Questions: Froude Number and Gravity Wave Propagation

Question 1

A dam spillway discharges flow at Fr = 2.4 into a downstream channel. An operator partially closes a gate far downstream. What effect does this have on the flow on the spillway face?

• A. The gate closure raises the water level on the spillway because it backs water upstream
• B. The flow on the spillway is unaffected — supercritical flow cannot receive information from downstream
• C. The gate closure increases the Froude number on the spillway by restricting outflow
• D. The gate causes a hydraulic jump to propagate back up the spillway

Answer: (B)

In supercritical flow (Fr > 1), the flow velocity exceeds the wave speed, so no gravity wave — which carries information about downstream conditions — can propagate upstream. The gate closure creates a disturbance that propagates upstream only at wave speed c, but the flow sweeps it back downstream faster. The spillway flow is therefore completely controlled by upstream conditions (reservoir head and spillway geometry) and is indifferent to the gate. This is the central practical consequence of the Froude number: it determines which end of a channel controls the flow.

Question 2

At critical flow (Fr = 1), surface gravity waves are stationary relative to the ground.

• T. True
• F. False

Answer: True

Critical flow is defined as the condition where the flow velocity V equals the surface wave speed c = √(gD). A small gravity wave propagating upstream at speed c against a current moving downstream at speed V = c has zero net velocity relative to the ground — it is stationary. This is the open-channel analogue of a sonic condition in compressible flow. It is also why critical flow is the control point in many structures: at a weir, sluice gate, or channel contraction, flow passes through Fr = 1 and the structure 'controls' discharge independently of downstream depth.

Question 3

In subcritical flow, downstream boundary conditions have no influence on the upstream flow profile.

• T. True
• F. False

Answer: False

This is precisely backwards. In subcritical flow (Fr < 1), the flow velocity is less than the wave speed, so gravity waves can propagate upstream. Downstream conditions — such as a dam, a gate, or a change in channel slope — send wave signals upstream that modify the water surface profile. This is why backwater curves in subcritical channels are calculated starting from the downstream boundary and working upstream. In supercritical flow (Fr > 1), the statement would be true: downstream conditions cannot communicate upstream and have no influence on the upstream profile.

Question 4

Why must the transition from supercritical to subcritical flow occur abruptly as a hydraulic jump rather than gradually?

Model answer: In supercritical flow, waves cannot propagate upstream, so the downstream subcritical region cannot send information upstream to signal 'slow down gradually.' Without this upstream communication, no smooth deceleration profile can be established. The transition must therefore occur abruptly and locally — the hydraulic jump — where depth increases sharply, velocity drops, and energy is dissipated as turbulence. The inability to communicate upstream is the same reason a supersonic aircraft produces a shock wave rather than gradually decelerating to subsonic: the physics of signal propagation prevents a smooth transition across the critical threshold.

The Froude number analogy to the Mach number is deep here. In both cases, the transition across the critical value (Fr = 1 or M = 1) is violent because the mechanism that allows gradual adjustment — wave propagation upstream — is disabled. Engineers exploit hydraulic jumps deliberately in stilling basins to dissipate the kinetic energy of high-velocity spillway discharge, converting it to heat and turbulence rather than erosive force on the downstream channel.

Question 5

A river transitions from a steep gorge (fast, shallow) to a flat floodplain (slow, deep). Which Froude regime applies in each reach, and what flow feature likely occurs at the transition?

• A. Subcritical in the gorge, supercritical on the floodplain; a hydraulic jump occurs at the gorge entrance
• B. Supercritical in the gorge, subcritical on the floodplain; a hydraulic jump occurs at the transition
• C. Critical flow in both reaches because total energy is conserved
• D. The Froude number is the same in both reaches because discharge is constant

Answer: (B)

Steep channels with high velocity and shallow depth produce high Froude numbers (supercritical). Flat channels with low velocity and deep water produce low Froude numbers (subcritical). When the supercritical flow from the gorge meets the subcritical conditions on the floodplain, a hydraulic jump forms at the transition point. The jump abruptly raises the water depth, drops the velocity, and dissipates a large fraction of kinetic energy as turbulence. This is why hydraulic jumps are common where mountain streams enter flat valleys. Option D is wrong: equal discharge doesn't imply equal Fr — velocity and depth can change together while Q = VA·D remains constant.