Consider the sound wave propagating at speed v through a pipe aligned with the x axis and with a cross-sectional area of A . In time interval d t it moves length d x = v d t . In steady state, the mass flow rate m ˙ ˙ = ρ ρ v A must be the same at the two ends of the tube, therefore the mass flux j = ρ ρ v is constant and v d ρ ρ = − − ρ ρ d v . Per Newton's second law, the pressure-gradient force provides the acceleration: d v d t = − − 1 ρ ρ d P d x → → d P = ( − − ρ ρ d v ) d x d t = ( v d ρ ρ ) v → → v 2 ≡ ≡ c 2 = d P d ρ ρ