The Admiralty coefficient is derived from the relation that the power required to drive a ship is proportional to the displacement and the cube of the speed, and inversely proportional to the length:

P = Delta^(2/3) V^3 / C

where P is the shaft power, Delta the displacement, V the speed and C the Admiralty coefficient. The coefficient C is a measure of the efficiency of the hull and machinery; a higher C means a more efficient ship (less power for a given displacement and speed). The relation is derived from the assumption that the resistance is proportional to the wetted surface area (proportional to Delta^(2/3)) and to the square of the speed, and that the power is resistance x speed, giving P proportional to Delta^(2/3) V^3.

For a fast ship, the resistance rises more steeply with speed (the wave-making resistance increases rapidly at high Froude numbers), so the simple V^3 relation underestimates the power. The Admiralty coefficient is therefore modified for a fast ship by using a higher power of the speed, e.g. P = Delta^(2/3) V^4 / C, or by using a speed-dependent coefficient. The modified form accounts for the increased wave-making resistance of fast ships.

A 6 m model of a ship has a wetted surface area of 7 m2 and, when towed in fresh water at 3 knots, has a total resistance of 35 N. Calculate the effective power of the ship, 120 m long, at the corresponding speed. n = 1.825, f from the formula SCF = 1.15.

Scale = 120/6 = 20.

Corresponding speed: V_ship = V_model x sqrt(scale) = 3 x sqrt(20) = 3 x 4.472 = 13.42 knots.

Ship resistance: R_ship = R_model x scale^3 x (rho_ship/rho_model) x SCF

= 35 x 20^3 x (1.025/1.000) x 1.15 = 35 x 8000 x 1.025 x 1.15 = 35 x 8000 x 1.17875 = 330,050 N.

Ship speed in m/s = 13.42 x 0.5144 = 6.90 m/s.

Effective power = R x V = 330,050 x 6.90 = 2,277,000 W = 2277 kW.

Answer: the effective power of the ship is about 2280 kW.