Conjugate depth
# Conjugate depth

In fluid dynamics, the conjugate depths refer to the depth $(y_1)$ upstream and the depth $(y_2)$ downstream of the hydraulic jump whose momentum fluxes are equal for a given discharge (volume flux) q. The depth upstream of a hydraulic jump is always supercritical. It is important to note that the conjugate depth is different from the alternate depths for flow which are used in energy conservation calculations.

## Mathematical derivation

Beginning with an equal momentum flux M and discharge q upstream and downstream of the hydraulic jump:

    $M={\frac {y_{1}^{2}}{2}}+{\frac {q^{2}}{gy_{1}}}={\frac {y_{2}^{2}}{2}}+{\frac {q^{2}}{gy_{2}}}.$

Rearranging terms gives:

    ${\frac {q^{2}}{g}}\left({\frac {1}{y_{1}}}-{\frac {1}{y_{2}}}\right)={\frac {1}{2}}\left(y_{z}^{2}-y_{1}^{2}\right).$

Multiply to get a common denominator on the left-hand side and factor the right-hand side:

    ${\frac {q^{2}}{g}}\left({\frac {y_{2}-y_{1}}{y_{1}y_{2}}}\right)={\frac {1}{2}}(y_{2}-y_{1})(y_{2}+y_{1}).$

The $(y_2−y_1)$ term cancels out:

    ${\frac {q^{2}}{g}}\left({\frac {1}{y_{1}y_{2}}}\right)={\frac {1}{2}}(y_{2}+y_{1})\qquad {\text{where }}q_{1}^{2}=y_{1}^{2}v_{1}^{2}=y_{2}^{2}v_{2}^{2}.$

Divide by $y_1^2$

    ${\frac {v_{1}^{2}}{g}}\left({\frac {1}{y_{1}y_{2}}}\right)={\frac {1}{2y_{1}^{2}}}(y_{2}+y_{1})\qquad {\text{recall }}Fr_{1}^{2}={\frac {v_{1}^{2}}{gy_{1}}}.$

Thereafter multiply by $y_2$ and expand the right hand side:

    $Fr_{2}^{2}={\frac {y_{2}^{2}}{2y_{1}^{2}}}+{\frac {y_{2}}{2y_{1}}}.$

Substitute x for the constant $y_2/y_1:$

    $Fr_{1}^{2}={\frac {x^{2}}{2}}+{\frac {x}{2}}\Rightarrow 0={\frac {x^{2}}{2}}+{\frac {x}{2}}-Fr_{1}^{2}.$

Solving the quadratic equation and multiplying it by ${\tfrac {\sqrt {4}}{2}}$ gives:

    $x={\frac {-{\tfrac {1}{2}}\pm {\sqrt {(1/2)^{2}+4(1/2)(Fr_{1}^{2})}}}{2(1/2)}}.$

Substitute the constant $y_2/y_1$ back in for x to get the conjugate depth equation

    ${\frac {y_{2}}{y_{1}}}={\frac {1}{2}}\left({\sqrt {1+8Fr_{1}^{2}}}-1\right)$

Note that this equation is only applicable to hydraulic jumps over flat beds.

