Gradually varied theory, Eq. (10.49), neglects the effect of width changes, db/dx, assuming that they are small. But they are not small for a short, sharp contraction such as the venturi flume in Fig. P10.113. Show that, for a rectangular section with b = b(x), Eq. (10.49) should be modified as follows: Investigate a criterion for reducing this relation to Eq. (10.49).

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dx 1-Fr2