Consider a shallow stream of depth , uniform width, and uniform flow velocity . Suppose that there is a very shallow bump of height on the (horizontal) bed of the stream, as shown in Figure 4.5. Suppose, further, that, at the point where the stream passes over the top of the bump, its velocity is , and its surface rises a height above the unperturbed surface.

Fluid continuity yields

Furthermore, application of Bernoulli's equation to a streamline lying on the surface of the water (where the pressure is atmospheric) gives

(4.30) |

The previous equation reduces to

Eliminating between Equations (4.29) and (4.31), we obtain

(4.32) |

which can be rearranged to give

(4.33) |

or

Here,

(4.35) |

is the Froude number of the unperturbed flow. Finally, given that and , Equation (4.34) reduces to

(4.36) |

It follows, from the previous expression, that if the flow is super-critical, so that , then is positive. On the other hand, if the flow is sub-critical, so that , then is negative. Thus, if a super-critical shallow stream passes over a very shallow bump on its bed then the surface of the stream becomes slightly elevated. On the other hand, if a sub-critical stream passes over such a bump then the surface of the stream becomes slightly depressed. A similar effect occurs when there is a narrowing of the channel in the horizontal direction. A more sophisticated version of the previous calculation, which does not necessarily assume that the stream is shallow, can be found in Section 11.10.