As time evolves, the motion of water becomes more and more complicated. The added mass changes over time, too: the added mass describes the total kinetic energy of water, but this kinetic energy is changing because (a) a larger and larger fraction of the cylinder becomes immersed, and (b) the free surface of the water becomes distorted. However, things are relatively simple at the very first moment; the cylinder is immersed only to a negligible depth (as compared to its diameter), and the water surface is still almost perfectly flat. The problem of finding added mass in this configuration can be reduced to another added mass problem by noticing that at the first moment, the water parcels at the free surface acquire strictly vertical velocities because the top surface is subject to atmospheric pressure and hence, there cannot be horizontal pressure gradients.
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