In the time dt which the wave takes to travel over MN the particle displacement at N changes by QR, and QR= - udt, so that QR/MN = - u/U.
Generally, if any condition in the wave is carried forward unchanged with velocity U, the change of 4 at a given point in time dt is equal to the change of as we go back along the curve a distance dx = Udt at the beginning of dt.
Construct a small cylinder inside the gas, having dS as base and edges such that the projections of each on the coordinate axes are udt, vdt, wdt.