fluid dynamics In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids— liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) an ...

, 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 Discharge may refer to Expel or let go * Discharge, the act of firing a gun * Discharge, or termination of employment, the end of an employee's duration with an employer * Military discharge, the release of a member of the armed forces from serv ...

(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 Energy conservation is the effort to reduce wasteful energy consumption by using fewer energy services. This can be done by using energy more effectively (using less energy for continuous service) or changing one's behavior to use less service (f ...

calculations.

Mathematical derivation

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

M=\frac+\frac=\frac+\frac.

Rearranging terms gives: :

\frac\left(\frac-\frac\right)=\frac\left(y_z^2-y_1^2\right).

Multiply to get a

common denominator In mathematics, the lowest common denominator or least common denominator (abbreviated LCD) is the lowest common multiple of the denominators of a set of fractions. It simplifies adding, subtracting, and comparing fractions. Description The low ...

on the left-hand side and factor the right-hand side: :

\frac\left(\frac\right)=\frac(y_2-y_1)(y_2+y_1).

The (''y''₂−''y''₁) term cancels out: :

\frac\left(\frac\right)=\frac(y_2+y_1)\qquad\textq_1^2=y_1^2 v_1^2=y_2^2 v_2^2.

Divide by ''y''₁² :

\frac\left(\frac\right)=\frac(y_2+y_1)\qquad\textF r_1^2=\frac.

Thereafter multiply by ''y''₂ and expand the

right hand side In mathematics, LHS is informal shorthand for the left-hand side of an equation. Similarly, RHS is the right-hand side. The two sides have the same value, expressed differently, since equality is symmetric.quadratic equation and multiplying it by

\tfrac

gives: :

x=\frac.

Substitute the constant ''y''₂/''y''₁ back in for ''x'' to get the conjugate depth equation :

\frac=\frac\left(\sqrt{1+8F r_1^2} - 1\right)

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