A stationary hydraulic jump occurs in a rectangular channel with initial and sequent depth being equal to 0.40 m and 1.60 m respectively. Calculate initial Froude number and energy loss.

Concept:

To determine the initial Froude number and energy loss in a hydraulic jump, we use specific formulas based on the depths before and after the jump.

Given Data:

• Initial depth (\( y_1 \)) = 0.40 m
• Sequent depth (\( y_2 \)) = 1.60 m

Calculation of Initial Froude Number (\( F_1 \)):

The Froude number (\( F_1 \)) can be calculated using the relationship between initial and sequent depths:

\( \frac{y_2}{y_1} = \frac{1}{2} \left(\sqrt{1 + 8F_1^2} - 1\right) \)

Substituting the given values:

\( \frac{1.60}{0.40} = \frac{1}{2} \left(\sqrt{1 + 8F_1^2} - 1\right) \)

Simplifying:

\( 4 = \frac{1}{2} \left(\sqrt{1 + 8F_1^2} - 1\right) \)

\( 8 = \sqrt{1 + 8F_1^2} - 1 \)

\( \sqrt{1 + 8F_1^2} = 9 \)

\( 1 + 8F_1^2 = 81 \)

\( 8F_1^2 = 80 \)

\( F_1^2 = 10 \)

\( F_1 = \sqrt{10} \approx 3.16 \)

Calculation of Energy Loss (\( \Delta E \)):

The energy loss in a hydraulic jump is given by:

\( \Delta E = \frac{(y_2 - y_1)^3}{4 y_1 y_2} \)

Substituting the given values:

\( \Delta E = \frac{(1.60 - 0.40)^3}{4 \times 0.40 \times 1.60} \)

\( \Delta E = \frac{1.20^3}{4 \times 0.40 \times 1.60} \)

\( \Delta E = \frac{1.728}{2.56} \)

\( \Delta E = 0.675 \, \text{m} \) or 27/40

Hence, the most appropriate answer is option 1.