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.
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.
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.