The specific speed ($N_s$) of a water turbine is a characteristic parameter that relates its speed (N), power output (P), and head (H) or discharge (Q). It represents the speed at which a geometrically similar turbine would run to produce unit power at unit head. The formula for specific speed is: $N_s = \frac{N \sqrt{P}}{H^{5/4}}$ (for power)For a Pelton wheel, the power developed is proportional to the discharge multiplied by the head: $P \propto Q \cdot H$. The discharge ($Q$) from a jet is proportional to the square root of the head ($Q \propto \sqrt{H}$). So, $P \propto \sqrt{H} \cdot H = H^{3/2}$.Substituting this into the specific speed formula: $N_s = \frac{N \sqrt{k \cdot H^{3/2}}} {H^{5/4}} = \frac{N \cdot k' \cdot H^{3/4}} {H^{5/4}} = N \cdot k' \cdot H^{3/4 - 5/4} = N \cdot k' \cdot H^{-2/4} = N \cdot k' \cdot H^{-1/2}$. Thus, $N_s \propto \frac{N}{\sqrt{H}} $. This is the specific speed defined in terms of speed and head only, which is also a valid form.However, let's consider the impact of multiple jets. If a Pelton wheel has 'n' number of jets, and each jet produces the same amount of power, then the total power output of the turbine will be 'n' times the power produced by a single jet (assuming no interference). $P_{total} = n \cdot P_{single\_jet}$. The head (H) remains the same. The speed (N) at which the turbine operates for optimal efficiency also remains the same for geometrically similar turbines under the same head.Using the specific speed formula $N_s = \frac{N \sqrt{P}}{H^{5/4}}$: For a single jet Pelton wheel, let the specific speed be $N_{s1}$. $N_{s1} = \frac{N \sqrt{P_1}}{H^{5/4}}$For a multi-jet Pelton wheel with 'n' jets, let the specific speed be $N_{s\_multi}$. The power will be $P_{multi} = n \cdot P_1$. $N_{s\_multi} = \frac{N \sqrt{n \cdot P_1}}{H^{5/4}} = \frac{N \sqrt{n} \sqrt{P_1}} {H^{5/4}} = \sqrt{n} \cdot \frac{N \sqrt{P_1}} {H^{5/4}} = \sqrt{n} \cdot N_{s1}$.So, the specific speed of a multi-jet Pelton wheel is $\sqrt{n}$ times the specific speed of a single-jet Pelton wheel, where 'n' is the number of jets. $\sqrt{n} = n^{1/2}$. Therefore, the specific speed for a multi- jet Pelton wheel is $n^{1/2}$ times the specific speed for a single jet Pelton wheel.