tractable and the restriction to two dimensions enables complex variable theory to be used. Some errors in the existing literature are pointed out and a closer look is taken at certain properties of the solution which appear hitherto to have been accepted without proof. For example, the question of whether or not the free surface is convex to the fluid is examined in detail and it is shown that this convexity is essential for the validity of results which depend on the use of the Wagner function. An independent proof is given of Garabedian's result that the angle of contact between the free boundary and the wedge lies between 0 and π/4 and this angle is shown to be related in an interesting way to the curvature at the point of contact. Although most of the analysis is concerned with the symmetric two-dimensional problem, some of the results are applicable to unsymmetric two-dimensional and to three-dimensional water entry patterns.