Flow
| q = V₁·y₁ | |
| Q = q·b | |
| V₁ = Fr₁·√(gy₁) | |
| V₂ = q/y₂ | |
| Fr₂ (subcritical) | |
| Critical depth y꜀ | |
Jump geometry
| Sequent depth y₂ | |
| Height h_j = y₂ − y₁ | |
| Relative height h_j/E₁ | |
| Length L = 9.75·y₁·(Fr₁−1)^1.01 (empirical) | |
| Relative length L/y₁ | |
STEADY JUMP
Momentum (conserved across the jump)
| M₁ = q²/(gy₁) + y₁²/2 | |
| M₂ = q²/(gy₂) + y₂²/2 | |
| Momentum check |M₁ − M₂|/M₁ | |
Energy (destroyed by the roller)
| E₁ = y₁ + V₁²/2g | |
| E₂ = y₂ + V₂²/2g | |
| h_L = E₁ − E₂ | |
| h_L = (y₂−y₁)³/(4y₁y₂) | |
| Energy dissipated h_L/E₁ | |
| Efficiency E₂/E₁ | |
Bélanger: y₂/y₁ = ½(√(1+8Fr₁²) − 1)
M = q²/(gy) + y²/2 (specific force ÷ ρg, per unit width)
h_L = (y₂−y₁)³/(4y₁y₂) · L/y₁ = 9.75(Fr₁−1)^1.01
E₂/E₁ = [(1+8Fr₁²)^{3/2} − 4Fr₁² + 1] / [8Fr₁²(2+Fr₁²)]
Self-checks: M₁ and M₂ are computed independently each frame — their
equality is the momentum principle the Bélanger equation comes from. The two h_L rows
(energy balance vs momentum-derived formula) must also agree exactly, and do.