lntrod uct ionThis paper contains work relating to the vertical impact of a cone on a water surface which was carried out at the Applied Mathematics Group, New York University, during the year 1945. If a cone enters water vertically at constant speed, similitude of the flow at different instants makes it possible to determine the force of impact in an exact manner by means of an iteration scheme involving an integral equation. This computation for a cone of vertex angle 120' was presented to Dr. A. Hillman, then with the Mathematical Tables Project, who In the first section we develop an exact theory for the vertical entry of a cone at constant speed under the assumption that the fluid is an ideal incompressible one, and we present the results of Dr. Hillman's computation for the 120' cone. Since the computation is difficult, it seems unlikely that it will be repeated for various cone angles and, for this reason, it is desirable to give a simpler approximate theory from which the impact force can easily be computed for any cone angle. This is done in section 2 by taking, as first approximation, the velocity potential arising from an ellipsoid circumscribed about the cone and then making wetting and free surface corrections in the manner described in the report [2a] for the sphere.Finally we compare our results with the experimental measurements of the Japanese investigator Watanabe [5a,5b].Final results are presented in the form of a graph (Graph 1) which gives the dimensionless virtual mass k of the fluid as a function of the vertex angle e of the cone. The actual virtual mass M is e M = kuB3 tan3 -2 where u is the fluid density and B is the depth of penetration below the initial water level. The impact force P is computed using the formula 3M
= + ( M / M , ) ]~where M , is the mass of the cone and U, the initial entry velocity.
JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.