Abstract
An analytical solution of the nonlinear equations of the theory of a directed fluid sheet (which includes vertical inertia) is used to prove that for constant depth at the waterfall's brink and variable far upstream depth, the energy head in steady flow in a free waterfall is minimized when the flow is critical. This result provides additional physical understanding of the special nature of critical flow in a waterfall and supplies further justification for using the free waterfall as a self-calibrated flow meter, as suggested by Rouse (1936). Also, this proof should not be confused with the more common proof of minimum energy head in a uniform stream because here the flow rate is not maintained constant and the fluid depth contracts as gravity accelerates the flow near the waterfall's brink.
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