Added Masses and Forces on Two Bodies Approaching Central Impact in an Inviscid Fluid

In several papers, which will be referenced, a procedure based on integral equations has been developed and applied for determining the interaction forces on two bodies approaching central impact in an inviscid fluid. The present work was undertaken to evaluate the accuracy of the results from that procedure by applying it to a pair of circles and a pair of spheres with which one could obtain solutions, as accurate as desired, by the method of successive images. A second purpose was to refine the procedure so that accurate solutions could be obtained at closer distances than heretofore. Solutions by the method of images, given by Hicks and Herman over 100 years ago, are not very clear, and since we have significantly extended their theory in the present work, it seemed appropriate to include a new derivation which we consider more rational. The extensions of the theory consists of 1 a truncation correction of the infinite series of the doublet strengths for the added masses and their derivatives, which can then be calculated accurately with a moderate number of terms even when the gap between the bodies is very small 2 asymptotic formulas for the added masses and their derivatives at small gaps which show that, for circles, the derivatives with respect to a parameter asymptotically proportional to the square root of the gap, are finite, and that derivatives with respect to the gap approach infinity inversely as the square root of the gap 3 a treatment of the case of a circular cylinder or a sphere, or bodies of arbitrary shape approaching a wall, showing that the forces on the body and wall are repulsive and of equal magnitude.