Specific speed Ns of a turbine is defined as the speed of geometrically similar turbines or a family of turbines having identical characteristics working under a head of 1m to produce 1 Kw of power output we know that

$u=\frac{\pi DN}{60}$

i.e $u \alpha$DN

But u $\alpha V d\sqrt{H}$

from Equation (i) and (ii) $\sqrt{H}\alpha DN$

$D\alpha \frac{\sqrt{H}}{N}$

Dsicharge Q=$\pi D D Vf$

But B$\alpha D\alpha \frac{\sqrt{H}}{N}$ and Vf$\alpha V \alpha \sqrt{H}$

Q$\alpha \frac{\sqrt{H}}{N}{N}\times \frac{\sqrt{H}}{N}\times \sqrt{H}$

Q$\alpha \frac{H^{3/2}}{N^{2}}$

Power P= P,g.QH

On substituting the value of Q

P$\alpha \frac{H^{3/2}}{N^{2}}\times H; p\alpha\frac{H^{3/2}}{N^{2}}$

or N=K$\times \frac{H^{5/4}}{\sqrt{P}}$

When p=1kw and H=1 m, N= $N_{9}$

$N_{s}$=K

From the above equation

NS=K

N=$N_{s}, \frac{H^{5/4}}{\sqrt{P}}$ specific speed

$N_{s}=\frac{N\sqrt{P}}{H^{5/4}}$

Above expression for specific speed combines all variables like speed power and H. Thus, the turbine can be better classified based on their specific speeds instead of individual variables of Q,H and N