Show that, if the friction loss in a pipe-line is proportional to the square of the velocity, the oscillatory motion of the level in a simple, open, cylindrical surge tank following complete shut-down of the turbines in a hydro-electric plant is given by an equation of the form

in which H represents the instantaneous height of the surge tank level above the reservoir level, and α and β are constants, α being positive or negative according as the flow along the pipe-line is towards or away from the surge tank. Determine α and β for a surge tank diameter 30 m, a pipe line diameter 4.5 m and a length of pipe-line from reservoir to surge tank 730 m. Immediately before the turbines are shut down the rate of flow along the pipe-line is 42.5 m³ ·s^−1 and the level in the surge tank is stationary and 1 m below the water level in the reservoir.