The ancients undoubtedly discovered the ideal shape of a water clock by trial and error. In examining some ancient water clocks I notice the shape is different depending on the size. For example, a 9-inch water clock will have a different shape than a 5-inch water clock. Obviously the size of the orifice is very important also, and for that reason water clocks were made from metal, even gold, or from hardened ceramic, so that the orifice could be sized very exactly.
Should a modern person wish to make a water clock without going through the agony of many hundreds of hours of experimentation, what theory could be used to determine the ideal shape using the principles of physics alone?
The type of clepsydra I am envision is one that would provide the height of the water as a linear function of time.