Problem Diagram I

The upstream Froude number (Fr₁) can be determined from the relationship between the upstream and downstream depths across the hydraulic jump as follows:

     

The upstream velocity is obtained from the Froude number:

The continuity equation,

     Q = V₁by₁ = V₂by₂

gives
     V₂ = V₁ (y₁/y₂) = (21.23)(0.5/3.5) = 3.03 ft/s

The downstream Froude number is then calculated to be:

Specific Force Diagram

Specific Energy Diagram

The momentum function is defined as

     M = Q²/gA + zA

where A = by and z = y/2 for a rectangular channel, and the volumetric flow rate Q is determined as follows:
     
     Q = V₁by₁ = (21.23)(10)(0.5) = 106.15 ft³/s

The momentum function thus becomes

     M = Q²/gby + by²/2     
         = (106.15)²/(32.2)(10)y + (10)y²/2
         = 34.99/y + 5y²

The plot of y versus M is given in the specific force diagram. Note that the momentum functions are the same at sections 1 and 2 before and after the hydraulic jump (for a flat-bottomed channel).

The specific energy is given by

     E = y + (1/2g)(Q/by)²
           = y + (1/64.4)(106.15/10y)² = y + 1.75/y²

The plot of y versus E is given in the specific energy diagram. The hydraulic jump process is indicated by the dashed line. Note that the specific energy lost across the hydraulic jump equals the change in E between points 1 and 2. It is given by

     ΔE = E₁ - E₂
           = 0.5 + 1.75/0.5² - 3.5 - 1.75/3.5²
           = 3.86 ft