Introduction

In a horizontal rectangular channel, a hydraulic jump occurs when there is a sudden change in flow conditions, resulting in a rapid increase in water depth and a decrease in velocity. This phenomenon is commonly observed in open channel flow, and it is important to understand the relationship between the depths before and after the jump, as well as the critical depth.

Explanation

1. Definition of terms:
- y1: Depth of water before the hydraulic jump
- y2: Depth of water after the hydraulic jump
- yc: Critical depth at the hydraulic jump

2. Bernoulli's Equation:
The relationship between the flow depth, velocity, and pressure is described by Bernoulli's equation. For horizontal flow in a rectangular channel, the equation can be simplified to:

P/γ + y + V^2/2g = constant

Where:
- P: Pressure head
- γ: Specific weight of water
- y: Depth of water
- V: Velocity of water
- g: Acceleration due to gravity

3. Conditions before the hydraulic jump:
Before the hydraulic jump, the flow is supercritical, meaning that the flow velocity is greater than the critical velocity. At this point, the flow depth (y1) is greater than the critical depth (yc). Applying Bernoulli's equation before the jump:

P/γ + y1 + V1^2/2g = constant

4. Conditions after the hydraulic jump:
After the hydraulic jump, the flow transitions to a subcritical state, where the flow velocity is less than the critical velocity. The flow depth (y2) increases significantly, while the velocity decreases. Applying Bernoulli's equation after the jump:

P/γ + y2 + V2^2/2g = constant

5. Relationship between depths:
From the above equations, we can equate the constants on both sides:

P/γ + y1 + V1^2/2g = P/γ + y2 + V2^2/2g

Canceling out the pressure terms and rearranging the equation:

y1 + V1^2/2g = y2 + V2^2/2g

6. Relationship with critical depth:
At the hydraulic jump, the flow depth (yc) becomes the critical depth. The critical depth can be determined using the specific energy equation, which states that the specific energy (E) remains constant throughout the flow:

E = y + V^2/2g = constant

Substituting the critical depth (yc) into the equation:

E = yc + Vc^2/2g = constant

Since the specific energy is constant, we can equate the specific energies before and after the jump:

y1 + V1^2/2g = yc + Vc^2/2g = y2 + V2^2/2g

Therefore, in a horizontal rectangular channel, the relationship between the depths before the jump (y1) and after the jump (y2) is given by:

yc^3 = y1 * y2 * y1 * y2

Conclusion

In conclusion, the hydraulic jump in a