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

The negative alternate depth for a given flow with the same specific energy as the subcritical depth mode of the flow or its corresponding supercritical depth mode of flow can be obtained by using either of the following equations:

Negative Alternate Depth= yneg= {0.5*(Vsub^2)/(2*g)}{1-sqr(1 + 8/Frsub^2)}ysub

where Frsub=Vsub/(g.ysub)^0.5 , such that Vsub is the velocity for subcritical depth and ysub is the corresponding subcritical depth of flow. 

or

Negative Alternate Depth= yneg= {0.5*(Vsup^2)/(2*g)}{1-sqr(1 + 8/Frsup^2)}ysup

where Frsup=Vsup/(g.ysup)^0.5 , such Vsup is the velocity for supercritical depth and ysup is the corresponding supercritical depth of flow. 

The Negative Depth is the algebraically negative value of the alternate depth energy equation solution. It has been discarded as an imaginary depth or rather a depth corresponding to a situation that cannot be imagined. It was assumed to have no bearing on the physical outcome of the flow in the channel.

 The Fayad Alternate Negative depth equation is:

Negative Depth/Positive Depth=0.25Fr{Fr-(Fr² +8)^0.5 }

Here, Fr represents the value of the positive Froude number of either of the supercritical or the subcritical depth for an idealized sluice gate.

• It is self-evident from the alternate depth equation for the subcritical depth and from the alternate depth equation for the negative depth that their algebraic sum equals the kinetic energy component for the  specific energy of the same flow with the supercritical depth regime.
• It is self-evident from the alternate depth equation for the supercritical depth and from the alternate depth equation for the negative depth that their algebraic sum equals the kinetic energy component for the  specific energy of the same flow with the subcritical depth regime.

 The Belanger form of the negative alternate depths ratio formula in a horizontal rectangular open channel is:

 yneg/ysub =1/{0.5[-1-(8Frsub^-2 +1)^1/2 ]}

 The above form can be readily validated via equating with the aforementioned formula and algebraic simplification.