To derive the Turbine specific speed equation we first start with the Power formula for water then using proportionalities with η,ρ, and g being constant they can be removed. The power of the turbine is therefore only dependent on the head H and flow Q.
:so 
let:
: = Diameter of the turbine runner
: = Width of the turbine runner
: = Speed of the turbine (rpm)
: = Tangential velocity of the turbine blade (m/s)
: = Specific Speed of the Turbine
: = Velocity of water at turbine (m/s)
Now utilising the constant speed ratio at the turbine tip, the following proportionality can be made that the tangential velocity of the turbine blade is proportional to the square root of the head.
:Speed ratio 
:so 
But from rotational speed in RPM to linear speed in m/s the following equation and proportionality can be made.
:so 
The flow through a turbine is the product of flow velocity and area so the flow through a turbine can be quantified.
:with 
and as shown previously:
So using the above 2 , the following is obtained
By combining the equation for diameter and tangential speed, with tangential speed and head a relationship between flow and head can be reached.
Substituting this back into the power equation gives:
To convert this proportionality into an equation a factor of proportionality, say K, must be introduced which gives:
Now assuming our original proposition of producing 1 kilowatt at 1m head our speed N becomes our specific speed . So substituting these values into our equation gives:
Now we know  we have a complete formula for specific speed,:
So rearranging for Specific Speed give the final following result:
where:
*  = Wheel speed (rpm)
*  = Power (kW)
*  = Water head (m)