Image solution for vertical motion of a point source towards a free surface

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Abstract
The vertical constant-speed motion of a constant-strength point source towards a horizontal free boundary is analysed. A procedure based on expansions in even powers of the Froude number is employed. The asymptotic expansion of the potential is found to satisfy a simple differential equation, which, when integrated, yields an image-type solution valid for all Froude numbers. Froude-number effects are contained in a distribution of sources along the vertical line from the image of the submerged source with respect to the undisturbed free surface upward to infinity. The solution is valid for arbitrary values of the density ratio across the free surface.

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