Ali S. Fayad, Discoverer of The Explicit Alternate Flow Depth in Rectangular Open Channels

Constant Specific-Force Transition:

The water flow transition, in an open channel, from a rapid low profile into a slow high profile is designated as a hydraulic jump. The specific force of the flow is preserved in the process while a portion of the energy is dissipated within the transition even for an assumed frictionless flow.

The Belanger form of the conjugate depths formula for a hydraulic jump in a horizontal rectangular open channel is:

Final/Initial =0.5{-1+[8Fr² +1]^1/2 }

Here, Fr represents the value of the Froude number of the flow initiating an idealized hydraulic jump.

yf/yi = 0.5 [(8Fri² +1)^1/2 -1]

The subscript "i", in the above equation, corresponds to the initiating flow of the jump and the subscript "f" corresponds to the final flow of the jump. Hence, yf is the depth of flow following the hydraulic jump while the depth of flow initiating the jump is yi. The conjugate depths ratio of the hydraulic jump are also referred to as the sequent depths.

In Chapter 15 of his book "Open Channel Hydraulics" (Copyright 1959 by McGraw-Hill Inc), Ve Te Chow provides a treatment for the hydraulic jump in a rectangular horizontal open channel. Chow's Eq.3-21, derived earlier in Chapter 3, provides a relationship between the final and initial depths of the jump in terms of initiating flow's Froude number. A modern source for information about the Hydraulic Jump is at FHWA

The Fayad Conjugate Depth:

By defining:   Fri'  = 1/Fri  

The Bélanger form of the conjugate depth can be manipulated into the Fayad form of the conjugate depth. 

yf/yi =1/{0.25Fri' [Fri' + ((Fri'^2)+8)^0.5]}

This obfuscated form is a mere mathematical variation of the same hydraulic relationship. Both forms share the exact utilitarian function of providing the same output for the same input.

Converse Flow of a Hydraulic Jump Flow

The Converse Flow through a sluice gate for the given depths of a hydraulic jump flow is derived by the manipulation of the following equation: 

 yi/yf={0.25Fri' [Fri' + ((Fri'^2)+8)^0.5]} 

 and by relating it to the hydraulic jump flow of interest.